Junction Journal 2015

MIT Educational Studies Program (ESP)
esp.mit.edu
Editing by Herng Yi Cheng

Cover page design by Lucia Lam
Typesetting by Sara Freed Sussman and Herng Yi Cheng
The views expressed in this journal represent those of the student, and
not necessarily those of ESP.
Contents
P REFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
J UNCTION P ROJECTS AND PARTICIPANTS . . . . . . . . . . . . . . . . . . . . . . . . . vii
Hackenbush and its Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ahaan S. Rungta, mentored by Lane Gunderman
Inquiry Is Elementary: Science Experiments for Young Students . . . . . . . . . . . . . . .

Anusha Datar, mentored by Elizabeth Berg
The Magical Number: e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Ariel Azbel, mentored by Herng Yi Cheng
Applications of Knot Theory to DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
David Li, mentored by Herng Yi Cheng
Teaching Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Garrett Mallinson, mentored by Lane Gunderman
On Quantum Mechanics and Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Junlin Mo and Trevor Pennypacker, mentored by Lane Gunderman
On the Configurations of Hats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Karen Ying, mentored by Herng Yi Cheng
DNA Sequence Analysis Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Karen Zhou, mentored by Lane Gunderman
Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Keren Shao, mentored by Christopher Harmon
Visualizing Light Patterns of Flat-Folding Origami . . . . . . . . . . . . . . . . . . . . . . . 65
Ria Das, mentored by Herng Yi Cheng
Understanding and Preventing the Formation of Suicide Pacts
Among Younger Population in both China and the United States . . . . . . . . . . . . . 77
Tianyu (Christina) Lin, mentored by Evan Kuras
Humanity throughout the Holocaust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Vineetha Yadlapalli, mentored by Elizabeth Berg
A BOUT THE MIT E DUCATIONAL S TUDIES P ROGRAM . . . . . . . . . . . . . . . . . 101

i
Preface
What is Junction?
Junction is a program organized by MITs Educational Studies Program (ESP) where 41
middle and high school students worked on summer-long independent study projects that they
had proposed themselves. Students completed their projects under the guidance of their Junction mentor. Junction 2015 students were between the ages of 13 and 18 from diverse educational backgrounds, coming from areas as close as the city of Boston and as far as Fujian, China.
Some students came to Junction as aspiring novelists, researchers, painters, educators,
game designers, civil engineers or conservationists; others sought meaning within poetry, computer science, design, linguistics, music or mathematics. Some students proposed Junction
projects in fields that were completely new to them, curious about a pattern that they noticed
or a topic that kept appearing in the books they read. Others found project inspiration in their
attempts to understand the experiences of a friend. Still others saw Junction as an opportunity
to express passions that they had already harbored for years.
Junction began with two months of student project development between each student
and their individual mentor as they communicated via email and video chat. To culminate the
summers efforts, from August 10 to 21 all mentors and students assembled on MITs campus for
an intensive program of events. Each student dove into further work on their project, met daily
with their mentor to discuss ideas, progress and feedback, and participated in four multi-day
Seminar classes and ten one-shot Sprinkler classes held on topics such as Russian Language
and Culture, Cheesecake Crafting and Climate Change Science, among many others. Most
classes were taught by the Junction mentors and directors; other teachers included undergraduates from MIT and Northeastern.
The six mentors of Junction 2015 Elizabeth Berg, Herng Yi Cheng, Lane Gunderman,
Christopher Harmon, Evan Kuras and Lucia Lam are current undergraduate and graduate
students pursuing various fields of study. Each mentor is profiled in the Junction Projects and
Participants section of the Journal. In addition to their mentoring and teaching responsibilities, the team used their ingenuity to think up and lead additional activities for Junction students throughout the program.
The mentors devised a puzzle hunt for the first day where students worked in teams to
solve puzzles that featured each mentors areas of expertise. During program down time, they
got to know students through card games and frisbee. Students satisfied their curiosity about
the mentors by having their anonymous questions answered at the mentors Ask Me Anything
lunchtime activity.
The mentors encouraged Junction students to venture out into their surroundings and
iii
P REFACE
bond as a community. A group of students constructed their projects at MITs Edgerton Center
Student Laboratory. The Urban Ecology Sprinkler class took students outside to investigate
the campus and the Gentrification and the Future of Cities Seminar traveled to South Boston
on a walking tour. Students, mentors and directors came together for lunchtime and afterprogram activities such as impromptu five-minute Firestorm classes, a jam session and walks
by the Charles River.
To prepare students for presenting their projects, mentors also developed workshops in
communication, as well as typesetting and editing papers. Also, mentors encouraged students
to practice public speaking and thinking on their feet through improvised presentation games.
On August 21, Junction held its Final Project Showcase at MITs Stata Center. At the showcase,
20 students gave formal talks on their projects to family and friends of the Junction participants, and many more presented informational posters, art pieces and demonstrations that
represented their projects.
What is the Junction Journal?

Every Junction student had the option to submit excerpts of their work to the Junction
Journal, a formal compilation of the 2015 projects that aims to portray the student diversity,
passion and individual commitment that inspired our mentor and director team. Though the
Journal represents the final product of some students work, behind each entry lies the students unique process of investigation, struggle and refinement, which we also ask the reader to
keep in mind.
It is important to note that not every student submitted to the Journal. Many students
undertook impressive projects that were not text-based and were therefore not as easy to represent on the printed page. We encourage the reader to look over the full list of student projects in
the Junction Projects and Participants section. We are especially grateful to Junction mentor
Herng Yi for editing the Journal, and to Junction mentor Lucia for designing the journal cover
art.
Junction focused on maximizing the learning experiences of middle and high school students mentored by college students. As such, all results from student projects should be interpreted as exploratory in nature.
Acknowledgements
First and foremost the directors wish to thank the 2015 Junction mentor team: Elizabeth
Berg, Herng Yi Cheng, Lane Gunderman, Christopher Harmon, Evan Kuras and Lucia Lam. The
amount of work that they each put into Junction was at times nothing short of astounding, and
we continue to appreciate their perspectives and contributions. We wish to acknowledge the
other Junction Seminar and Sprinkler teachers for their energy and creativity: Crystal Wang,
Taylor Sutton, Arkadiy Frasinich and William Navarre from MIT, as well as Gina Asipenko from
Northeastern. We wish to thank ESP members Emily Tencate, Anthony Lu and Favyen Bastani
for helping out at Junction almost every day. We are grateful to the 2015 ESP chairs, Megan
Belzner and Lisa Ho, for being sources of expertise and advice from the time we began to deiv
P REFACE
velop this years program in January 2015 until Junctions last day in August. We are also indebted to Dr. James Bales for offering mentors and students the use of MITs Edgerton Center
Student Laboratory to work on projects involving electronics and light mechanical tools, and to
librarian Forrest Larson for offering time and materials so that a Junction student could learn
about sound engineering and music production at MITs Lewis Music Library.
We are ultimately grateful to the Junction Class of 2015 and their families for providing us
with much to learn about, reflect on and enjoy. Thank you for sharing your outstanding work
with us this summer. We hope that you take pride in reading the array of explorations ahead.
Sara Freed Sussman and Corinn Herrick
MIT Educational Studies Program
Cambridge, MA
Junction Projects and Participants

Projects Listed by Mentor
Christopher Harmon studies Mechanical Engineering at the Massachusetts Institute of
Technology (MIT). He has worked on a number of projects involving robotics, electrical engineering, prototyping, and design analysis. He also enjoys composing music.
Alex Yu
Da-Jin Chu
Joel Saint-Eloi
Keren Shao
Long Do
Rudd Dantes
Sarah Kim
Viraz Mahanti
Making Music From Scratch

Learning CAD
Building a RC Plane
Control Theory
Building a Quadcopter
Building a RC Car
Bridge Building
The Fruit Chopper
Elizabeth Berg studies Earth, Atmospheric, and Planetary Sciences, as well as Literature
at MIT. Her pursuits span race and gender, atmospheric and oceanic physics, science writing,
environmental science and policy, teaching and cooking.
Anusha Datar
Inquiry is Elementary: Science

Experiments for Young Students
Claire August
For Lack: On Ecopoetry
Tina Lu
Tyler Wolfe
Vineetha Yadlapalli
Wendy Matt
Racism and Ethnic Prejudice

Smog in China
Humanity throughout the
Holocaust
The International Banquet
vii
J UNCTION P ROJECTS AND PARTICIPANTS

Evan Kuras is enrolled in a Masters program in Environmental Conservation at UMass
Amherst, and has conducted biodiversity research in the Amazon rainforest. His interests include ecology, public health, urban studies and sociology.
Christina Lin
Understanding and Preventing the Formation of

Suicide Pacts Among Younger Population in both
China and the United States
Danielle Gillerin
How Do You Define Success?
Jorie Coe
Malaria and Climate Change
Mady Eori
Bobcat Conservation of Western Massachusetts
Olivia Garrahan
Boy Band Breakups
Alexander
Lee-Papastavros
Music and Social Change: Transcending Conflict

Through Song
Haveesh Viswanatha
The Benefits of Detroit
Herng Yi Cheng is a Mathematics major at MIT who likes investigating the mathematics
behind origami, the art of folding paper. His interests span mathematics, origami, their respective histories, as well as theoretical computer science and science communication.
Ariel Azbel
David Li
Farid A. Azar Len
John Namgung
Karen Ying
Ria Das
The Magical Number: e

Applications of Knot Theory to DNA
Notes on Set Theory and Functions
Machine Learning in Relation to Number Sets
On the Configurations of Hats
Visualizing Light Patterns of Flat-Folding Origami
Lane Gunderman studies Chemistry and Physics at MIT and has done research on biological simulations. He enjoys reading plays and learning about computational, physical and
theoretical chemistry, together with game theory and physics.
Ahaan S. Rungta
Charlie Kip
Garrett Mallinson
Junlin Mo &
Trevor Pennypacker
Hackenbush and its Variations

Designing a Mythology Based Card Game
Teaching Coding
On Quantum Mechanics and Computation
Karen Zhou
DNA Sequence Analysis Program
Logan Troy
Bridging the Gap Between Empirical and

Theorized Game Behavior
viii
J UNCTION P ROJECTS AND PARTICIPANTS

Lucia Lam is a Mechanical Engineering major at MIT who enjoys writing, as well as finding trends and common themes throughout history, linguistics or other forms of data. She is
especially intrigued by trends across cultural, geographic and disciplinary boundaries.
Davon Dowie
Jennifer Yu
Hannah Parrott
Owen Tellalian
Pallavi Krishnamurthy
Shashvat Srivastava
William Hu
Anavrin: Book One

Juggling for Brains: Achieving Effective Focus
Linguistic Similarities
Effects of Smell on Taste
Gender and the Criminal Mind
The Perception of Time
Dysphemisms: An Overview
The Directors
Corinn Herrick is a senior at MIT majoring in Computer Science. She has been involved
in ESP since her freshman year, serving as Splash Director, Spring HSSP Director and ESP Chair.
She really enjoyed getting to know all of the students at Junction and seeing all of their impressive work.
Sara Freed Sussman is a current sophomore visiting undergraduate at Harvard College
studying physics and math, originally from Lesley University. She likes learning about particles
and studying abstract algebra. She has worked at the Food Project and City Year Boston in
Dorchester, and has also directed Spring HSSP for ESP.
ix
Hackenbush and its Variations

Ahaan S. Rungta
mentored by Lane Gunderman
Abstract
Hackenbush is a mathematical game concerning graphs and thus falls under
the categories of game theory and graph theory. The below study was conducted to further understand and formalize past research on the strategies and
mathematical notation used for Hackenbush, with hopes to perform extensive
research in the future.
1 Introduction
We, first, introduce the notion of a legal initial position, on which the game of Hackenbush
is played. We draw a ground line and a graph G under the following conditions, under the
assumption that all edges are line segments:
All the segments of G are either blue or red.
We say that G is a union of n disconnected graphs, and we call them I 1 , I 2 , . . . , I n , where
n 1.
For each I i , exactly one of its vertices must lie on the ground line.
For example, the graph shown in Figure 1 consists of three disconnected graphs and is a
legal initial position G for blue-red Hackenbush.
Figure 1: Legal Hackenbush starting position
Ahaan S. Rungta
H ACKENBUSH AND ITS VARIATIONS
On the other hand, the graph in Figure 2 is not legal, since one disconnected graph is not
attached to the ground line.
Figure 2: Illegal Hackenbush starting position

There are two players. One owns the color red call this the red player and the other
owns the color blue call this the blue player. A legal move is one where a player removes at
least one edge of the color that they own. After a legal move, if the resulting graph contains a
disconnected graph that does not share a vertex with the ground line (an illegal initial position),
then that disconnected graph is removed altogether. The game goes as follows:
One player begins the game by making a legal move.
The other player makes a legal move on the new graph.
The players alternate moves until one of the players have no legal moves to make. This
player is declared the loser and the player who made the last move is declared the winner.
Finally, we define a Hackenbush value v(G) of a position G to be a numerical value given to
a Hackenbush game such that it satisfies some mathematical properties of surreal numbers,
described below.
2 Known facts
It is known that two-player Hackenbush is a combinatorial game, because it satisfies the
four conditions for a combinatorial game:
i. Two players alternately move, the owner of each color.
ii. There is no factor of luck involved and both players have perfect information; in other
words, it is a fair game whose result is determined based on the moves that are made.
iii. The game must terminate, since there are a finite number of vertices and edges.
iv. There is no draw, since the winner is determined by the last person with a move, and we
know that this exists because of condition (iii).
2
Ahaan S. Rungta
VARIATIONS
3 Variations
3.1 3-color Hackenbush
Consider, now, a game of Hackenbush, also with two players and with the same set of legal
moves. However, now suppose there are also black edges, which either player can remove. A
special case of such a three-player game is one where there are no edges owned by either the
red player or the blue player; i.e. all the edges are black. In this case, at any point in the game, the
two players have the same set of legal moves available to them. A black-red-blue Hackenbush
game with only black edges is also called impartial Hackenbush.
We denote n to be the impartial game that starts with an initial position of n black segments chained from the ground, as shown in Figure 3.
Figure 3: The starting position for the impartial Hackenbush game 3
Theorem. Given any finite impartial game position G, there is a unique integer n 0 such that
v (G + n) = 0.
4 Relation to Nim
4.1 The game of Nim
In the game of Nim, we begin with m piles of stones. The i th stone has n i heaps. For
example, in the case of m = 3, we have three piles, each with n 1 , n 2 , and n 3 stones. A move is
defined as a removal of m i stones from pile i , where 1 m i n i . Two players alternate moves
until all the stones have been removed. The last player to take a stone is declared the winner.
The other player is declared the loser.
3
Ahaan S. Rungta
4.2 Winning Nim

Definition. Let m and n be two integers such that, in binary, we have
)
a k a k1 a 0 2 ,
(
)
n = b k b k1 b 0 .
m =
Then, the Nim sum of m and n is defined as

(
)
m n = c k c k1 c 0 2 ,
where c i {0, 1} is the the remainder when a i + b i is divided by 2.
For example,
13 7 = 11012 01112
= 10102 .
The above is also known as bitwise addition.
The default starting configuration of a Nim game usually has a Nim sum of zero. The winning strategy is well-known. The player that starts the game must always leave a non-zero Nim
sum after the first move, which the second player can take advantage of. Therefore, you are
always at a disadvantage if you start the game first at the default configuration.
Towards the end of the game, the method does break down For example, if you have two
piles with one stone each, this position has a Nim Sum of zero, but the player who moves first
in this position wins. More precisely, the Nim strategy changes only when normal play would
result in at most one stone in each pile. In that case, the correct move is to leave an odd number
of piles with one stone (in normal play, the correct move would be to leave an even number of
piles with one stone).
4.3 Nimbers
A nimber is a position n for some ordinal number n, where 0 = {} and (n + 1) = n
{n}. Let G, H be positions positions and let G + H be a position in a combined game where a
current player can choose either to move in G or in H . Computation of G+H is done by repeated
application of the rule
G + H = {G + h | h H } {g + H | g G}.
In other words, position addition is commutative and assosciative.
Theorem. Let the nimber value of game positions G, H be N (G), N (H ). Then,
N (G + H ) = N (G) N (H ).
Ahaan S. Rungta
QUANTIFYING GENERALIZED H ACKENBUSH
4.4 Analogy to Hackenbush

Consider a union of graphs of the form n.
Figure 4: The Hackenbush game consisting of 1, 3, 5, and 7 is equivalent

to a Nim game with original piles with 1, 3, 5, and 7 stones, respectively.
We claim that there is a bijection between a game of Nim with piles of stone with counts
n 1 , n 2 , . . . , n m and a Hackenbush position consisting of chains n 1 , n 2 , . . . , n m . consistent with
the following result.
Theorem. (Sprague-Grundy) Every impartial game under normal play convention is equivalent
to a nimber.
5 Quantifying generalized Hackenbush

We use a similar type of arithmetic as we did for Nim and impartial Hackenbush. Again,
we let v(G) be the Hackenbush value assigned to a position G, such that v(G) satisfies nimber
properties and the following rule holds.
Definition. (The Simplicity Rule) Let b be the largest value of any position to which blue can
move. Let r be the smallest value of any position to which red can move. (Due to convention,
b < r is always true.) If there is an integer n satsifying b < n < r , then v(G) is the closest such
integer to 0. Otherwise, v(G) is the unique rational number x satsifying b < x < r whose denominator is the smallest power of 2.
Again, v(G + H ) = v(G) + v(H ) for positions G, H . In other words, we think of the Hackenbush value v exactly as we did for Nim values N .
6 Future goals
Ideally, it would be helpful to find a direct relation between some properties of an initial
position and the numerical Hackenbush value of the graph, since the value is immediately related to which player has the winning strategy for the game.
5
Ahaan S. Rungta
We would also like to be able to tackle specifically complicated cases, using the facts we
know about simple ones. For example, cycles in graphs are harder to deal with and can affect
the way the value of a graph relates to just the degrees of vertices.
Also, we could consider infinite games using set theory and perhaps find a homomorphism
between Hackenbush and other more popular mathematical games.
Acknowledgements
Thank you to MIT ESPs program Junction for providing the opportunity to conduct the
above studies and to my mentor Lane Gunderman for assistance, advice, and guidance with the
project. Also thank you to Richard Stanleys Transparencies lecture collection available online
for reference materials.
Inquiry Is Elementary:
Science Experiments for Young Students
Anusha Datar
mentored by Elizabeth Berg
About
After the frustrating experience of being taught science through rote instruction and labs
that were like recipes, many students find themselves annoyed at and confused by the nature of
the subject itself. Students typically do not design their own experiments and solutions until the
later years of their high school education. I created a blog at http://inquiryiselementary.
blogspot.com/ to combat this custom; scientific inquiry should be an important part of curricula for all students, as it will inspire them to pursue and appreciate science and systematic
thought processes. Hopefully, this site will be a valuable resource for educators and parents
hoping to inspire future scientists.1
1 Ice Cube Challenge!

Grades: K2, 35
Subject: Physics, Engineering
Time: 30 minutes
Adapted from http://inquiryiselementary.blogspot.com/p/about.html
Anusha Datar
I NQUIRY I S E LEMENTARY
1.1 Background for Implementation

In this activity, students will learn about conductors and insulators in an exciting setting.
With the element of competition, they will be motivated to work effectively and tweak their
designs in order to come up with the best solution.
This is an exciting engineering challenge in which students will learn about insulators
and heat transfer in a competitive and exciting setting. While this experiment works with any
amount of students at any age level, it works best with ten to twenty kids separated into teams,
though it also works with just an individual.
1.2 Background for Students

Some materials are very good at deflecting heat, allowing them to keep cold objects cold
and hot objects hot. These materials are called insulators, and they are an essential part of
everyday technology, from coolers to lunchboxes. In contrast, some materials are effective at
allowing heat to pass through them, such as metals.
1.3 Materials
(Should have enough for each group to use):
Aluminum Foil
Masking or Duct Tape
Printer Paper
Newspaper
Parchment/Wax Paper
Cardboard Boxes
Cotton Balls
Rubber Bands
Additional conductors/insulators
Ice Cubes (1 per group/person and 1 separate control)
First, ask the students what they know about ice and what happens as ice cubes melt.
Where does the heat go? What happens to the ice cube? Is this similar to how humans react to
heat?
Then, distribute all of the materials except for the ice cube to the student(s).
After these materials have been handed out, present them with the problem they have to
solve: Using the given materials, they need to build a place where the ice cube could be kept
so that it would not melt. It would be ideal to either weigh or take pictures of the initial cubes.
Afterwards, give the students time to build their structures. Encourage them to plan what they
8
Anusha Datar
O N A R OLL
want to build and which materials they want to use before they start. After they have finished
building, give each group an ice cube to place into their structures. Additionally, place an ice
cube on a bowl or plate without the structure so that students can compare their experimental
results with a controlled group that has no structure. After about thirty minutes, have students
compare the ice cubes in their structures to the ice cube in the bowl.
After deciding who the winner is, discuss the strengths and weaknesses of the different
designs. What materials were useful? Why were they useful? What could we do to make the best
structure? Where is this applicable in real life?
Covers MA 35 Physical Science Standards #6
2 On a Roll
Grades: K2
Subject: Physics
Time: 1520 minutes

This activity is designed to help students understand the properties of different objects
based on their size, shape, and distribution of weight. By experimenting with different ideas,
students will be able to clearly understand the effects of an objects physical properties on how
it acts. This activity is ideal when students work in small groups so that they can effectively
discuss their ideas, but its also okay if they work alone.

Many different three-dimensional objects react to force, or a push or pull, in different ways.
For example, dropping a tennis ball on the ground is very different from dropping a bowling ball
on the ground. In this lesson, we will discover the way that different shapes respond the forces
when they are rolled.
9
Anusha Datar
I NQUIRY I S E LEMENTARY
2.3 Materials
(Should have enough for each group to use):
Construction Paper
Glue/Tape
After introducing students to the materials, present them with the problem that they have
to solve: They need to design one object that rolls in a straight line, and one object that rolls in a
circle. Allow them to try many different combinations in order to decide which one works best.
If some students finish quickly, ask them to design more objects that also meet these conditions.
When all of the students have come up with their shapes, have them discuss why their
designs were effective. What is different about the two objects? Where is the weight centered?
How does this affect how each of the objects roll?
Covers MA K2 Physical Science Standards #3
3 Be a Beaver
Grades: 35
Subject: Biology, Engineering
Time: 30 minutes

This activity integrates both biology and engineering so that students can effectively learn
about the amazing structures animals are able to construct. Students can either work in groups
or individually, though it may be better for them to work in groups so that they can discuss their
ideas.

Beavers are rodents that live both in and out of water. They are threatened by predators
such as wolves and coyotes, and they need a safe place to store their food during the winter.
For this reason, beavers work together to build huge structures made of wood, rocks, and mud.
These structures give them a protective home. Dams also block the rivers that they are built on,
but they allow some water through. In this activity, we will construct our own beaver dams.
10
Anusha Datar
B E A B EAVER
3.3 Materials
Container for sand
Plastic Storage Bin or similar
Aquarium or similar
Sandbox
Sand
Water
Building materials
Popsicle sticks/Skewers
Straws
Rocks
Etc.
After explaining the background information to the students, distribute the materials to
them.
Have the students fill their containers with sand. Then, have them carve a river in the sand
by digging a path through it that is at least two or three inches deep and two or three inches
wide. Have them each pick a location for their dam on the river. Ideally, the dam should only let
a little bit of water through. Students can look at photographs of actual beaver dams for ideas.
Next, let each group construct their dams. Remind them that water pressure is higher as
the water gets deeper, so the bottom of their dam will have to be stronger than the top of the
dam. When students believe that they have finished, allow them to test their dams by pouring
water down the river. If the dam is not effective, let them decide on and accordingly make the
appropriate changes.
After the students have finished with the dam, ask them what they think they did well and
what they think they could have improved on. What was the best shape for the dam to be? What
materials worked best? Where was the best location to build the dam? How do real beaver dams
reflect this?
Covers MA 35 Earth and Space Science #12
For more activities, visit: http://inquiryiselementary.blogspot.com/ or
tinyurl.com/junctioninquiry.
11
The Magical Number: e

Ariel Azbel
mentored by Herng Yi Cheng
1 Introduction
For centuries mathematicians have been fascinated with the vast applications of one particular number. This famous number is known as e. A Scottish physicist, John Napier, first
presented this irrational constant in the world of mathematics four hundreds years ago, and to
this day it is still widely used in calculus. This constant is approximately equivalent to 2.718, but
it goes on infinitely. Other than compound interest, I was not aware of any other application of
the number e. I became fascinated in this mysterious number when I started to learn calculus.
The number appeared in many areas of calculus, causing me to expand my interest in it.
2 Compound Interest and Natural Logarithms

When most math students think of the number e they think of its application in compound
interest and continuous interest formulas. The formula for calculating the compound interest
(
j )nt
is S = P 1 + n . This formula is deceptively simple because the variables have very straightforward meanings. P is the starting amount of money you have, S is the value after t periods, j
is the annual interest rate, n is the number of times the interest is compounded per year, and t
is number of years the money is borrowed for. When you plug in one for t and j while n is approaching infinity the limit is equivalent to the number e. This fascinating mathematical fact is
how John Napier discovered this irrational constant. Surely enough, the adventure of the number e does not stop here. Another common citing of the number e is in the base of the natural
logarithm. When mathematicians were deriving logarithms they would not get nice numbers,
rather they would get strange decimals and fractions. After years of testing they discovered one
number that could be used as a base for the logarithm, and once it was differentiated, it would
give you a good answer. This number was e, and when you differentiate log base e, also known
as the natural logarithm, you get 1/x because the natural logarithm of e is equal to 1. This astounded the math community, since such a strange, irrational number could give you such a
good answer.
13
Ariel Azbel
T HE M AGICAL N UMBER : e
3 Derivative of e x and Proving Eulers Identity

One of the first concepts most students learn in calculus is how to calculate a derivative.
To clarify, a derivative of a function is the slope at a particular point on that graph. The number
e has an immense role in derivatives because once you differentiate the function e x you get e x .
This is the only function in all of mathematics where this happens. If you were to graph the
function e x and find the slope of each point on that graph it would give you that same function.
Moreover, the derivative of C e x is also C e x for any constant C , and when x = 0 the function
needs to evaluate to 1. Therefore the only function whose derivative is itself, and satisfies f (x) =
1, is f (x) = e x . The function e x has many other interesting applications, such as the Taylor and
Mclaurin series. The Taylor series is function that is represented by an infinite sum of terms
that are based on the derivatives of that function at a certain point. The Mclaurin series is
equivalent to the Taylor series, except the point that you are calculating the functions derivative
at is 0. Here are a few important examples of the Mclaurin series:
(1)n
x2 x4
x 2n
= 1
+
2! 4!
n=0 (2n)!
(1)n 2n+1
x3 x5
sin x =
x
=x
+
3! 5!
n=0 (2n + 1)!
cos x =
for all x
for all x
From these series we can derive one of the most beautiful formulas in all of math known
as Eulers Identity e i + 1 = 0, which is equivalent to Eulers Formula e i = cos + i sin , except
x is substituted by . The way you get this identity from the three Mclaurin series previously
mentioned is by using the following steps: first, you take the series with the function e x and
(i )2
(i )3
plug in i for x, and you would get: e i = 1 + i
1! + 2! + 3! + . We still want to prove how the
sum of other two series is equal to the first series, so you take the sin x function, multiply it by
3
5
+ i 5!
. Finally, you take the cos x
i , and input , so it should look like this: i sin = i1! i 3!
2
function, and plug in , making it equal the following: cos = 1 2! + 4! . Now, if you add
the cos and i sin functions the outcome would be equivalent to the e i . We have proven that
e i = cos + i sin , so there is only one more step to prove that e i + 1 = 0. You need to plug in
for , and you will receive e i = cos + i sin . If you simplify this you will get e i = 1 + 0,
and now it is very clear that e i + 1 = 0. You may be confused why all this work was done in
order to prove some small formula, but e i +1 = 0 has great value in it because it is know as the
most beautiful formula in mathematics. Most mathematicians claim this because the formula
is made up of every important constant used in math.
4 Conclusion
The number e has bewildered the math and science community with its application, since
the time of its discovery. In the future I hope to learn the application of e in other sciences more
in depth. I also hope to some how apply my research to another project that can be used in a
science fair. So far I have conducted research on the number e, but I have not applied it to a real
world problem.
14
Alex Yu (left) working on his project, Making Music from Scratch,

guided by mentor Christopher Harmon (center). The completed
music is available at https://soundcloud.com/yushuf/
yushuf-chris-harmon-junction.
Charlie Kip (left) and Keren Shao

(right) testing a mythology-based
card game designed by Charlie
in his project.
Applications of Knot Theory to DNA

David Li
Abstract
DNA topology is an unavoidable characteristic when dealing with virtually any
natural DNA molecule. It is an essential part of DNA replication, gene expression, non-homologous end joining, and other processes in molecular biology.
Its study using knot theory has led to the development of several important
medicines and provided insight into many of these fundamental processes.
This report briefly explains some of the basics of knot theory, DNA, and their
intersection. Some of the interesting results we obtained are also included in
this report.
1 Introductory Knot Theory

Topology is the study of properties unaffected by the continuous deformations of shape or
size of figures. Knot theory is a subfield of topology that studies mathematical knots. Conventional knots are not mathematical knots because the ends must be joined together in a mathematical knot. This means that mathematical knots cannot be undone. In mathematical jargon,
a knot is an embedding of a circle in R3 with no self intersections. Any modification to the knot,
as long as the strand doesnt break or pass through itself, is equivalent to the original knot. The
simplest knot is just an unknotted circle, which is named the unknot or the trivial knot. The first
real knot is the trefoil knot (Figure 1). The real trefoil knot is in three dimensional space, but
studying two dimensional images is easier so mathematicians use pictures like the one below.
These pictures are called projections.
Figure 1: A Projection of the Trefoil Knot (Adapted from The Knot Book by Colin
Adams [1])
17
David Li
A PPLICATIONS OF K NOT T HEORY TO DNA
The crossings of a given projection are the places where the knot crosses itself in the image.
The trefoil knot above has three crossings. We say that the trefoil knot has a crossing number
of three because there is not a projection of the trefoil knot with less crossings. Officially, the
crossing number is defined as the smallest number of crossings in any projection of the knot.
To distinguish if a projection is equivalent to the trefoil knot or any other kind of knot, deformations must be used. One basic deformation is planar isotopy which consists of stretching
the length of sides without altering the crossings. In 1926, a set of three very important deformations on knots were defined and listed in a paper written by Kurt Reidemeister [2]. They are
referred to as Reidemeister moves. A Type I Reidemeister move is the addition or removal of
a twist in the knot. A Type II Reidemeister move is the addition or removal of two crossings
(Figure 2). A Type III Reidemeister move is the sliding of a strand of the knot from one side of a
crossing to the other side of the crossing. Using these three moves and planar isotopy, you can
determine conclusively if two projections are equivalent.
Figure 2: The 3 Reidemeister Moves (Adapted from Wikimedia Commons)

Crossing number is a knot invariant because Reidemeister moves and planar isotopy do
not affect it. Currently, all knots with a crossing number of sixteen or less have been identified.
These knots are denoted in Alexander-Briggs notation, which consists of its crossing number
and then a subscript.
Figure 3: All of the Knots with Crossing Number 7 or Less (Adapted from The
Knot Book by Colin Adams [1])
18
David Li
DNA
A link is a set of knotted loops all tangled together. Two links are considered to be the same
if one link can by deformed to the other without ever having any of the loops intersect itself. For
simplicity, all links in this report will have two components.
One specific topological invariant of links is known as the linking number. To calculate the
linking number, you first choose an arbitrary orientation on each component of the link. Then
at each of the crossings, one of the pictures in the figure below will hold. You count a 1 for each
crossing of the third and fourth type and a +1 for any crossing of the first or second type. Next,
you take the sum of the +1s and 1s over all the crossings between the components of the link
and divide the sum by 2. This is the linking number. All linking numbers must be integers. One
example is shown in Figure 5.
Figure 4: Computing Linking Number (Adapted from the Wikimedia Commons)
Figure 5: The Linking Number of this Link is 2 (Adapted from the Wikimedia
Commons)
2 DNA
DNA, deoxyribonucleic acid, is the molecule that stores genetic information in all living organisms. It is a polymer of repeating subunits called nucleotides. Each nucleotide is composed
of a 5 carbon 2-deoxyribose sugar, which contains a 3 hydroxyl group, 1-3 phosphate groups,
and one of four heterocyclic bases: Guanine, Adenine, Thymine, or Cytosine. A phosphodiester
bond links together the phosphate groups and hydroxyl groups of two adjacent nucleotides
to form a single strand of DNA. Hydrogen bonding between corresponding bases allows two
strands of DNA to base pair, which can form various structures. Adenine forms two hydrogen
bonds with Thymine and Cytosine forms three hydrogen bonds with Guanine.
In cells, DNA is most commonly found as the famous double helix structure [4]. This is
also known as B-DNA. This double helix structure is stabilized by pi-pi interactions between
the cyclic rings in each of the nucleotides and hydrogen bonding between corresponding bases
19
David Li
Figure 6: Primary Structure of DNA (Adapted from DNA Topology by Bates and
Maxwell [3])
[10]. B-DNA is a right-handed helix with 10.4 base pairs per turn of the helix when fully relaxed and a repeat distance between each nucleotide of 3.4 Angstroms. This means that one full
turn of DNA is approximately 35 Angstroms long [14, 26]. The phosphate backbones of the two
strands of single stranded DNA form what looks like the rails of a spiral staircase and the bases
look like the steps up the staircase. The true conformation of B-DNA in vitro depends on the
hydrophobicity of bases, the solution pH, temperature, and many other factors.
Figure 7: B-DNA (Adapted from Molecular Biology of the Gene by Watson et. al.)
20
David Li
S UPERCOILING AND T OPOISOMERASES
Other structures can also form as a result of base pairing including cruciforms [8], triplexes
[11, 13], and even quadruplexes [5, 13].The first structure in Figure 8, is a type of DNA triplex
called H-DNA. It consists of a triple-stranded region and a single-stranded region. It is strongly
favored in solutions with low pH [12]. A Holliday Junction is shown in the second image. It is a
result of homologous recombination and can also form in strands with palindrome sequences
[15]. G-Quadruplexes and i-Motifs form in single stranded DNA with long sequences of only
guanine and only cytosine respectively [13].
Figure 8: Alternate DNA Structures (Adapted from Biophysical Studies on DNA

Micromechanics by Qu Yuanyuan)
3 Supercoiling and Topoisomerases

DNA is normally a very long molecule compared to its diameter, and in order for it to be
replicated the strands must be separated to serve as templates for daughter strands at the replication fork. However, separating the two strands requires the DNA double helix to spin ahead
of the spitting strands. The twisting of the helix will tighten ahead of the replication fork which
will induce some strain. This strain leads the helix to coil on itself. This is known as supercoiling. There are two types of supercoiling, negative and positive, as shown in Figure 9. When the
DNA is overwound, or has less than 10.4 base pairs per turn, it will positively supercoil. When
the DNA has more than 10.4 base pairs per turn, it is underwound and will negatively supercoil.
Within each type, the DNA can coil either plectonemically or toroidally [16, 17].
DNA supercoiling is crucial for some processes and inhibits others. Supercoiling is the
first step in DNA compactification in both eukaryotes and prokaryotes. After the DNA is supercoiled, it is wrapped around histones before being assembled into nucleosomes and packaged
into chromatin. Supercoiling also helps proteins access the major and minor grooves of DNA.
Supercoiling is energetically unfavorable for most DNA structures so supercoiled DNA has more
energy that proteins or other molecules can harness [26].
To measure DNA supercoiling quantitatively, F. Fuller [7], G. Calugareanun [25], and James
White [9] developed the Calugareanu-White-Fuller Theorem:
Lk = T w + W r
21
(1)
David Li
Figure 9: Supercoiling (Adapted from Stanford Biochem 201 Slides 1999)

In the Calugareanu-White-Fuller Theorem, Tw stands for Twist, Wr stands for Writhe,
and Lk stands for Linking Number. The Twist describes the number of times the individual
strands of DNA coil around the central axis of the helix. The Twist Equation is shown in Figure 10. The Writhe is a measure of how many times the double helix coils around itself. It is
given by the equation in Figure 11 [26]. The Linking Number is the same as the Linking Number
invariant from Section 1. Because it is an invariant, any change in Twist must also manifest as a
change in Writhe.
1
Tw =
2
ds +
[] X
=T +N
2
Figure 10: Twist Equation

1
Wr =
4

C C
d r1 d r2
r1 r2
|r1 r2 |3
Figure 11: Writhe Equation

Once we have calculated the linking number, we just need a couple more equations to
quantify the supercoiling. We define Lk to be the linking number of relaxed DNA. The equation
for Lk is given in Equation 2. It is calculated by dividing the number of base pairs N by the
proper number of base pairs per turn for a piece of relaxed DNA h. This means that Lk is
defined with a base unit of turns. Because Lk does not have to be an integer, it isnt a true
linking number. From this definition we have Lk which is equal to the difference between Lk
and Lk.
N
h
Lk = Lk Lk
Lk =
(2)
(3)
From above, we know that the introduction of supercoils into a DNA molecule corresponds
to the introduction of torsional stress. The larger Lk is, the more torsional stress there will be.
22
David Li
S UPERCOILING AND T OPOISOMERASES
Additionally, a given Lk will generate less torsional stress the larger the number of bases N is,
so we must normalize it. This gives us the specific linking difference .
=
Lk Lk Lk
=
Lk
Lk
(4)
We now have a specific number that we can use to compare the supercoiling of different
DNA molecules. For example, natural covalently-closed circular DNA molecules from E. Coli
have on average a specific linking difference of around 0.06 [3].
DNA molecules with the same length and sequence but differ in Lk are known as topoisomers. From some thermodynamics equations, we can calculate the concentration of a certain
topoisomer. The concentration of DNA with a linking number Lk is:
(
)
K (Lk Lk )2
1
[Lk] = exp
Z
RT
(5)
where R is Boltzmanns constant, T is the temperature in Kelvin, and Z is a normalization constant. The distribution of topoisomers follow a Gaussian distribution with Lk as the center [18].
To maintain this distribution, cells have certain enzymes called topoisomerases that can interconvert between topoisomers. There are two classes of topoisomerases, known as Type I and
Type II. Within each of the types of topoisomerases, there are subclasses of topoisomerases.
Type I topoisomerases have 3 distinct subclasses (A, B and C) and Type II topoisomerases have
2 distinct subclasses (A, and B).
Type IA topoisomerases can only relax negative supercoiling. Type IA topoisomerases resemble a padlock and operate through a strand-passage mechanism. First, a single DNA strand
is cleaved and opened by a catalyic tyrosine. A second DNA strand is navigated through the gap
and then the broken strand is resealed. Topo 1, Topo 3, and Reverse Gyrase are all in this class.
Interestingly, Reverse Gyrase can positively supercoil DNA away from topological equilibrium
using ATP hydrolysis. This is extremely common in organisms that thrive in environments with
extreme heat because the positive supercoiling helps protect the DNA.
Type IB topoisomerases (swivelases) can relax both negative and positive supercoiling.
Type IB topoisomerases pretty much just nick the DNA and let it spin around itself. Friction
between the enzyme and DNA keeps it from spinning too much and the DNA is resealed.
Type IC topoisomerases act very much like Type IB topos but have a different protein fold.
Suprisingly, there is only one protein in this class (Topo V).
Type IIA topoisomerases can relax both positive and negative superoiling. They function
much like Type IA topoisomerases except they require ATP or an alternate energy source. They
also employ a strand passage mechanism but cleave both strands of a double stranded DNA
duplex and pass a second duplex through the break. This class includes Gyrase, Topo II, and
Topo IV. Gyrase is particularly special in the fact that it can introduce negative supercoiling
while other topoisomerases can not.
Type IIB topoisomerases act very much like Type IIA topoisomerases even using the same
mechanism, but the structural arrangement differs greatly. There is only one topoisomerase in
this class, Topo VI. [19]
Topoisomerases are the target of many cancer medicines because without topoisomerase,
DNA replication cannot occur. For example, one specific class of antibiotic drugs, quinolones,
23
David Li
block the reformation of the covalent bonds after a Type II topoisomerase has split a DNA segment. This leads to fatal double-stranded breaks which make quinolones potent antibacterial
agents [20]. Some other topoisomerase inhibition mechanisms are shown in Figure 12. Knot
Theory can also be used to study other enzymes [2124, 26].
Figure 12: Various Topoisomerase II inhibition Mechanisms (Adapted from All

Tangled Up: how cells direct, manage and exploit topoisomerase function [19])
4 Rational Tangles
A tangle is a region of a knot or link projection that is surrounded by a circle such that
the knot or link crosses the circle exactly four times. We will think of the four points where the
strands cross as occurring in the four compass directions NW, NE SW, and SE [1]. As with knots,
two tangles are equivalent if we can get from one to the other by a series of Reidemeister moves
and planar isotopy.
The tangle in the upper left corner of Figure 13 is the 0 tangle. It is one of the most basic
tangles. The tangle in the upper right corner of Figure 13 is the 2 tangle because it is only different from the 0 tangle by 2 counter clockwise twists. If they were clockwise twists it would have
notation 2. Another way to determine whether the twist is positive or negative is to look at the
overstrand. If the overstrand has a negative slope, it is a negative twist. If the overstrand has
a positive slope, it is a positive twist. To continue constructing the tangle, you must reflect the
tangle over the NW and SE diagonal line. Next, you take the two new NE and SE strands of the
24
David Li
R ATIONAL TANGLES
Figure 13: Different Tangles (Adapted from Wikimedia Commons)

tangle and twist how ever many times you want. In Figure 13, the strands are twisted two times.
Any tangle that can be constructed in this manner is known as a rational tangle. With rational
tangles there is a very easy way to tell if two rational tangles are equivalent. All you have to do
is calculate the continued fraction corresponding a tangles tangle notation. For example, the
final tangle in Figure 13 has a tangle notation of 3, 1, 2. The continued fraction corresponding
1
to 3, 1, 2 is 2 +
which simplifies down to 72 . The continued fraction corresponding to
1
1 + 3
1
4, 5, 7, 1 is 1 +
which simplifies down to 156
137 .
7+ 11
5+ 4
The closure of a tangle leads to the formation of a knot or a catenane. There are two types
of closure operations, numerator and denominator, which are commonly denoted by N and D
respectively. K is also used to denote the numerator closure of a tangle [1, 26]. When you write
down the Conway notation for a knot, you use the tangle that forms the knot when under a
numerator closure operation. Both operations are shown in Figure 14. To check if two closures
are the same knot requires the use of the tangle fraction. Suppose you have two rational tangles
(p )
( p )
p
p
with simplified tangle fractions q and q . If K q and K q denote the corresponding rational
(p )
( p )
knots obtained by taking the closures of these tangles, then K q and K q are topologically
equivalent if and only if p = p and either q q (mod p) or q q 1 (mod p) [26, 27].
We developed a program that can calculate all two twist rational tangles that will form a
p
certain knot using Python. The program first uses a given q to generate all tangle fractions
( )
( )
p
p
p
. This is calculated using modular arithmetic and the
that
fit
the
equation
K
=
K
q
q
q
constraint that p must be greater than q for the tangle fraction to be a valid tangle fraction.
Once this list has been calculated, the program iterates through this list and finds solutions for
25
David Li
Figure 14: Closure Operations on a Tangle T (Adapted from Classifying and

Applying Rational Knots and Rational Tangles by Kaufman et. al.)
each specific tangle fraction. If a specified tangle fraction is equivalent to an integer , there are
always three solutions. These solutions are (0, ), (1, 1), and (1, +1). Solutions for fractions
that arent integers are calculated using the floor and ceiling functions. The program was ran
on all the knots from 31 to 814 . Interestingly, on one of the knots the program ran on, 62 , we
managed to produce a solution, 4, 3 that was even simpler than the one in a standard knot
table. We created a physical model of the closed tangle and tried to deform it into the knot 62 . It
took multiple attempts and quite a bit of time, but in the end we managed to deform it into the
knot 62 . It turns out that you have to rotate the closed tangle 180 to see the desired knot. It was
also special because the solution, (4, 3), has a total of seven crossings at the beginning which
is more than our target knot. All of the solutions we found are in Figures 15 and 16 below.
Figure 15: Solutions for Knots with Crossing Numbers of 1-7
26
David Li
R ATIONAL TANGLES
Figure 16: Solutions for Knots with a Crossing Number of 8

You might be wondering how this is related to molecular biology. We attacked this problem
because two twist DNA substrates can be designed and synthesized. Most of the biological
background is based off of one paper Construction and Electrophoretic Migration of SingleStranded DNA Knots and Catenanes written by Alexander Bucka [28]. The synthesis process is
shown in Figure 17. It is suprisingly effective and efficient. It starts with the formation of a DNA
loop by base pairing of homologous sequences. Upon formation, this loop is bound by the two
ends of the DNA. DNA ligase then joins the ends with a phosphodiester bond which effectively
closes the knot.
27
David Li
Figure 17: Single-stranded DNA Knot and Catenane Synthesis (Adapted from
Electrophoretic Migration of Single-Stranded DNA Knots and Catenanes by
Alexander Bucka [28])
28
David Li
B IOLOGICAL C ONSEQUENCES
5 Biological Consequences
DNA supercoiling is an attribute of almost all in vitro DNA. It helps with DNA compaction,
replication, and directly influences the interactions between specific proteins and DNA. Many
interactions can be modeled by individual steps, each impacted by supercoiling. These include
the binding of topoisomerases, recombinases, and strand-transfer proteins. Because rewinding
negatively supercoiled DNA brings it toward equilibrium, separation of strands and increased
twisting are both energetically favorable. This makes DNA replication favorable in negative
supercoiled DNA. Supercoiling also affects the secondary structure of DNA and will encourage
the formation of Z-DNA.
In all biological systems, replication of DNA proceeds from specific sites known as origins of replication. In bacteria, only one origin is present in the entire genome. In eukaryotes,
there are multiple replication origins. Negative supercoiling opens up the origin for binding
and actually stabilize the region. In yeast, the presence of negatively supercoiled DNA activates
unwinding of various autonomously replicating sequences. Normally, at the end of replication,
the end product is a catenane of some sort. Type II topoisomerases will then come over and
decatenate the structure resulting in two identical DNA rings.
DNA topology is also a factor that controls gene expression. The binding of transcription
factors or RNA polymerase to DNA will also significantly impact the topology of the DNA. In
transcription initiation, a RNA polymerase holoenzyme is formed and binds to the DNA which
stabilizes the negative supercoiling and promotes strand seperation. Initiation is a complicated
process that involves a number of phases. This equates to many opportunities to implement
control mechanisms. Various regulatory proteins also depend on the topology of local DNA.
6 Future Research
In the future, maybe the formation of various catenanes could be modeled and synthesized in real life. Deriving a equation for the electrophoretic mobility of a molecule is also a
interesting pursuit. Studying the structure of topoisomerases and analyzing the results of topoisomerases on various knots would also be possible areas to delve deeper into. Performing wet
lab work involving DNA topology also seems interesting.
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of Knots. Providence, R.I.: American Mathematical Society.
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David Li
[5] Lam, E., Beraldi, D., Tannahill, D., & Balasubramanian, S. (2013). G-quadruplex structures
are stable and detectable in human genomic DNA. Nature Communications, 4, 1796.
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doi:10.1093/nar/30.6.e24
31
Mentor Evan Kuras (red cap) exploring the urban ecology of MIT with students.
Teaching Coding
Garrett Mallinson
1 Introduction
I plan to start a coding club at my school. I will need to learn more python in order to
teach it to others. I plan to create class curriculum and I will also learn effective methods to
teach coding to small groups. Over the summer I took coding classes, in which I expanded
my programming skills and looked at their teaching methods. At Junction, I planned my class
curriculum. In the beginning of the school year I will implement my curriculum when I start
my coding club.
2 Lesson Plans
Day 1
Create list of everything the students think coding is
Create a list on how coding is used in everyday life
Show coding video https://www.youtube.com/watch?v=uEdyTlI3BAA
Explain what python and coding are
(if time permits) show how to download python
Create another list on what students think coding is now
Day 2
teach students basic python (if statements, strings, for loops etc...)
Day 3
have the students try to fix a text adventure
33
Garrett Mallinson
T EACHING C ODING
Day 4
have the students try to fix the syntax errors in droids
Day 5
have students create their own text adventure (contest for who can make the best text
adventure)
Resources
http://www.cheatography.com/davechild/cheat-sheets/python/
http://www.dummies.com/how-to/content/python-for-dummies-cheat-sheet.html
http://www.astro.ufl.edu/~warner/prog/python.html
http://learnpythonthehardway.org/
34
On Quantum Mechanics and Computation

Junlin Mo and Trevor Pennypacker
1 What is Quantum Mechanics?

In everyday life, we intuitively understand how the world works. Throw an object up, and
it will come down, as per the law of gravity. If you attempt to walk through a wall you will fail,
as two objects cannot occupy the same space at the same time. Until the turn of the twentieth
century, scientists believed that these basic laws applied to everything in the universe. However,
as technology improved, scientists realized that many of these principles fail on the atomic
level. Thus, scientists developed a new branch of physics known as quantum mechanics to
better understand the way tiny particles move and interact. Quantum mechanics, put simply,
is the study of very, very, very small things.
1.1 The Schrdinger Equation

The fundamental equation of quantum mechanics is the Schrdinger Equation. It reads
i
= H ,
t
where H is the Standard Hamiltonian operator

H =
2 2
+ V.
2m
For a given eigenstate, we also define that

H = E ,
where E is the eigenvalue.
The Schrdinger Equation, when given certain parameters and solved, gives the wave
function of a particle. Denoted by , the wave function provides information as to the density of the particle. Specifically,
b
a
|(x, t )|2 d x = {probability of finding the particle between a and b at time t }

35
Junlin Mo
O N QUANTUM M ECHANICS AND C OMPUTATION

Since the total probability of an event is always 1, we know that

|(x, t )|2 d x = 1.
2 The Infinite Square Well

Take a particle that is completely free, except at two ends where an infinite potential prevents it from escaping. This scenario is also called the Particle in a Box", and is shown in Figure 1.
Figure 1: Particle in a Box

So what would the wave function look like for this particle? Lets first start with the timeindependent Schrdinger equation.
2 d 2
+ V = E .
2m d x 2
Like the name connotes, the time-independent Schrdinger equation is a variant of the
Schrdinger equation that remains constant as time progresses, making the equation much
easier to solve. Since the particle is free-floating, we can substitute 0 for V (x) to get
2 d 2
= E ,
2m d x 2
or
d 2
= k 2 where k =
d x2
2mE
.
This is the simple harmonic oscillator equation, which has the general solution
(x) = A sin(kx) + B cos(kx).
It is possible to solve for a particular solution when given some boundary conditions, but
well leave the solution in its general form for simplicity.
36
Trevor Pennypacker
F OURIER T RANSFORMATIONS
2.1 Particular Solutions

When given a set of boundary conditions, it is possible to solve the general simple harmonic oscillator equation to get a particular solution. Take Figure 1 for example, where the
bounds are at x = 0 and x = a. (0) = 0, and thus
(0) = A sin(0) + B cos(0) = B,
so B = 0, and therefore (x) = A sin(kx). The boundary conditions also maintain that (a) = 0,
and thus ka is an integer multiple of . Another way of writing this is
kn =
n
, with n = 1, 2, 3, . . .
a
Next, a process called normalization is used, which constrains A to a specific value to

make the total probability density equal to 1.
a
a
2
|A|2 sin2 (kx)d x = |A|2 = 1, so |A|2 = .
2
a
0
The solutions then are
n (x) =
( n )
2
sin
.
a
a
2.2 A Useful Property

The solutions to the infinite square well problem are mutually orthogonal, or orthonormal, meaning
m (x) n (x)d x = 0, whenever m = n,
which can be condensed to
m (x) n (x)d x = mn ,
where mn is the Kronecker delta, defined as

{
mn = 0 m = n
mn = 1 m = n.
3 Fourier Transformations
Unlike the simple harmonic oscillator, some signals are not so easy to work with. However, any continuous periodic function can be modeled by a linear combination of sinusoidal
functions, and the Fourier Transformation provides a method of doing so.
Let g (t ) be the Fourier approximation of a function f (t ). The general Fourier decomposition for g (t ) is given by
)
(
)
(
2nt
2nt
+
A m cos
.
g (t ) =
A n sin
T
T
m=0
n=1
37
Junlin Mo
The constants A n and A m determine the relative weights for each of the sinusoids, and become
infinitely small as m and n become large, and can be solved for using the following equations:
2
An =
T
)
2nt
dt
f (t ) sin
T
2
Am =
T
T
0
)
2mt
dt.
f (t ) cos
T
Or, utilizing Eulers Identity e i x = cos(x) + i sin(x), the Fourier decomposition equation
can be rewritten as
g (t ) =
cn e i
2nt
T
where c n =
n=
1
T
f (t )e i
2nt
T
dt.
3.1 The Sawtooth Function

Take for instance a saw function, as shown in Figure 2. A function like this is difficult
to work with, and thus it is helpful to use a Fourier transform. As seen in the following diagrams, the Fourier decomposition models the sawtooth function more precisely as the number
of terms increases.
As with the solutions to the simple harmonic oscillator, the sinusoids in a Fourier series
are orthogonal, making them ideal to work with.
(a)
(b)
(c)
(d)
Figure 2: (a) Saw function. (b) Approximation with three terms. (c) Approximation with seven terms. (d) Approximation with 41 terms.
38
Trevor Pennypacker
I NTRODUCTION TO QUANTUM C OMPUTATION
4 Introduction to Quantum Computation

To properly study Quantum Computation, one must study the fundamentals of classical
computation. A classical computer is a device that can store and process data. Computers process data by executing algorithms. These algorithms provide a set of instructions for the computer to follow. In addition to being well-defined, algorithms need to be able to be translated
into a physical task.
Currently, classical computers have been developed to the point where billions of processes per second can occur within commercial computers.
When studying computers, it is important to remember the relationship between computation and physics. As we dive deeper into the field of quantum computing, the laws of physics
will begin to devise and set limits to principles of computation. Classical and realistic computers exist in a physical space and must obey the laws of that physical space, as opposed to
common idealized computational models .
5 Classical Computing
The foundation of all classical computing is a simple logic gatea system capable of being
switched on and off, true and false, 1 and 0, etc. From this basic unit we begin to develop and
build upon more difficult concepts. Modern computing is built upon hundreds of layers of
systems, from binary to the coding languages used to program the applications that we use.
5.1 Gates
A logic gate performs a logical operation on one or more logical inputs and produces a
single logical output. Logic gates are the fundamental building blocks to circuits, which may
contain strings of gates in succession.
The NOT gate is a standard gate of classical computation. The NOT gate operates on one
input. For input A The NOT gate will output the opposite value of the input. (See Figure 3)
Figure 3: This is an example of the NOT gate.

The OR gate requires 2 inputs. From these 2 inputs, OR will check if any of them is 1. If so,
it outputs a 1. (See Figure 4)
Figure 4: This is an example of the OR gate.

39
Junlin Mo
Likewise, the AND gate also require 2 inputs. The AND gate will only output a 1 of both
inputs are 1. (See Figure 5)
Figure 5: This an example of the AND gate.
6 Qubits
Qubits are essentially a pair of orthogonal states. Unlike regular bits, Qubits can represent
0, 1, or a superposition of 0 and 1. A bit can only represent 0 or 1, or 21 pieces of information.
A qubit is able to represent 0, 1, or a superposition of 0 or 1. This can be represent 22 pieces of
information. With this property, qubits are able to transmit more information at one time. In
essence, a qubit bears remarkable similarity to the particle in a box problem presented earlier
(see Figure 1). Similar to how the particle in a box can be decomposed into sine and cosine
waves using the Fourier Transform, a qubit can be decomposed into superpositions of binary
states.
Figure 6: The Bloch Sphere

This is a common representation of the qubit.
7 Quantum Gates
One of of the most common gates used in quantum computation is the Hadamard Gate.
The Hadamard gate, represented by H , takes a a vector (bit) of 0 or 1 and produces a superposition with it:
H |0 = p1 (|0 + |1)
H |1 =
2
1
p (|0 |1)
2
40
Trevor Pennypacker
T HE QUANTUM F OURIER T RANSFORM
In addition to producing a superposition of the vector 0 or 1, the Hadamard Gate is its own
inverse. This means that H = H 1 . Therefore, when applied to the created superposition of the
vector 0 or 1, it will revert the superposition back into the binary vector of 0 or 1:
(
)
H p1 (|0 + |1) = |0
( 2
)
H = H 1 =
H p1 (|0 |1) = |1
2
8 The Quantum Fourier Transform

The Quantum Fourier Transform, also known as Quantum Phase Estimation, is a linear
transformation on qubits. The Quantum Fourier Transform obtains a good estimate of the
phase parameter, , of a periodic state. Consider the two-qubit state:
1
1 2
e 2i y |y
p
2 y=0
2
x 2 can be determined from the first qubit by applying a Hadamard gate. In turn, x 1 has
to be determined from the second qubit. If x 2 = 0, then the 0 is negligible because it does not
affect the binary decimal value of . Therefore, we can determine x 1 by applying a Hadamard
gate second qubit since would just equal 0.x 1 . If x 2 = 1, we will need to apply a phase rotation
operator, R 2 :
[
] [
]
1
0
1
0
R2 =
2 =
0 e 2i (0.01)
0 e 2i /2
Notice the 0.01 in the exponent. This exponent is written in base 2 decimal form, (i.e. 0.01 =
22 ). The inverse of R 2 also will use base 2 in the exponent.
[
]
1
0
1
R2 =
0 e 2i (0.01)
In order to execute the phase estimation, the binary representation of the decimal is taken.
Then, the right most digit of the decimal is checked. If said digit is 1, it activates the controlled
rotation gate. If the digit is 0, it does not activate the rotation gate. Either way, it returns the 0
or 1 and it tells you the coefficient of the negative power of 2. The rotation gate will essentially
change the right most digit to 0. This will iterate until the frequency is obtained or one runs out
of qubits. (See Figure 7)
Figure 7: A circuit for the 3-qubit phase estimation algorithm.
41
Junlin Mo
9 Time Complexity
The time complexity of any given algorithm is the amount of resources used by a computer
to complete the algorithm. Two very important resources for computers are time and space.
We measure the amount of a resources used in a computation for solving a given problem as
a function of the length of the input of an instance of that problem. For example, a computer
may take 4n 2 +2n units of time to solve the problem. We would say that the running time of that
algorithm for these computer is O(n 2 ). Time Complexity is often referred to in O notation. An
algorithm is generally considered efficient if the amount of the resource used is O(n k ) for some
k. We call these algorithms polynomial with respect to the resource. If an algorithms running
time is in O(n) or O(log n), we call these linear and logarithmic, respectively. These algorithms
are all considered efficient.
As a note, O( f (n)) represents an upper bound of the running time. This means that algorithms in O(n) or O(n 2 ) are also in O(n 3 ). is used for lower bounds. Algorithms that use (c n )
for some constant, c, resources are considered exponential and inefficient. This is because, at
some point (or amount of resources), an exponential algorithms will exceed the running time
of even the largest polynomial algorithms.
10 Modern Implications
One of the largest subject of algorithms of interest in the field of quantum computing are
prime factorization algorithms. These algorithms are of large interest because of their applications in the RSA. The RSA cryptosystem is a public key protocol commonly used in government
and corporations in order to encrypt sensitive information. The security of RSA relies upon the
assumption that that factoring large numbers on computers is a difficult and strenuous process.
That being said, currently, there is no known classical computer prime factorization algorithm
that has a polynomial running time, O(n k ). If an algorithm were to be found, it wound undermine the security of RSA.
However, there are quantum prime factorization algorithms that are vastly more efficient
than classical computation algorithms (See Figure 8). Both Shors algorithms and a minimization algorithm have been used to factor large numbers in a timely manor. For example, Shors
algorithms running time is O((log n)2 (log log n)(log log log n)). One thing to note is that these
algorithms have also only run on ten or less qubits.
10.1 Moores Law

Moores Law is essentially an observation on the rate of change in silicon-based processors.
Gordon E. Moore, co-founder of Intel, observed that the number of transistors in a dense integrated circuit had doubled approximately every two years. After 50 years of evidence, Moores
Law has been a reference point in the advancement of computing (See Figure 9). However, in
accordance to Moores Law, there are only so many transistors that we can fit into a dense integrated circuit before it becomes physically impossible to add more. Quantum computers already use subatomic particles as bits because they rely on the quantum mechanical properties
of subatomic particles, such as superposition, to process information. Looking into the future,
42
Trevor Pennypacker
M ODERN I MPLICATIONS
Figure 8: Table of prime factorization algorithms and their results

society may change their reliance from silicon-based computing to other forms of computing,
quantum computing being a possible alternative in x amount of years.
Figure 9: Graph of Moores Law in history.
43
Junlin Mo
Sources
Kaye, Phillip, Raymond Laflamme, and Michele Mosca. An Introduction to Quantum Computing. Oxford: Oxford UP, 2007. Print.
Griffiths, David J. Introduction to Quantum Mechanics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005. Print.
44
On the Configurations of Hats

Karen Ying
1 Introduction
In this paper, well be exploring a collection of logic puzzles in which a number of people
are given colored hats and asked to guess the color of the hat on their head (which they cannot
see) based on the information that is presented to them.
The solutions to these puzzles relate to information theory and data transmission. Additionally, these puzzles are common interview questions at large companies such as Google and
Jane Street Capital.
2 Hats in a Circle with One Collective Strategy

n people stand in a circle, each with either a white or a black hat and are given
a deterministic strategy (i.e. the strategy cannot be to guess randomly) to guess
the color of his own hat by observing the other n 1 hats. All players must use the
same strategy, which they agreed on beforehand. Let us represent a black hat as
1 and a white hat as 0 [1].
Theorem 1. For n = 2 and a particular strategy, there will always be a hat assignment that guarantees no one guesses correctly.
Proof. For n = 2, there are only the following four strategies:
1. Guess 0.
2. Guess 1.
3. Guess the same color as what the other person is wearing.
4. Guess the opposite color of what the other person is wearing.
45
Karen Ying
O N THE C ONFIGURATIONS OF H ATS
For each of these strategies, we can assign the hats in a way that no one guesses their hat
color correctly:
1. [1, 1]
2. [0, 0]
3. [0, 1] or [1, 0]
4. [0, 0] or [1, 1]
What if n > 2? How many people can guess correctly? We find that at least one person can
always guess their hat correctly.
Theorem 2. n > 2 a strategy where at least one person guesses correctly.
Proof by Construction. We will use the following two rules which form our strategy:
1. When n is even and you see an alternating pattern of hats, guess the hat color that fits in
the alternating pattern (i.e. guess the color of the person sitting two spaces to the right of
you).
2. In all other cases, guess the color of the hat to your immediate right.
With this strategy, we have the following cases:
Case 1. n is odd: We need the following lemma.
Lemma 2.1. If n is odd, there is always at least one pair of adjacent people with the same
color hats.
Proof. Assume there are n = 2m + 1 people and that there are no pairs of adjacent same
color hat people. Pick a person, without loss of generality, assume his hat is black. Then
the people immediately next to him have the same color hats, white. More generally, the
people k spaces to the left and k spaces to the right have the same colors. However, the
people m spaces to the left and m spaces to the right sit next to each other and have the
same color hat.
Since there is at least one adjacent same color pair in any odd hat assignment, rule 2
guarantees that at least one person guesses correctly.
Case 2. n is even and the hat assignment is alternating: By rule 1, everyone guesses correctly.
Case 3. n is even and the hat assignment is alternating with one error, e.g. 0101010111: In this
case, the error will see an alternating pattern of hats and guess wrongly. The other n 1
people will use rule 2, and guess the color of the hat to their right. The person sitting to
the left of the error will be the only one who guesses correctly.
46
Karen Ying
H ATS IN A C IRCLE WITH O NE C OLLECTIVE S TRATEGY
Case 4. n is even and the hat assignment is not Case 2 or 3, so every other hat assignment: Since
the pattern is not alternating, there is at least one pair of people with the same color hat
sitting right next to each other. Thus rule 2 guarantees that at least one person guess
correctly.
Theorem 3. n no strategy that gives more than
n
2
correct guesses for any hat assignment.
Proof. Let us calculate the expected value of the number of correct guesses given out in a random hat assignment, N :
N=
number of people that guess correctly in x
x{0,1}n
=
=
1
2n
1
2n
number of correct guesses p gets in x
x{0,1}n ppeople
number of correct guesses p gets in x
ppeople x{0,1}n
Going through all x, we see y {0, 1}n1 , 0 y or 1 y . For each y, an assignment of the rest of
the hats, one time ps hat is guessed correctly based on the strategy and one time it is not. Thus,
N=
1
2n
2n1 =
ppeople
1
n
n 2n1 = .
n
2
2
In general, the minimum of a set is less than or equal to the expected value, or average,
with equality when, in this case, all hat assignments give the same number of correct guesses.
Consider the assignment of all white hats without loss of generality. This assignments means
everyone guesses the same thing. Either n people guess correctly or no one guesses correctly.
Thus the minimum does not equal the average which means that the minimum number of hats
is less than n2 .
Theorem 4. 2n a strategy such that at least n 1 people guess correctly.
Proof by Construction. Among everyone, there is either an odd or even number of black hats.
If one knows the parity of black hats, he is able to figure out the color of his hat and guess
accordingly. Thus if n people think that theres an odd number of black hats and the other
n people think that theres an even number, exactly n people will guess the color of their hat
correctly.
Since theres an even number of people standing in a circle, we are able to pair everyone
up with the person standing directly across from them, forming n pairs. Within each pair, the
two people will either have the same color hats, call them same color pair or different color
hats, different color pair. Suppose there are m different color pairs.
The strategy works as follows:
1. If you observe the rest of the n 1 pairs as same color pairs, then guess your partners
color hat.
47
Karen Ying
2. If you observe at least one different color pair out of the n1 pairs, find the person nearest
clockwise to you who is part of a different color pair. If his hat is black, judge the total
number of black hats as odd and make your guess accordingly. If his hat is white, judge
the total number of black hats as even and guess accordingly.
Thus we have the follow 3 cases:
Case 1. m = 0: By rule 1, 2n people are correct.
Case 2. m = 1: By rule 1, the two people in the one different color pair will both be wrong.
However for the rest of the n 1 pairs, exactly one out of the two people will be correct
since the two people will make different observations about the parity of the black hats.
Thus n 1 people will guess correctly.
Case 3. m > 1: By rule 2, n pairs, two people in each pair will make different judgments about
the parity of black hats. Exactly n people will be correct about the parity and thus guess
the color of their hat correctly.
This strategy works for even numbers of people. However, what if theres an odd number?
Can we generalize, or do even better?

Theorem 5 (From [2]). n a strategy in which n2 1 people guess correctly.
Proof by Construction. We define two configurations of hats adjacent if flipping a single hat in
one results in the other. In addition, we define a configuration of hats to be symmetric if rotating
each hat m people around the circle for some m < n produces the same configuration. We are
going to need the following lemma to construct our strategy which begins our transition into
graph theory.
Lemma 5.1. Given a multigraph G with minimum degree 2m, for each edge we can choose one
of its vertices such that every vertex is chosen at least m times.
Proof. First if there are any cycles of edges v 1 v 2 , v 2 v 3 , . . . , v r 1 v r , v r v 1 , we can choose the vertices of the cycle in a direction along the cycle. Do this until the edges left to be assigned have
no cycles and thus form a forest.
Let degT (x) be the degree of vertex x after cycle deletion i.e. the number of edges coming
out x that are not part of cycles. For each tree T , pick a vertex v T such that degT (v T ) = 1 and
orient T so that every edge in the tree points away from v T . Every vertex with degree one after
cycle deletion originally had degree at least 2m + 1 and has been already chosen by at least m
edges. Every other vertex u T , is chosen at least degT (u) 1 times. If u is not part of any
cycles then degT (u) = deg(u) 2m and it is chosen degT (u) 1 times which is at least m times.
However if u is part of c cycles in addition to to T , u is chosen degT (u) 1 + c times total, which
is m:
degT (u) + 2c
m
deg(u) = degT (u) + 2c 2m =
2
Simplifying the LHS we have:
degT (u) + 2c degT (u)
=
+c
2
2
48
Karen Ying
H ATS IN A C IRCLE WITH O NE C OLLECTIVE S TRATEGY
We know,
degT (u)
2
degT (u) 1
Thus,
degT (u)
+ c degT (u) 1 + c = degT (u) 1 + c m

2
Every vertex in a multigraph is either only part of cycles or part of cycles and a tree. If a
vertex is only part of cycles, its degree is 2m and is chosen m times. If a vertex u is part of
cycles and a tree, weve shown above that u is chosen at least m times as well. This concludes
our lemma.
Back to the strategy. We deal with the symmetric hat assignments by having the people
implement the following strategy: if there is a color and a person such that the configuration
would be symmetric if that person has that color hat, the person guesses that color. This ensures
that for symmetric configurations, every person will guess correctly.
And now for non-symmetric configurations, we construct a multigraph whose vertices are
equivalence classes, v i , of n-bit non-symmetric strings under rotation. First connect every adjacent hat configuration with a line. We can summarize and simplify these lines connecting
adjacent configurations in different rotation classes by an edge in this way: The number of
edges connecting v i to v j is equal to the number of adjacent hat configurations a i v i has in
v j and a j v j has in v i . Then label these edges with (n 1)-bit strings which represent the hat
configurations one person would possibly see if he looks clockwise around the circle such that
he knows which two equivalence classes the actual hat assignment may belong to.
E.g. for n = 5:
Figure 1: Multigraph for n = 5
49
Karen Ying
Orient the multigraph as described in the lemma above, and then simplify the equivalence
classes into vertices.
E.g. for n = 5:
Figure 2: Simplified multigraph for n = 5

Finally, our strategy is: Have everyone agree on a multigraph representation for n people
beforehand and memorize it. Look clockwise around the circle. You see an (n 1)-bit string
which corresponds to an edge in the multigraph. This edge chooses a vertex in the multigraph.
Thus the chosen vertex represents the equivalence class you guess the hat configuration is in.
Guess the hat color that would make the configuration in the equivalence class complete. By
the previous
will be chosen
n lemma, we can assign a vertex to each edge G such that each
nvertex
at least 2 times
n which agrees with Theorem 3. This means at least 2 people guess correctly because 2 people chose that the hat configuration equivalence class correctly and thus
guessed correctly.
There is also the case of a symmetric hat configuration with one error. The error looks
around and sees a symmetric configuration and therefore guesses the wrong color. The other
n 1 people see a non-symmetric
configuration and use the multigraph strategy. This is why
n
we can only guarantee that 2 1 people guess correctly.
50
Karen Ying
F INITE H ATS IN A L INE
3 Finite Hats in a Line

n people stand in a line, wearing either a black hat or a white hat. These people
can see all the hats in front of them but not the hat on their head nor the hats
on the heads of the people behind them. They guess in the order in which they
are standing, starting with the person in the very back. Everyone can hear each
others guesses. Again, assign a black hat the number 1 and a white hat, 0.
Theorem 6. n a strategy such that n 1 people will always guess their hat color correctly.
Proof by Construction. We build the following strategy: The person all the way in the back cannot gain any information about the color of their hat, so he sacrifices himself for the greater
good of mathematics. However, he is able to convey information to the rest of the n 1 people.
He takes the sum of the number representation of the hats in front of him mod 2 and that is his
guess. This guess conveys the parity of the black hats to the people in front of him. The other
n 1 people can deduce their hat color because they know the parity of the black hats in front
of them from the guesses of the people behind them. Thus the other n 1 people are able to
guess their hat color correctly.
4 Infinite Hats in a Line

A countably infinite number of people stand in a line, wearing either a black hat
or a white hat. These people can see all the hats in front of them but not the hat
on their head nor the hats on the heads of the people behind them. They guess in
the order in which they are standing, starting with the person in the very back.
Everyone can hear each others guesses. Assign a black hat the number 1 and a
white hat, 0.
For any n > 0 we proved before that only one out of every n people fails to guess correctly.
Thus for an infinite number of people, we can just divide everyone into groups of size n and
have them implement the finite strategy sown above. However, we can do better and have only
a finite number of people guess incorrectly.
Theorem. For an infinite number of people standing in a line, all but finitely many people can
guess their hat color correctly.
Proof by Construction. Let S = {(a 1 , a 2 . . .) | a i Z2 }, the set of hat assignments. We need to define an equivalence relation on the hat assignments. For s, s S, we say s s if they eventually
agree in every place (e.g. (11111111 . . .) 01111111 . . .).
Before everyone lines up, they agree to invoke the axiom of choice and choose a single
element from each equivalence class in S and memorize each element.
Person i in the line sees:
s = (. . . a i +1 , a i +2 . . .)
51
Karen Ying
He doesnt know the first i elements but he can determine the equivalence class of s by
observing the hats in front of him. He then recalls from his memory the element that everyone agreed to memorize from the equivalence class of s and guesses the i th element of the
sequence.
If everyone in the line implements this strategy, only a finite amount of people will guess
incorrectly since two hat assignments in the same equivalence class will only differ in a finite
number of places before converging.
5 Applications
Hamming codes are a type of efficient single error correcting codes, giving 2k 1-bit messages, where k-bits are redundant and the rest of the 2k k 1-bits are data. A property of single
error correcting Hamming codes is that it either is a valid message or one error away from being
a valid message [3] .
Tying this back to our puzzles, suppose we have n = 2k 1 people in a circle wearing either
a black (1) or a white color hat (0). They guess the color of their hat by observing the other n 1
people. We have everyone assume that the assignment is not a valid Hamming encoded message, e.g. if a person knows that him having a black hat makes the assignment a valid message,
he guesses white. Unfortunately, the strategy breaks down when the configuration is a valid
Hamming encoded message. But for every valid message there are n invalid messages (since
there are n bits that you can change in the string to result in an invalid message) which
means
n
n
n2
that the probability of success is for every person is n+1 , averaging n+1 n = n+1 correct
guesses.
6 More Directions
Now that you have a feel for some of the types of hat logic puzzles that exist and their
solutions, what if we have many different color hats instead of just black and white? n colored
hats? What if theres an infinite number of people in a circle? What if people standing in a line
had to make their guesses simultaneously? What generalizations can you make about these
puzzles?
Bibliography
[1] Canada/USA Mathcamp, cited 2015: Qualifying Quiz,
http://www.mathcamp.org/prospectiveapplicants/quiz/index.php
[2] Jack Gurev, cited 2015: On the Symmetric Groups of Hats

[3] Alex Zorn, cited 2013: Colored Hats and Logic Puzzles,
http://mathcircle.berkeley.edu/archivedocs/2012_2013/lectures/
1213lecturespdf/BMC_Int_Jan22_2013_HatsAndLogicPuzzles.pdf
52
DNA Sequence Analysis Program

Karen Zhou
Abstract
Bioinformatics is the application of mathematics and computer science to develop methods of analyzing and studying biological data. Programs that compare and analyze DNA sequences are important for the sake of scientific and
medical progress. They are useful in researching genetics, identifying evolutionary history, studying epidemiology, and more. This purpose of this project
was to develop a basic DNA sequence analysis program using the C++ programming language. Specifically, it would input two files of DNA sequences, use the
word method to compare their sequences, align the files and display the results, and provide more information about the sequences. Additional features
and functions will be added after the duration of Junction.
1 Introduction
There exist several DNA sequence alignment and analysis programs; one such program is
called basic local alignment search tool, or BLAST. BLAST was first implemented in 1990 (Kumar, 2005). It works by parsing nucleotide sequences into query words, performing character
string comparison against all sequences in the target database, and identifying statistically significant matches. BLAST uses an identity matrix and applies a scoring system to get a raw score
of how similar two sequences are. Then it uses statistics to convert the raw score into a bit score
(BLAST, 2007). Programs like BLAST are useful for studying gene families, analyzing evolutionary patterns, identifying species, and more. BLAST served as inspiration for this project.
The purpose of my research project was to develop a C++ computer program that analyzed
similarities in DNA sequences. The program would read in two FASTA files of nucleotides. This
program would first assign values of 0, 1, 2, and 3 for A, C, T, and G respectively. Then, similarly
to the blastn program of BLAST for nucleotide-nucleotide comparison, it would parse the nucleotide sequences in both files into query words of size 8. The parsed words would be hashed
and stored in a hash table. Then the hash tables of both files would be compared to each other.
A function would align the sequences based on similar sequences and their positions. Finally,
the program would show additional information about the files, such as the percentage of similarities and the number of different base pairs.
53
Karen Zhou
DNA S EQUENCE A NALYSIS P ROGRAM
I was inspired to do this project because I had previously used bioinformatic programs
like BLAST and Seaview, and phylogenetics programs like BEAST. Because bioinformatics combined two things I enjoyed, biology and programming, I decided to look more into it. Junction
was the perfect opportunity to further explore bioinformatics, especially because it is not typically taught in a high school setting. I wanted to gain insight into how the programs worked by
creating one of my own. It is my intention to continue this project after the duration of Junction,
potentially for science fair, and to improve my computer program. I hope to eventually use my
program for actual genetics research.
2 Methods
The materials required for this project include:
C++ Programming
Computer/Laptop
Programming software (Eclipse or Windows Visual Studio 2013)
Seaview (to check alignment)
Two FASTA files of nucleotide sequences
USB drive or other method of backing up work
(optional) BitBucket or other code repository site
Before beginning the project, I researched existing DNA analysis programs, like BLAST.
Then I downloaded the necessary programming software: Eclipse for C++, as well as Seaview.
I opened the Eclipse workspace and began a new C++ project. In the source folder, I added a
header file, named Hash.h, which was then included in the main.cpp. The majority of functions
were written in the header file. In Hash.h, I declared a class called Hash. Then I declared and
created functions that read in FASTA files, converted the sequences in the files (A 0, C 1, T
2, G 3), and parsed the converted files into words of size 8 bp. Then I wrote a function that
hashed the parsed words. The hash function converted each base-4 parsed word into a base-10
integer. This integer value, known as the hash value, was then stored in a hash table (see Figure 1). The hash table was generated in another function. Each hash value, which represented
a parsed sequence, was stored in a space in the hash table that corresponded with its position
in the original sequence. Then I created functions that compared the hash tables of each file
by searching for matching hash values (see Figure 2). Matching hash values meant the both
sequences contained that particular sequences. The differences in locations of matching hash
values were then used to align the files. More functions converted the hash values back into
sequences of nucleotide bases, and displayed the final alignment. In the main.cpp, I created
a new Hash variable, so that I could apply the functions from the header to the sequence files.
When the program ran, it asked the user to input the file names. Then it displayed the aligned
result, along with base-pair differences and percentage similarity. I backed up my work onto a
USB drive every time I made a major update (e.g. after creating a successful function, before
creating a new function). Debugging occurred throughout this whole process.
54
Karen Zhou
R ESULTS
Position
0
1
2
...
n
Hash Value
10062
40248
29923
...
7523
Figure 1: Hash table visualized. The hash value is stored relative to its position
in the sequence.
Figure 2: A visualization of the method of comparison. Red arrows compare

the corresponding positions of the hash tables and show mismatches. Green
arrows show matching hash values. When matches are found, the differences
in position of the hash values are used to align the sequences. For example, the
difference in position for hash value 10062 is 1 0 = 1, which implies a shift of
1 in the alignment.
3 Results
Figure 3: This screenshot of the console shows the resulting alignment after
two FASTA files are inputted. The differences are highlighted in yellow.
55
Karen Zhou
Figure 4: This screenshot shows the end of the alignment. The dashes represent gaps where deletions or insertions may have occurred.
Figure 5: This screenshot shows the additional info provided by the programs
analysis of the two DNA sequences. The program counted the number of different base pairs and calculated the percentage of similarity between the two
files.
Figure 6: This screenshot of the header file shows the various functions that
were created to ultimately display the alignment and other information. There
were several hundred lines of code in this program.
56
Karen Zhou
D ISCUSSION
4 Discussion
There were several challenges that I faced while conducting this project. Brainstorming
and implementing the most efficient methods was a time-consuming process. Many bugs rose
along the way, especially as the code got longer, as it got harder to determine the source of
error in the multitude of lines. There were also software issues; for a while I had to switch to
Windows Visual Studio because Eclipses compiler was not functioning correctly. Furthermore,
my original hash table was of size 29,425,664, and running the program while comparing each
value of the two 29,425,664 sized hash tables took over 40 minutes.
Hash Value + 48 (Pos)
0
...
10062 + 48 (1) = 75, 598
...
(48 1) + 48 448 = 29, 425, 663
Exists? (Yes = 1, No = 0)
1
...
1
...
0
Figure 7: Original hash table visualized. The size of the entire hash table was
29,425,664. If a particular sequence was contained in a specific position in a
file, then a 1 was stored in that hash table index. If it was not contained in the
file, a 0 was stored.
Then the hash table was shortened to 48, or 65,536. This increased the speed significantly.
However, because this new hash table stored positions relative to hash values, it was difficult to
analyze the locations of sequences and to align them accordingly.
Hash Value
0
1
...
10062
...
48 1 = 65, 535
Positions where this hash value

is found (if not found 1)
1
1
...
1, 17
...
35
Figure 8: Revised hash table visualized. Now the hash table is of size 48 or
65,536. Because certain strings of bases can be found in multiple areas of a
sequence, multiple positions could be stored relative to hash value.
Finally, the hash table was changed to store hash values relative to their position. So the
size was now the total number of parsed words obtained from the file (see Figure 1). The files I
used each yielded 441 parsed words, so their hash tables were of size 441. This not only ensured
that the program would be faster, but it also simplified the alignment process. The time limit
was another obstacle that prevented me from adding more features to my program. Some additional features I could add in the future include displaying locations of differences, making the
57
Karen Zhou
alignment more accurate, and identifying specific genes. Furthermore, I used the word method
of comparison in my project, but dynamic programming is another method of sequence alignment that I could explore in the future. Ultimately, this project was challenging, but rewarding,
and I gained a lot of knowledge and insight into bioinformatics.
Acknowledgements
Id like to thank my mentor, Lane Gunderman, the Junction directors, the other Junction
mentors and students, and my friends and family for all their contributions to these amazing two weeks. Id also like to thank my AP Biology teacher, Dr. Offner, for first introducing
me to BLAST and thus opening my world to bioinformatics, and my programming teacher,
Ms. Knecht, for teaching me the coding skills I used for my project.
References
Altschul, S. F., Gish, W., Miller, W., Myers, E. W., and Lipman, D. J. (1990). Basic local alignment
search tool. Journal of molecular biology, 215(3), 403-410.
Beringer, K. (2014, June 21). C++ tutorial: Intro to hash tables. Retrieved from
http://pumpkinprogrammer.com/2014/06/21/c-tutorial-intro-to-hashtables/
BLAST [PowerPoint slides]. (2007, November 5). Retrieved from http://www.ncbi.nlm.nih.

gov/Class/MLACourse/Modules/BLAST/slide_list.html
[Rolling Hash (Rabin-Karp Algorithm)] [Lecture notes]. (2011, February 18). Retrieved from
http://courses.csail.mit.edu/6.006/spring11/rec/rec06.pdf
Suresh Kumar (2005). Bioinformatics web. Retrieved from: http://www.geocities.com/

bioinformaticsweb/
58
Control Theory
Keren Shao
mentored by Christopher Harmon
Introduction by Christopher Harmon

Keren was interested in the analysis of real, dynamic systems and how to control them. These systems include many things, from a cruise controller in a
vehicle to the stabilization of large aircraft. But with any of these systems, there
remains exactly one goal: using an input to have a desired effect on an output.
For simplicity, he focused on SISO (single input, single output) systems, and
conducted tests using different controllers to command a copper flywheel to
spin at various speeds. The goal was to devise a method of achieving a system
that would respond to changes in its state quickly and accurately when given a
command signal. Then the goal was to improve that method using feedback
control analysis techniques. The following exposition by Keren outlines the
theory and general process he used.
1 Block Diagram
It wont be extremely complicated to describe a system if we use the following:
Figure 1: Block Diagram

Since there is an ordinary differential equation in our plant, we transfer the system into
the frequency domain (and do easy algebra!)
59
Keren Shao
C ONTROL T HEORY
2 Laplace Transform
Using Partial Integral,
L( f (t )) = F (s) =
e st f (t ) d t
0

L( f (t )) =
e st f (t ) d t = sF (s) f (0)
0
Since this is a LTI (Launch-Terrible-Instrument?) System

L( f (t )) = F (s)
L( f (t )) = sF (s)
L( f (t )) = s 2 F (s)
Now we transform the ODE in the flywheel
J + B = T
J s(s) + B (s) = T
1
=
T
Js +B
Hooray! The time of graphs and algebra! Based on the block diagram, we can get the Error
Function E (s) = (s)K t .
KQ
=
T (transfer function) =
d J s + B + KQ
(
)
Kt Ka Km
Kd
Q=
and K = K p + K i s +
N
s
If K =
Kp
a
Ki s
b
Kd
sc
then
T=
K (as + b + cs 2 )Q
J s 2 + B s + K (as + b + cs 2 )Q
3 Analysis
3.1 Response Curve
Since
(
T = KQ
c
J +KQc
s2 +
s2 +
a
J +K Qc
s + J +Kb Qc
B +KQa
K Qb
J +K Qc s + J +K Qc
Wn =
K Qb
J + KQc
B + K Qa
(4KQb)(J + K Qc)
where W n is the natural

frequency, is the damping ratio and W d is the damping frequency
satisfying W d = W n 1 2 ,
60
Keren Shao
A NALYSIS
Figure 2: O A is the time constant, OB is the rise time, OC is the peak time, DF
is the overshoot, and OE is the settling time.
Important parameters:
= 1 e 1
Tr =
(1 cos1 )
2
1
Tp =
1 2
p
2
OS = e / 1
I n(%)
Ts =
W n
3.2 Root Locus

T=
K Q(cs 2 + as + b)
(J + K Q)s 2 + (B + K Qa)s + K Qb
When the numerator is 0, we get zeros. When the denominator is 0, we get poles. Assumption:
for the sake of god and algebra, we set a = 2, c = 1, b = 5, and J = B = Q = 1. The numerator is 0
when s = 1 + 2i or 1 2i , and the denominator is 0 when s = 0 or 1.
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Keren Shao
C ONTROL T HEORY
Figure 3: Root Locus

Is this system stable...? (Probably) Does it respond quickly? (No)
4 Improvement
1. Try to move the Zeros to more negative area with respect to The Real Axis (Asjust a, b, c)
2. Try to use P/PI/PD controllers for this system? (In fact, P seems to be the best controller
for this system if we do not have a bad motor?)
62
Keren Shao
I MPROVEMENT
Figure 4: Lab Results
63
Above: Long Do with a quadcopter that he built for his project. Below: A cheesecake-making class!
Visualizing Light Patterns

of Flat-Folding Origami
Ria Das
1 Background and Motivation

1.1 Origami and its importance in mathematics and science
Origami is the ancient Japanese art of paperfolding. Thousands of years old, it has given
us such wonders of art and physics as the paper crane, paper elephants, and many other impressive models. Yet, besides being a form of art, origami also has a lot of intersection with
mathematics and science. The mathematics behind origami models allows one to come up
with novel origami designs, to explain the foldability of a given design, and to create algorithms
for use in computer programs that make it much easier to design origami models and visualize
the folded state of a given design. The precise specification of crease patterns in origami has led
scientists to explore its application in diverse scientific pursuits, such as in the folding of large
space telescopes that could make it possible to transport them in size-limited spacecraft [1],
and in manufacturing synthetic DNA using folding techniques that best resemble the folding
involved in natural DNA synthesis [2].
1.2 Flat-folding origami tessellations and light patterns

Tessellations are repeating patterns of geometric shapes that cover the plane with no gaps
or overlaps. These are used in wallpaper designs, mosaics, and so on. Origami tessellations
are repeating patterns of geometric shapes folded by using pleats and twists from a single
sheet of paper [3]. A popular type of basic flat-folding origami tessellations is considered the
brainchild of Japanese origamist Shuzo Fujimoto. The design produced after folding the pleats
and twists is flat that is, it can be expressed using two-dimensional geometry. At the same
time, different points in the folded paper may have a varying number of layers of paper on it.
Thus, if the sheet of paper is semi-transparent, then the folded design, when held against bright
light, will have different amounts of light coming through its different parts.
65
V ISUALIZING L IGHT PATTERNS OF F LAT-F OLDING O RIGAMI
Ria Das
The following picture shows a Five-and-Four folded origami tessellation against the outside light. Some of the gaps between darker bands were caused by slight inaccuracies in folding.
Figure 1: Light pattern of a folded Five-and-Four tessellation.
1.3 A computer program for visualizing light patterns

This MIT ESP Junction project was an independent study with the aim of understanding
the mathematics behind origami and origami tessellations, as well as an attempt to write algorithms based on computational geometry and graph theory that would allow a computer
program, given a crease pattern, to show on a monitor or in a printout the light pattern that
would result if the folding was carried out. The availability of such a tool makes it easier and
faster to visualize the light patterns of origami tessellations.
2 How to use the computer program

2.1 Input: Vertices
The vertices are specified in a text file with each line being of the form:
<vertex-label-name> <x-coordinate> <y-coordinate>
66
Ria Das
H OW TO USE THE COMPUTER PROGRAM
The following excerpt shows three lines of a sample vertex file:

1A 0 0
2X -2 0
3P -2 1
2.2 Input: Faces

The faces of a crease pattern are polygons bounded by the crease patterns lines that do not
overlap each other and together cover the entire paper. Each face is specified by a face name
(for reference) followed by a list of vertices specified in a single line. Each vertex in the list is
identified by its position (calculated as the line number minus 1) in the vertex input file. The
first line of the face input file, however, is used just for specifying the fixed, starting face for
folding and hence contains only a single face name.
Specifically, the first line of the face text file has the following format:
<face-label-name-to-be-used-as-the-fixed-face-in-the-model>
Each subsequent line has the following format (the vertex-index-number is the line number of the vertex in the vertex file minus 1, to allow for 0-based indexing):
<face-label-name> <vertex-index-number> <vertex-index-number> ...
The first few lines of a sample face input file are shown below:
Face-AXMV
Face-ABC 0 6 7
Face-BCYST 6 7 8 13 5
Face-ADB 0 3 6
2.3 Input: Transformations

The information in the transformation input file is used for positioning and making duplicate copies of the main origami pattern on the screen. Each line of the file contains a label
followed by one or more transformations, as follows:
<label> <reflection or translation or scale> ...
The reflections are specified in the following format:

m b <isVertical>
where m and b are numbers representing the slope and y-intercept of the reflection line y =
mx + b.
The <isVertical> component can either be true or false. If true, m can take any numeric
value, and b is the x-intercept of the vertical line x = b.
Translations (x, y) 7 (x + a, y + b) are specified in the following format:
67
Ria Das
a b translate
A scale (x, y) 7 (ax, a y) is represented by the following line:
a 0 scale
Finally, the three possible labels at the start of a line are old, new, and all.
old applies the following transformations to the crease pattern/folded model specified by the
vertex and face info files.

new makes a duplicate copy of the old-transformed unit, and apply the following transforma-
tions to it.
all makes duplicate copies of the old-transformed and all the new-transformed units on the
screen, eliminate the common boundary lines between those transformed units, and apply the transformations specified in this line to them.
(Note: There must be exactly one old label in the transformation file, followed by zero or
more new-labeled lines and at least one all-labeled lines, in that order.)
Here are the lines in a sample transformation input file:
old
new
new
new
all
all
all
all
all
all
all
all
2 2 translate 0.5 0 scale

0 0 true
0 0 false
0 0 true 0 0 false
5 0 translate
-5 0 translate
0 5 translate
0 -5 translate
5 5 translate
-5 -5 translate
-5 5 translate
5 -5 translate
3 Sketch of the algorithm: design and implementation

3.1 Overall algorithm
1. Read the following from the input files: vertices (label and coordinates), faces (fixed face
name, each face name and its list of vertex numbers), and positioning information (for
replication, translation, and scaling).
2. Determine the list of adjacent faces for each face.
3. Starting from the designated fixed face, compute a Hamiltonian path (if one exists); otherwise, find a spanning tree.
68
Ria Das
S KETCH OF THE ALGORITHM :
DESIGN AND IMPLEMENTATION
4. For each face, traverse back along the path to the fixed face (root). Multiply the sequence
of transformation matrices corresponding to the edges being traversed to compute the
final conversion matrix for the face.
5. For each face, use its final conversion matrix and its original vertices to obtain the coordinates of the vertices after the folding.
6. Display the faces on the screen, before folding (show crease pattern) or after folding
(show folded model), using semi-transparent fill of the interior of the faces. Areas where
multiple faces overlap become darker in appearance.
7. If the user specifies a test points coordinates, compute the number of layers at that point
by using a variation of PNPoly algorithm that considers proximity to sides and vertices of
a face.
3.2 Major steps

1. Finding an adjacency list for the faces: The adjacency of a pair of faces is determined by
checking to see if they share a line segment. This is determined by checking for common
subsequence in the two faces circular sequences of vertices, possibly reversed. For example, faces ABCD and DCEF are determined to be adjacent because they share the CD line
segment. This adjacency information is used in modeling a crease pattern as a graph and
generating the folding sequence and transformation matrices for each face, as explained
below.
2. Modeling a crease pattern as a graph: In graph theory, a graph is a mathematical structure made up of nodes and line segments called edges that connect them. The specified crease pattern is modeled as a graph in the program with each face represented as
a node, and the line segment shared between each pair of faces represented as an edge
connecting two nodes. This representation allows use of graph theoretical algorithms and
analysis, in particular for the path finding purposes described below.
3. Hamiltonian path finding: A Hamiltonian path in a graph is a simple path that visits
every vertex exactly once [4, p. 1066]. Upon modeling the crease pattern as a graph, the
program attempts to find a Hamiltonian path starting from the fixed face specified in the
face input file. If one exists, then it performs transformations for folding the faces in the
reverse path order. The line containing the line-segment shared between a consecutive
pair of faces is used as the line of reflection that represents the folding action. The specific
algorithm used for finding the Hamiltonian path, if one exists, is shown later.
4. Spanning Tree finding: A spanning tree connects all of the nodes of a graph without any
cycles, such that there is exactly one path connecting every pair of nodes. If a Hamiltonian
path cannot be found starting from a fixed node (face), a spanning tree is computed using
an algorithm shown later.
5. Determining the transformations for each face: Consider the sequence of edges, representing the shared line-segments between each consecutive nodes on the path from each
69
Ria Das
node to the fixed node. A face undergoes a fold, that is, a reflection, on each such crease.
Such reflections are represented as affine transformations [5]. Left multiplication of the
33 identity matrix with the sequence of affine transformation matrices provides us with
the single conversion matrix to be used to left-multiply the vertices of the face.
6. Compute the new coordinates of the face vertices: The conversion matrix computed
above for each face is used to left-multiply the matrix representing the vertices of the face
to obtain the coordinates for the vertices after folding.
7. Finding the number of layers of paper present at a given point: Due to folding, the number of layers of paper at a given point may vary from 0 (if outside the folded area) to n,
where n is the total number of faces in the crease pattern. For a given point, the number
of layers is determined by using a variation of the PNPoly algorithm [6] that allows specification of an error tolerance value to handle approximation of values in double arithmetic
computation.
3.3 Combined algorithm for finding Hamiltonian path (if present) and
finding a spanning tree
The following stack-based algorithm finds a Hamiltonian path starting from a fixed face,
if one exists. While looking for a Hamiltonian path, it also computes a spanning tree with the
fixed face as the root, by recording a (single) parent node for each node in the graph.
String HpSt (Node N, Stack<Node> Path) {
P <- Path.peek(); // find parent of node N - may be null (if N is root)
if (not(visited(N))) {
visited(N) <- true;
parent(N) <- P;
}
if (Path.contains(N)) return false; // LOOP detected
else {
Path.push(N);
if (Path.length() == nodeCount) return true;
// found Hamiltonian path
// iterate over each adjacent node except parent // SUCCESS if any child succeeds
for (each node M in adjacencyList(N) AND M != P)
if (HpSt(M, Stack Path)) return true; // found Hamiltonian path
// backtrack - FAILURE because all children failed
Path.pop();
return false;
}
}
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Ria Das
S KETCH OF THE ALGORITHM :
DESIGN AND IMPLEMENTATION
3.4 Computing conversion matrix that represents the sequence of folds

that a face goes through
Each reflection that a face goes through is of one of two types: against a non-vertical line
and against a vertical line. The affine transformation matrix used for these two cases is shown
below. The general formulas in the two cases differ only in 1st and 2nd rows of the 3rd column.
Table 1: Affine transformation matrices from fold line equations
Non-vertical line: y = mx + b
angle A = tan1 m
Non-vertical line: x = b
angle A = /2
Affinetransformation matrix:
cos 2A sin 2A
b sin 2A
sin 2A cos 2A b cos 2A + b

0
0
1
Affinetransformation matrix:
cos 2A sin 2A
2b
sin 2A cos 2A
0
0
0
1
If an N -vertex face, represented by a 3 N matrix, undergoes a sequence of folds denoted

by fold1 , fold2 , . . . , foldm and the corresponding affine transformation matrices are denoted by
M (fold1 ), M (fold2 ), . . . , M (foldm ), then the final conversion matrix M (foldfinal ) is computed as
M (foldfinal ) = M (foldm )M (foldm1 ) M (fold2 )M (fold1 ).
To compute the coordinates of the transformed face, the 3 3 conversion matrix, denoted
by M (foldfinal ), is multiplied with the 3 N matrix representing the sequence of vertices of the
original face. If the vertices of the face are (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x n , y n ), then the original face
matrix M (faceoriginal ) is
x 1 x 2 x N 1 x N
M (faceoriginal ) = y 1 y 2 y N 1 y N
1 1
1
1
The matrix M (facetransformed ) representing the vertices of the transformed face is then
M (facetransformed ) = M (foldfinal )M (faceoriginal )
3.5 Calculating the number of layers of paper at a certain point

To calculate the number of layers of paper at a given point in the crease pattern or folded
model, a slight modification of W. Randolph Franklins PNPoly algorithm is used [6].
The PNPoly algorithm employs a ray-casting method, looping over each edge of the face,
to determine whether a given point is strictly inside a polygon. In each iteration of the loop, if
the horizontal ray extending from the test point to the right intersects the edge of the polygon
(at a single point), then it toggles a Boolean variable which gets set to true if this is the n th time
we found such a case and n is odd.
In this light pattern visualization tool, a special requirement added to compensate for
slight inaccuracies in the values of tan1 m made while computing transformation matrices
71
Ria Das
was to consider a point that was on or very close to a side of a face as within the face. In order to
implement this, if a point is not strictly within a face, it is checked if the point is within a certain
ERROR distance from any edge of the face, for the very small ERROR value of 106 units (see
Figure 2). If the point is indeed inside this error margin, it is considered to be inside the face,
and an additional layer is added to the layer count.
Figure 2: Error checking in modified PNPoly algorithm if point is not inside

the polygon, check if it is within ERROR distance of each edge (blue area), where
ERROR = 106 units (diagram not to scale)
4 Results
The outputs generated by the computer program upon clicks of the show crease pattern
and show folded model buttons are shown below.
Figure 3: Five-and-four Tessellation

72
Ria Das
F UTURE W ORK
The extent of darkness of an area reflects the number of layers covering that area of the
paper. The exact number of layers covering a point may be found by entering the coordinates
of the point and clicking the Calculate number of layers button.
Figure 4: Square Twist
5 Future Work
In the future, I would like to improve the program in several ways, including:
using a dynamic programming based algorithm for finding Hamiltonian path [7] to reduce its worst case running time from O(n!) to O(n 2 2n )
showing generated transformation matrices
showing the layers visually with the face labels
enhancing the GUI: coordinate plane zoom in/out, changing the paper color
From a conceptual perspective, I would like to explore the following areas:
additional mathematics behind origami tessellations
3D origami tessellations
3D computer graphics and affine transformations
73
Ria Das
Bibliography
[1] Robert J. Lang. Origami: Science, mathematics, and technology.
http://www.langorigami.com/science/science.php
[2] Erik D. Demaine and Joseph ORourke. Geometric Folding Algorithms. Cambridge University Press, 2007.
[3] Eric Gjerde. Origami Tessellations: Awe-Inspiring Geometric Designs. Boca Raton, FL: CRC
Press, 2009.
[4] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction
to Algorithms. MIT Press, 2009.
[5] John Choate. Reflections.
http://www.zebragraph.com/Geometers_Corner_files/Reflection.pdf
[6] W. Randolph Franklin. Pnpoly point inclusion in polygon test.

http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html
[7] ShreevatsaR. Answer to What is the dynamic programming algorithm for finding a Hamiltonian cycle in a graph?, last modified September 7, 2009. http://stackoverflow.com/
a/1387823
74
William Hu explaining his project, Dysphemisms: An Overview.
Final Project Showcase at the MIT Stata Center
Wendy Matts project:

a cookbook with recipes
from around the world.
Understanding and Preventing

the Formation of Suicide Pacts
Among Younger Population in
both China and the United States
Tianyu (Christina) Lin
mentored by Evan Kuras
Abstract
A suicide pact is an agreement between a small group of people to commit suicide together.
Cases of suicide pacts were found through searching for news articles in Chinese searching engine Baidu and American searching engine Google. In the investigation, cases of suicide
pacts in China and the United States between those who were under 30 years old were gathered
(5 in the United States and 10 in China). Demographics, traditional or cybersuicide, method of
suicide pact formation, outcome of suicide, and geographic location of victims were taken note
of. Categorization and statistical analysis were used to process the data. For demographics, it
was found that victims of suicide pacts in China were generally in an age group from 20 to 25
and were older than victims in suicide pacts in the United States. There were no cybersuicide
cases in the United States. In fact, suicide pacts in the United States were formed between people who were familiar with each other in real life but never blood-kin; on the other hand, the
number of suicide pacts formed through QQ chat rooms was significantly higher than other
methods, possibly due to the herd mentality that instigated participants in the chatroom to
commit suicide. Since the victims in many suicide pacts in China formed suicide pacts under
the influence of other on the Internet, the number of cases with victims that backed out was significantly higher than those in the United States. Geography seemed to play an important role
at the same time. The provinces in which there were many suicide victims were provinces that
produced many immigrant workers, indicating the importance of improving migrant workers
mental health in preventing the formation of suicide pacts. In order to better understand suicide pacts in the younger population in both China and the United States, more cases would be
necessary, and such cases could be possibly obtained through research institutions and local
law enforcements.
Key Words: suicide pact, suicide prevention, cross-continental comparison
77
U NDERSTANDING AND P REVENTING THE F ORMATION OF S UICIDE PACTS Tianyu (Christina) Lin
1 Introduction
Suicide is the act of intentionally causing ones own death. Such events are tragic and
even more so when people take such actions together. There are two recognized categories of
group suicides. The first is mass suicide, in which a larger number of people kill themselves together for the same ideological reason. One example would be the mass suicide in Jonestown in
1978, when Jim Jones, the churchman who founded Peoples Temple, led his followers to commit suicide in Guyana, a nation in South Africa (Mass suicide at Jonestown). The other type
of group suicide is the suicide pact, which involves a small group of people whose motivations
are typically non-ideological. Those who commit suicide in the same subdistrict, a defined administration area, and within three days of each other are considered parties in a suicide pact
in England and Wales (Rajagopal).
There are two types of suicide pacts. One is traditional: the participants are already familiar with each other in real life. The other is a cybersuicide pact: the participants meet over
the Internet. What made suicide pacts known to the general public was two separate pacts that
killed nine individuals in October 2004. Participants in both suicide pacts were strangers who
met in suicide-related chatrooms online and then ended their lives together near Mount Fuji in
Japan (Huus).
Statistically, less than 1% of all suicides are suicide pacts. However, the prevention of suicide pacts can be crucial in saving lives, because usually one party of the pact is not as willing
to commit suicide and could be saved. In suicide pact cases where the instigator is more determined to end his or her life than the other victims of the suicide pact, the instigator is usually the
one who is deceased, and the survivor is usually the person who is compelled by the instigator
to participate in the pact (Rajagopal).
Past research has compared suicide pacts in Eastern and Western countries. In one particular study published in Journal of Clinical Psychiatry in 1985, 20 cases of suicide pacts in
Florida were compared to cases in England, Japan, and India. It was found that suicide pacts
occur more frequently in Japan. Lover pacts, spouse pacts, and friend pacts are more prevalent
in Japan, England, and India, respectively (International Comparisons).
Although states and cities in different cultures have been studied, countries, such as the
United States and China, are not systematically studied as a whole. Also, suicide pacts form an
integral, though not very well-known, part of the tragic deaths that occur each year in China.
The number of suicide pacts has been on the rise in both China and the United States in recent
years. In China, for instance, news articles concerning new suicide pact cases in Baidu have
risen from 66 (April to December 2014) to 194 (January to August 2015) (data gathered by the
author). Also, traditional suicide pacts have been between individuals within families or in
romantic relationships. However, with the rising popularity and prevalence of the Internet, now
cybersuicide pacts constitute a fair percent of all the suicide pact cases.
Although some research has been done to analyze the act of individual suicide, suicide
pacts remain as an area that is poorly investigated although their occurrence is increasing.
Also, despite some research on the cultural differences of suicide pacts, the research remains
limited. Most importantly, suicide pacts between young people, or those in their twenties or
younger, have not been studied, yet suicides are the most preventable form of death for teens.
Understanding and being able to prevent the formation of suicide pacts among the younger
78
M ETHODS
population can be effective in saving youth population in both countries.

In order to understand the formation of suicide pacts in young people in both China and
the United States, demographics, how the pacts were formed, outcome of the pacts, as well as
whether the suicide pact was a traditional or cybersuicide pact were factors that were taken
note of and analyzed in suicide pact cases. Finding ways to prevent the formation of suicide
pacts in two countriesChina and the United Statesis another part of the project.
2 Methods
2.1 Procedure
Cases of suicide pacts in China were gathered by searching the keyword ,
meaning suicide pact in Chinese, in the Baidu search engine, specifically under news. Cases
of suicide pacts in the United States were gathered by searching the keyword suicide pact in
the Google search engine, specifically under news. Only online news articles were used because
it was the venue in which the latest suicide pact cases were reported. Only cases that happened
in either China or the United States involving individuals in their twenties or younger were included in this investigation. All available cases within 5 years (20102015) were gathered since
the data collected in such time frame is current and relevant to the situation today. The following factors were studied in each case: demographics (age of the victims of suicide pact), whether
the pact was a traditional or cybersuicide pact, how the victims formed the suicide pacts, and
the outcome of the pacts. Also, for cases in China, the origin of victims was taken note of as well
as the place of suicide. For cases in the United States, only the pace of suicide was taken note of
because the origin information was largely unavailable.
2.2 Analysis
Based on notes, the total number of cases that fall under each category of each variable
for both China and the United States was compiled. For demographics, the categories are 15
and under, 16 to 20, 20 to 25, and 26 to 30. For the different types of suicide pact, there are two
categories-traditional suicide pacts and cybersuicide pacts. For the ways that victims formed
suicide pacts, the categories are classmates, family, friends, colleagues, couples, through QQ
groups, through online forums, and other. For outcomes of the suicide pacts, the categories
are all dead, one rescued, one backed out, all rescued, and all backed out. After obtaining the
total numbers, the percent of each category in each factor was calculated by dividing the total
number of each category by the total number of cases (5 for data on United States and 10 for
data on China) or total number of victims in the case of demographics.
Chi-square test was performed on all factors (Lancaster). The expected value was the average of all the categories, and the null hypothesis was that the cases in factor are evenly distributed throughout all the categories.
79
3 Results
3.1 Demographics
Figure 1: The percent of participants (sample size: 10 for the United States and
26 for China) of the suicide pact in each age group in both China and the United
States.
Although the victims in the United States are spread relatively evenly through the first three
age groups, none were in the oldest age group. There are significant differences between the
percent of people in the suicide pacts in China and the United States in the first two age groups,
which are considred as stages for teenagers and young adults in both countries. However, the
percent of victims who fall under the category of 21 to 25 is on average fifty percent higher
than the other three categories. Age of participants in the suicide act does not affect percent
of participants in the United States (p > 5%) but does affect percent of participants in China
(p < 5%).
80
R ESULTS
3.2 Incidents of traditional versus cybersuicide pacts

None of the suicide pacts in the United States are cybersuicides. More than half of the suicide pacts in China are cybersuicide pacts. Two cases that are worth noting in China. One is a
pact that was not formed via the Internet, but the participants in the pact are barely acquainted.
Another is one in which there is a combination of traditional and cybersuicide pacts. The category of the suicide act does affect percent of participants in the United States (p < 5%) but does
not affect percent of participants in China (p > 5%).
Figure 2: The percent of suicide pacts (sample size: 5 for the United States and
10 for China) in both China and the United States that are either traditional
suicide pacts or cybersuicide pacts.
3.3 Method used to form suicide pact

From Figure 2, it is already known that gathered suicide pacts formed in the United States
are all traditional. The specific breakdown of the formation show that there are equal numbers
of pacts that are formed between friends and couples. For suicides in China, however, there was
no pact that was formed between couples. The numbers of pacts formed between classmates,
family, and friends are even. The highest number of suicide pacts are formed through QQ chat
groups, chat groups formed on a popular web-based instant messenger in China. Some were
formed through suicide forum. The two methods in others were roommates who did not really
know each other and online but specific method was not mentioned. The method used to form
suicide pact does affect percent of participants in the United States (p < 5%) but does affect
percent of participants in China (p < 5%).
81
Figure 3: Percent of pacts (sample size: 5 for the United States and 10 for China)
that used different methods to form suicide pacts in both China and the United
States.
3.4 Outcome of suicide pact
Figure 4: Percent of different outcomes of the suicide pacts in both China and
the United States (sample size: 5 for the United States and 10 for China).
The mortality of the suicide pacts in the United States was significantly higher than that
in China. In fact, no one in the suicide pacts in the United States backed out. The percent of
situations in which one party in the suicide pact backed out was higher in China than the United
States. The number of rescues in China was also higher than the United States. The condition
of all participants after the suicide pact does affect percent of participants in the United States
(p < 5%) but does affect percent of participants in China (p < 5%).
82
R ESULTS
3.5 Geographic locations of suicide victims
Figure 5: The number of victims from different provinces in China. Provinces

were used to geographically distinguish cases because they are conventions
that are economically and culturally different. The location is not where the
location where the victims committed suicide but where the victims were originally from. The highest number of victims is in Sichuan, Shanxi, Shandong,
and Hunan. The cases were all in the general area of south-east of China. The
provinces with the most cases surround the other provinces that had just individuals of cases.
Figure 6: The number of victims that committed suicide in each province.

The provinces with the highest number of suicidesShanxi, Shandong, and
Zhejiangwere coastal provinces.
83
Figure 7: States in which the suicide pacts took place. There were no suicides in
the central area of the United States. Other than the suicide pact in Minnesota,
all the pacts were in coastal areas (East coast and West coast)
4 Results and Analysis

4.1 Demographics
The victims of suicide in China that were in the 21 to 25 age group were 50% more than
the victims in any other age groupe. It is important to note that 21 to 25 is around the time
that students in China graduate from college and start job-hunting; the high number of suicide
pacts in this age group indicates that the intense pressure of finding jobs in China has triggered
the formation of many suicide pacts, indicating the necessity of counseling effort for newly
graduated Chinese students. In the United States, on the other hand, the victims of suicide
pacts were younger than the victims in China.There were no victims of suicide pact in the age
group of 25 to 30. At the same time, Chi-square rejected null hypothesis, which shows that age
does not affect the percent of victims in suicide pacts in the United States.
4.2 Incidents of traditional versus cybersuicide pacts

The lack of cybersuicide in the United States suggested that those in the United States
prefered to die with someone he or she loves and is close to. However, the higher number of cybersuicide pacts in China demonstrated that some victims in suicide pacts in China just wished
to die while in the company of someone, no matter who his or her companion may be. The fact
that two of the traditional suicide pact cases also involved strangers or people who were barely
acquaintances further supported this conclusion.
4.3 Method used to form suicide pact

The highest percent of cases in the United States between classmates, friends, couples,
which were all people that were close to one another in real life. However, interestingly, none
of the cases involved people who were related by blood. This absence of suicide pact between
84
R ESULTS AND A NALYSIS
blood-kins may suggest that many of the suicide victims were aware of their suicides impact on
their families. Therefore, one possible method to persuade people from joining suicide pacts is
through the comfort and persuasion of the families.
In China, The highest percent method used to form suicide pact is QQ groups. First of all
QQ groups are extremely prevalent in the China and are not strictly controlled by Tencent. Inc,
the company that provides the service of the QQ app, thus there were suicide pact chatrooms
available (ji zhe wo de) as the easy way to find someone to accompany one to commit suicide.
According to a Chinese nationally recognized counselor, the pessimistic view of the world is
very easily spread through QQ groups (ji zhe wo de). Although committing suicide is an act
that requires a lot of determination, when even one person expresses such negative thoughts
or when a group of people discuss methods of committing suicide in the chatroom, the onlookers desire to commit suicide can grow stronger and can lead to eventual execution of the
suicide pact (ji zhe wo de). The herd mentality can also explain the prevalence of the formation
of suicide pacts via QQ groups. In a study regarding the herd mentality of college students, it
was noted that the herd mentality allows one to feel less guilty about the wrongful thing that
one has done, which in this case would be committing suicide (da xue sheng cong).
4.4 Outcome of suicide pact

The fact that there are more rescues in China is either caused by coincidence or the victims
usually choose places that are easier to find and methods that are reversible. Such a high possibility of termination of suicide pact mid way correspond to the fact that many suicide pacts
in China are formed with strangers and QQ groups thus they agreed to be in the suicide pact
because of factors such as herd mentality.
4.5 Geographic locations of suicide victims

In China, two of the provinces where the highest number of suicide pact victims were originally fromShanxi and Sichuanas well as other provinces, such as Jiangxi and Hubei, are
provinces that historically produce a lot of immigrant workers. Thus, the high number of suicide pact victims from these victims could be caused by migrant workers. In fact, in one of
the cases, three victims in a suicide pact in China were specifically listed as migrant workers in
the news article. This conjecture is supported by the fact that Zhejiang, one of the wealthiest
provinces in China (2014 nian zhong guo), became the province where the highest number of
suicide victims decided to commit suicide, suggesting that some migrant workers from other
provinces and work in Zhejiang committed suicide in where they work. Also, there had been research done on the mental health of immigrant workers, and it turned out that the state of mental health of migrant workers is worrisome due to financial pressure and the constant stress of
being from a different province (da gong zu xin). In the research, it was found that one in four
migrant worker has some degree of mental health issue. Also, 98.76% of the migrant workers
interviewed for the investigation are not happy with their life. Another mental health problem
that migrant workers created in their province of origin was the the children that the left behind.
In fact, according to a recent study, children who are left-behind were 18% more likely to have
some form of mental disorder than those who are not (da xue sheng cong).
85
In the United States, 80% of the victims are from the coastal states (either east coast or
west coast) and no victims from the central areas of the country. Trying to analyze by looking at
wealth of each state did not yield any noticeable patterns.
4.6 Problems encountered during the investigation

One major problem encountered during the investigation was the scarcity of cases reported in the news, especially in the United States. All 35 pages of Google searching results
were looked through and only 5 cases were found. In order to gather more cases in the future,
it is possible to contact local law enforcement or perhaps research facilities for suicide.
5 Conclusions
In conclusion, through data analysis of suicide pacts, several patterns were found regarding suicide pacts in younger people in China and the United States. For demographics, there
were many victims of suicide pacts in the age group of 20 to 25, suggesting the pressure of finding jobs after graduating college in China. There were more cybersuicide pacts in China and
many of these pacts were formed via QQ chatrooms, suggesting the herd mentality that could
cause victims to commit suicide when they do not fully want to. The fact that more victims
of suicides backed out of suicide pacts further support the proposition that many suicide pact
victims in China were not fully ready to commit suicide. In the United States, although many
people chose to form traditional suicide pacts with people they knew, the pacts were never between close kin. In terms of geographic locations, it is possible that migrant workers played a
big role in the existence of suicide pacts and counseling these migrant workers can play an important role in stopping the formation of future suicide pacts. In the United States, most of the
victims were from the east coast or the west coast. However, in order to more fully investigate
the formation of suicide pacts in China and the United States, a bigger sample size would be
necessary.
6 Acknowledgements
I would like to thank Evan Kuras, my mentor, who supported my idea from its initiation
and provided me with helpful feedback and ideas. He also had the right combination of compassion and humor that helped me through this serious yet sometimes mood-crushing investigation, and I could not ask for a better mentor. I would also like to thank MIT Junction for
giving me the opportunity to work on a project that I wanted to investigate for a long time and
the platform on which I can present my work to the world.
86
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87
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two females who met over the internet formed suicide pact
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89
Humanity throughout the Holocaust

Vineetha Yadlapalli
mentored by Elizabeth Berg
What qualities make one human? Does being human mean to always hold ones beliefs?
Does being human mean to abandon all in order to achieve survival? The meaning of being
human is debatable, and the definition depends on context and opinion. The humanity of people always changed throughout history, but one time, humanitys beliefs were put to the test:
during the Holocaust. It was a period of genocide led by German Nazi leader Adolf Hitler, who
targeted many groups of people he considered worthless. The victims included Jews, Gypsies,
the physically and mentally disabled, homosexuals, and many other groups of people, whether
targeted because of religion, ethnicity or political beliefs, resulting in 11 million deaths. These
deaths were the murders of innocent people (when excluding the criminals). In the time of the
Holocaust, the victims had to cope with the cruel environment of a concentration camp governed by SS officials. People died due to starvation, exhaustion from overworking, hangings,
gas chambers, and crematoriums. This situation forced each victim to face the struggles and
deal with that everyday life or to succumb to the escape of death. Thus, the humanity in them
changed.
While in the camps, people questioned parts of their humanity, such as why they were
placed in the concentration camps, the absence of Gods aid, and more. These questions need
to be answered in order to evaluate this difficult time in history. This paper will address various
aspects of humanity present or absent in camp life, and will answer questions like: How does
the peoples humanity change in a situation where it does not seem to exist? How harsh does
dehumanization get and what does it do? How was the Holocaust an example of how people
are affected mentally by the actions and beliefs of people around them? These questions will
be explored by examining memoirs written by Holocaust victims and survivors: Night by Elie
Wiesel, The Diary of a Young Girl by Anne Frank, and Survival in Auschwitz by Primo Levi.
Night, written years after the Holocaust by a Romanian Jew named Elie Wiesel, reflects
upon his struggle as he travels through various concentration camps at the age of fifteen with
his father. The Diary of a Young Girl, written by a victim who was a thirteen-year-old girl named
Anne Frank, gives a look into hiding in the upper parts of a building from discovery during the
Holocaust. Lastly, Survival in Auschwitz was written after the Holocaust ended by an Italian
chemist, Primo Levi. He was twenty five when he entered a concentration camp, and he spends
about a year in camps before his liberation. He explains his experience and thoughts in the
book.
Humanity has two different, but complementary definitions: being kind and compassion91
Vineetha Yadlapalli
H UMANITY THROUGHOUT THE H OLOCAUST
ate towards others, and the state of being human. Dehumanization is the psychological process
of making someone seem less than human, and unworthy of being treated as one, which results
in the victim themselves believing they arent human either. This certain concept will be delved
into later.
Different types of people who existed in the camps viewed humanity differently there. On
a literal level, the types of people included the oppressed and the oppressors. The oppressed
were victims like Jews, Poles, and prisoners of war, while the oppressors included the officials
that ran the genocide, ranging from Hitler and the Nazis, to the SS officials and Germans. How
the oppressors could orchestrate such a horrific event, and how they could hurt people without
regret or feeling is still something to be thought about today.
The Nazis and SS officials and German civilians, the main oppressors of the victims besides
Hitler, varied in positions. The SS were the people that controlled the German police forces and
the concentration camp system (USHMM para. 2) and the Nazis were the people of Hitlers
political party called the National Socialist German Workers Party . . . or Nazi party . . .
(USHMM para. 4). The German civilians were the people living in Germany under Hitlers rule.
The SS officials would deliver whipping blows to the working victims in the camp and order
them around, exhausting them to the final breath. But, they showed momentary hesitation in
rare scenarios, like when it came to painful situations and the victims impacted them in some
way. For example, in Night, the camp people are forced to watch hangings to enforce discipline.
One hanging that Wiesel sees is different from the others he has seen before. It is the hanging
of a young boy. The boy is tied in the gallows along with two older men while the camp people
stare in horror. Wiesel notices the SS seemed more preoccupied, more worried, than usual.
To hang a child . . . was not a small matter . . . This time, the Lagerkapo refused to act
as executioner (Wiesel 64). Kapo is a German term for a leader of a group of people in the
concentration camp. It could be a Nazi, SS, or a promoted prisoner. Despite the SS officials
cruel attitude, they could feel guilt for their actions. This is important because it was proof they
had sympathy. In someone that appears to be inhumane, refusing to hang an individual or hurt
someone contradicts their supposed beliefs.
Also, earlier in Night, a Polish official tells Wiesel, Dont lose hope . . . Have faith in life .
. . Hell does not last forever . . . Help each other. That is the only way to survive. (Wiesel 41)
They could also provide hope, something unexpected for these people. I think that one thing
to infer about the SS officers is they were psychologically affected by the Holocaust as they had
murdered people, while in reality they felt guilt and shame each time for their actions. They
may have not been fully compassionless, from showing pity on the victims or reluctance to
carry out tasks, showing that they may have taken on a certain look on humanity with prejudice
and racial purity, but really felt differently, despite being the oppressors. Though they cannot
be excused for their atrocious acts, it does not mean they all lacked compassion.
In his experience, Primo Levi states there were two kinds of people that exist: the saved
and the drowned. Other pairs of opposites . . . are considerably less distinct . . . this division is much less evident in ordinary life; for there it rarely happens that a man loses himself
(Levi 100). The saved were the people who created contacts with others, gained knowledge and
respect, and had a good way to survive. For example, the saved could mean the Jews or Poles
who survived by adapting, like learning camp rules or rationing their food supply. The saved
could even be the non-Jewish civilians, Nazi party members, and soldiers, because they were
not targeted. On the contrary, the drowned were the ones who did not find a way to live in the
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camp quickly enough and were doomed to die inevitably. They were labeled as the muselmann:
the weak, the inept, those doomed to selection. (Levi 100) The drowned were the victims, and
could be Jews, Poles, and more. Another way to categorize people would be the Holocaust
camp survivors and the victims. Yet, the term survivor would apply to people that physically
lived through the Holocaust, as most people did not survive inside. They lost who they used to
be before they were taken to the camps. Their past identities slipped away.
Many types of people were targeted, and when they were all placed together in a concentration camp, it was challenging to understand others linguistically and socially to survive. The
camp victims had to learn to understand the German commands the SS gave, and they had to
adjust to other cultures ways, whether the people they met be Italian or Romanian or more.
This constant reminder to survive led to the regression in the humanity of people there.
What exactly was lost in humanity because of the environment of inhumanity? Much was,
because to survive in the face of driving necessity and physical disabilities many social habits
and instincts [were] reduced to silence (Levi 93). Civilization enforces social habits to live by,
and even though civilization is not present in the concentration camp, social habits vanish due
to the need to survive, which surpasses the importance of social habits and humane respect.
One aspect of humanity in civilization that is always encouraged but is discarded here is morality. Morality is a system where right and wrong or good and bad is distinct. Being moral means
to uphold good and right behavior and living by the rules one sets. Yet in the camps, there were
unclear meanings of good and bad or just and unjust in there. For example, theft in Buna,
punished by the civil direction, is authorized and encouraged by the SS; theft in camp, severely
repressed by the SS . . . (Levi 92). The ordinary moral world could not survive in the camp because of the instinct of survival. Levi adapts to this lifestyle free of typical rules, and says that .
. . if I find a spoon lying around, a piece of string, anything which I can acquire without danger
of punishment, I pocket them and consider them mine by full right (Levi 33). This enforces the
necessity of giving up morals to live in a situation where one could either save himself or drown.
Because as Levi says, Survival without renunciation of any part of ones own moral world . . .
was conceded only to very few superior individuals . . . (Levi 99). Their humanity changes because their moral beliefs must be shed to live. But what is unacceptable is the mistreatment of
people, especially the women. Anne Frank writes in her diary about how she hears that Men,
women, and children all sleep together. One hears of frightful immorality because of this; and
a lot of the women, and even girls, who stay there any length of the time of expecting babies
(Frank 50). The camp breeds an environment of immorality, and raping someone would be
to invade ones privacy and dignity, something they have not lost, unlike everything else they
have. Part of being oneself is to have independence and the ability to control themselves. To
take that away by this mistreatment is taking away ones humanity. In addition, to rape and
have no shame doing so shows the uncivilized and immoral attitudes the people adopted; their
humanity had altered due to the land of inhumanity.
On another note, savagery erupts in the conditions they live in, as everyone regresses from
civilized to savage, turning on each other. For example, a full-scale battle for food bursts between the Jews on a train that Wiesel is traveling on. Some German civilians decide to toss
bread onto the train where the packed, starving Jews watch. In the wagon where the bread
had landed, a battle had ensued. Men were hurling themselves against each other, trampling,
tearing at and mauling each other. Beasts of prey unleashed, animal hate in their eyes . . .
and the spectators observed these emaciated creatures ready to kill for a crust of bread (Wiesel
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Vineetha Yadlapalli
101). The Jews are all fighting like uncivilized creatures in order to get food and survive, uncivilized here meaning without typical rules. Food was not given by the SS officials before the train
started, showing how uncompassionate the SS could be by not helping the victims, causing the
humanity of the Jews to diminish too like when they fight for example. In addition, family loyalty and connections cease to matter when it comes to survival in its worst. Everyone forgets
who they are fighting with, battling just to get food. Among them are a father and his son, Meir.
During the bread fight, Meirs father is able to acquire a piece of bread, and he intends to share
with Meir, when he is killed by his own son. The father cries Meir! My little Meir! Dont you
recognize me . . . Youre killing your father . . . I have bread . . . for you too . . . for you too. . . .
(Wiesel 101). Meir kills him for bread, and moments later, he dies as well when two other men
kill him, therefore killing his father for nothing. Here, Meir betrays his father and goes against
family loyalty to gain food for himself, showing that family connections do not matter anymore,
and one is mostly on their own in the situation.
Later, Wiesel discovers the difficulty of resisting the urge to abandon. He promises himself
to have the strength to not betray his father. However, after the train, he feels guilt whenever
he thinks he would rather keep food he gives to his father for himself, so he can survive. In
the beginning of the novella, Wiesel cares deeply for his father. He still does now, but because
of survival, Wiesel acquires a selfish side. For example, he does not cry when his father dies,
making him ashamed of his selfishness. Selfishness and survival win out over family, showing
how everything important from before must go to live. It is hard to keep civil aspects from the
past with one when the situation demands the loss of all that to survive.
Despite the loss of morality, civility, and savagery, some remained kind. Some camp inmates help out others by encouragement. For example, Wiesel tells how he is beaten up for no
reason by a camp official and lies in the corner, trying to recover from the blows, when he feels
a cold hand wiping the blood from my forehead. It was the French girl. She . . . slipped me a
crust of bread . . . she said Bite your lips, little brother . . . Dont cry. . . (Wiesel 53). This
demonstrates the seldom acts of kindness inmates did, despite all the cruelty they face. Some
remember to keep their kindness with them, which can go a long way. Levi also encounters
understanding when he converses with another person in the camp. He explains the situation
of his Italian mother in hiding, and says how the listener timidly embraces me . . . I have not
forgotten his serious and gentle face of a child, which welcomed me on the threshold of the
house of the dead (Levi 27). Levi can still recall this moment of sympathy, showing the power
kindness can have. These seldom outliers have great impact on others.
Levi also receives help from a German civilian on the other side of the barbed wire, saying
himself that without the Germans assistance, he would not have survived. Civilians did have
the kindness to help. Also, in Anne Franks diary, Frank writes of the German civilians she encounters in public. Jews could not drive, so they had to walk. She writes how You could see
by [the Germans] faces how sorry they were they couldnt offer us a lift; the gaudy yellow star
spoke for itself (Frank 27). The Jews were required to wear badges to be identified. These
stars were a psychological [tactic] aimed at isolating and dehumanizing the Jews . . . directly
marking them as being different . . . to everyone else. It allowed for the easier facilitation of
their separation from society . . . (Arsenault para. 3). The star had power over the Jews and
Germans: it degraded the Jews, and resulted in no help from Germans, even if they wanted to.
They could not in fear of discovery. Other Germans though, like Levis helper, gave assistance
though it was punishable. Only a few of the kind Germans had the courage to do the right thing
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Vineetha Yadlapalli
even if it disobeyed the system.

People do have the kindness to act and help others, one of the most vital traits to possess,
because kindness gives a reason to others to still believe in the good of humanity. Everyone
must help each other to survive together, although survival there did not occur mainly on that
trait: cleverness, wit, and connections were the way to pass selections, endure the days work,
bargain for materials and more. Because survival to them was more important than kindness,
Jews fought against other Jews for things. Though they should have been allies in the situation,
the kindness and respect for one another diminished, being replaced with natural survival instincts. The SS were the real issue. This is dangerous because if the victims turn on each other,
the Germans in charge have won, and they fulfill the twisted SS perspective of the Jews being
less than human.
The Nazis and SS viewed Jews and other targeted groups with disgust and hate; the victims
were not human to them. The SS made it their goal to enforce dehumanization on the victims,
because they believed they were not human to begin with. There are levels to this process,
and with each, the human is degraded further. In the Holocaust, dehumanization reached the
ultimate low, as victims lost who they used to be in the process. One thing it did was giving
no respect for the camp victims there. In Night, Wiesels father is suffering a colic attack, so
he got up and asked politely, in German, Excuse me . . . Could you tell me where the toilets
are located? The Gypsy stared at him . . . as if he wished to ascertain that the person . . .
was actually . . . a human being . . . he slapped [him] with such force that he fell down .
. . (Wiesel 39). Though Wiesels father addresses the guard in the language which the guard
uses and asks with respect, he is treated like he is not worthy of fulfilling basic human needs
and is instead treated with violence in return. The victims lose respect in the eyes of the SS,
and because of this, they build up fear of mistreatment and feel the pain of the incident. The
scar for Wiesels father to bear is physical and mental. Respect no longer exists for the people.
Something notable is that in normal circumstances, being slapped in such a way would bring
shame upon the victim of such violence. Yet here, Wiesels father shows no humiliation to his
son when he comes crawling back. This shows that dignity is not important here, as everyone
is equal and less than human. In addition, part of their identity is stolen when they are forced
to strip and wear new uniforms and shave their hair off. The new uniforms are thrown at the
people carelessly, further showing the lack of respect for them. Wiesel recounts on this moment
by saying Our clothes were to be thrown on the floor at the back of the barrack . . . For us
it meant true equality: nakedness . . . (Wiesel 35). Nakedness means true equality to them
because any past societal position associated with their clothing is stripped away, leaving them
equal. Clothes give one uniqueness and freedom of expression. Clothes can also represent
status in a class or society. But, the removal of their clothes symbolizes the loss of expression
and belonging in a place of civilized society. They all feel the same, as the hierarchy of society
from the past no longer exists here. But, this can make them feel distant from a familiar humanmade system that everyone belongs to, and that can take away from them.
One major aspect that is taken away from all the victims concerning identity is their name,
which can affect one greatly. Each person received a tattooed number on their arm, symbolizing the permanent reminder they are not worthy of being called by their real name, and they
are nothing but a number, even on them post-Holocaust, just like the SS and Nazis would have
liked. Some people even forget their real name while there in the camp. Levi writes of a fellow
person that worked with him during the day for some time, saying He is Null Achtzehn. He is
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not called anything except that, Zero Eighteen . . . everyone was aware that only man is worthy
of a name, and that Null Achtzehn is no longer a man, I think that even he has forgotten his
name . . . (Levi 41). Ones identity defines who they are, and for them to lose who they are because of others is the worst of dehumanization, because identity gives one purpose to live, to be,
to be human. This shows that the SS symbolically took away their actual names, which enforced
dehumanization further, because when one no longer has a name, they cease to be a human.
Also, the numbers each camp victim was branded with were prison-like, which would imply
guilt for wrongdoing. However, the irony is most of the people in the concentration camps were
innocent. This criminal branding could have changed the way they viewed themselves (as good
or bad, human or not), which makes dehumanization strike harder.
Another thing Holocaust victims faced because of dehumanization was the loss of their
possessions and themselves. When people entered the concentration camps, they were separated from family, causing one to lose hope and worry constantly. It wounded them to know
their beloved ones were dead or in pain, and they could not do anything to help them, causing
psychological effects on their minds through pain and guilt. When entering, they lost clothing
from their past lives and all their valuables, such as gold, only for the SS officials gratification
by gaining wealth. For example, Wiesel is forced to give up his tooths gold crown to a doctor.
Wiesel escapes the encounter . However, a fellow inmate demands he give the gold crown and
a ration of bread to him afterwards. Wiesel exclaims What? My ration of bread so that you can
have my crown? (Wiesel 56). This interaction shows the injustice that occurred. Having possessions gives a victim a sense of ownership as well as memories, but when it is stripped away,
they can feel empty without it, since possessions make one feel important of owning something
and being human by having that right to own it.
Holocaust victims also lost who they used to be by no longer holding individual identity.
After the Holocaust ends, Wiesel decides to look into a mirror for the first time since he lived in
the ghettos long ago. Wiesel says I had not seen myself since the ghetto. From the depths of
the mirror, a corpse was contemplating me. The look in his eyes as he gazed at me has never
left me (Wiesel 115). Wiesel saw a corpse, not the boy he used to be. He could not recognize
himself, showing how people would lose themselves because of dehumanization, so they were
degraded and eventually lost themselves.
A joy of being human is the freedom. Yet, in the Holocaust, freedom is taken away. For one,
all the victims were forced into an unsanitary, confined concentration camp, surrounded by
guards. They also lost the freedom of expression by stripping their old clothes. In addition, their
freedom was restricted when they lost their rights and had rules. According to Frank, she writes
of how Jews, including her, must wear a yellow star . . . Jews are banned from trans[portation]
and are forbidden to drive . . . Youre scared to do anything because it may be forbidden.
Our freedom was strictly limited . . . (Frank 14). Jews were denied from basic property and
rights, and were singled out like wrongdoers when they wore stars. Freedoms absence allowed
dehumanization to occur further.
The SS treated the victims like animals. The Jews and other sufferers were beaten frequently, whether for no reason or for punishment. They were even beaten by fellow Jew inmates. Wiesel says that Dozens of inmates were there to receive us, sticks in hand, striking
anywhere . . . Strip! Hurry up! Raus! (Wiesel 35). The fact that Jews beat fellow Jews shows
they were emotionally stripped, and without emotion, people would act like animals. They
were treated as animals too as they were caged by barbed wire in the small camp, and were
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Vineetha Yadlapalli
called items, not people. For example, the SS addressed them as items. Levi says how the Germans held the roll-call. At the end, the officer asked Wieviel Stuck? The corporal . . . replied
that there were six hundred and fifty pieces. . . (Levi 10). To the SS, the victims were to be used
to work or if unable, worthless and to die. He will be a hollow man . . . whose life and death
can be lightly decided with no sense of human affinity . . . on the basis of pure judgement of
utility . . . (Levi 22). They were treated like objects for work and like animals, not people.
The Jews and the other Holocaust targets reached the limit of dehumanization. Even Levi
acknowledges it, as he says we became aware that our language lacks words to express this
offence, the demolition of a man . . . We had reached the bottom . . . no human condition
is more miserable than this, nor could it be conceivably be so. Nothing belongs to us . . .
They will even take away our name . . . (Levi 22). Unfortunately, they had reached the limit,
losing everything. They suffered humiliation, isolation from society, denial of rights, and were
separated from possessions and their beloved.
Something vital to the Jews that they lost was religion. Throughout the Holocaust, as people kept praying for the end of their suffering, they gradually lost their faith in God because of
the lack of response to their prayers. Before the Holocaust, religion was everything to Wiesel.
But, during the stay in the concentration camp, he goes through so much he loses that faith. As
he sees the hanging of a child in the gallows, he hears someone say For Gods sake, where is
God? And from within me, I heard a voice answer: Where he is? This is where - hanging here
from this gallows. . . (Wiesel 65). Religion gives hope that somehow, everything will turn out
alright. Wiesel says that God died, which shows the point where his faith is shattered.
There could be three possible explanations to why God did not respond to the Jews. One
is that he did not care for his devotees, so he did not help. This could cause anger as well as loss
of faith. But, the second possible explanation is more dangerous: God simply could not help.
This would mean that the SS had more power than God himself, terrifying to the religious. This
shook their faith and brought the belief they would suffer a punishment by the SS more powerful
than Gods force. Also, the Jews may have thought God did not help because they deserved it: a
twisted, yet possible reason the devotees could have conjured up in their desperation to try to
explain their suffering. If God did not care and thought they did not deserve his help, they would
believe they were worthless and less than human, enforcing dehumanization more. Because
they lost their faith, many were shaken severely by this revelation. They lost a massive aspect of
their past lives and culture, so they lose hope and succumb to dehumanization.
The aspects of the victims previous lives came to use for camp survival. For example, they
made an economy called the Market, where chaos and noise occurred as people bargained for
items and food from many different types of people. Levi describes the Market as very active.
Although every exchange (in fact, every form of possession) is explicitly forbidden . . . the
northeast corner of the Lager . . . is permanently occupied by a tumultuous throng . . . (Levi
84). They were able to make an economy with currencies and bargaining. The system is similar
to the outside societys economy. Another aspect that helped was their past lifes profession.
Some would use their skills to their advantage. One person who does so is Engineer Kardos, a
man who moves around the bunks, tending to wounded feet . . . That is his trade. There is
no one who will not willingly renounce a slice of bread . . . in this manner, honestly, engineer
Kardos solves the problem of living (Levi 59). Many different skilled people were able to put
their knowledge to use so they could survive, which shows the importance of aspects of their
past lives in aiding their survival in the camp.
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Vineetha Yadlapalli
The past aspects also reminded them of what they did not have. The victims dreamt of
food and the desire to consume it, which was always interrupted by the reality of having to
consume few small portions of stale bread and watery soup instead. This aspect of recalling
food can lead to ones downfall, as they could go mad by recalling such fantasies.
Throughout this paper, humanity in the Holocaust was explored, from the change of peoples humanity, which resulted in the loss of morality, family loyalty, and kindness, and savagery
spreading, to the effects of dehumanization on people, and how cruel it got (identity loss, possessions, respect, themselves, animal regression, religious faith). Three books shed light with
first-hand accounts of the authors experiences: Night by Elie Wiesel, The Diary of a Young Girl
by Anne Frank, and Survival in Auschwitz by Primo Levi. The Holocaust was a dark and confusing time for many people, victims and non-victims alike. This paper has hopefully explained
various concepts from that time.
Despite it all, Germans could not take away the victims memories, as long as they held on
to them and kept remembering. As Levi says, We know where we come from. The memories
of the world outside crowd our sleeping and our waking hours; we become aware, with amazement, that we have forgotten nothing; every memory evoked rises in front of us painfully clear
(Levi 55). These memories were a blessing to recall, yet torture, because to remember meant to
regret. They had memories of their past lives and the Holocaust with them.
Holocaust memories bring up the importance of the Holocaust. What is the most important about this event is the necessity to remember that it happened and to prevent such a
tragedy from occurring again. Human hate should cease and instead all should respect each
other for who they are, by supporting one another. Also, it is crucial to acknowledge and understand the Holocaust, but to not empathize with the victims. The reason for this is that one in
the modern world is incapable to empathize. One cannot compare their ordinary experiences
to something else on a whole different level. By trying to empathize, one will minimize the situations severity. To reduce the level of what the event really was is the worse one could do to
victims. By avoiding empathy and discouraging hate, such an atrocity can remain an anomaly.
The world must remember the Holocaust, because like Wiesel said, if everyone forgets, we are
guilty, we are accomplices (Wiesel 118).
Works Cited
Frank, Anne. The Diary of a Young Girl. New York: Doubleday, 2001. Print.
Levi, Primo. Survival in Auschwitz: If This is a Man. New York: bnpublishing, 2008. Print.
Wiesel, Elie. Night. New York: Hill and Wang, 2006. Print.
SS. United States Holocaust Memorial Museum. United States Holocaust Memorial Museum, 20 June 2014. Web. 18 Aug 2015. http://www.ushmm.org/wlc/en/article.php?
ModuleId=10007400
Arsenault, Joshua. Holocaust Badges.Holocaust Memorial Center Zekelman Family Campus. Holocaust Memorial Center, n.d. Web. 18 Aug 2015.
http://www.holocaustcenter.org/holocaust-badges
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About the MIT Educational Studies Program

The MIT Educational Studies Program (ESP) is a student group at the Massachusetts Institute of Technology that recruits MIT students and community members to teach classes for
high school and middle school students from the Boston area and beyond.
We are all MIT undergraduates, graduate students, alumni, and community members who
volunteer for ESP out of a love for teaching, organizing, and service. We offer a variety of
programs that range in size, scope, and style from the single-weekend Splash and Spark,
to the longer HSSP and Junction. More information about all of our programs is available at
esp.mit.edu. If you have additional questions, please email us at esp@mit.edu.
The MIT
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Editing by Herng Yi Cheng

Inquiry Is Elementary: Science Experiments for Young Students . . . . . . . . . . . . . . .

The Magical Number: e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

A BOUT THE MIT E DUCATIONAL S TUDIES P ROGRAM . . . . . . . . . . . . . . . . . 101

What is the Junction Journal?

Junction Projects and Participants

Making Music From Scratch

Inquiry is Elementary: Science

For Lack: On Ecopoetry

Racism and Ethnic Prejudice

J UNCTION P ROJECTS AND PARTICIPANTS

Understanding and Preventing the Formation of

How Do You Define Success?

Malaria and Climate Change

Bobcat Conservation of Western Massachusetts

Boy Band Breakups

Music and Social Change: Transcending Conflict

The Benefits of Detroit

The Magical Number: e

Hackenbush and its Variations

DNA Sequence Analysis Program

Bridging the Gap Between Empirical and

J UNCTION P ROJECTS AND PARTICIPANTS

Anavrin: Book One

Hackenbush and its Variations

Figure 1: Legal Hackenbush starting position

H ACKENBUSH AND ITS VARIATIONS

Figure 2: Illegal Hackenbush starting position

Figure 3: The starting position for the impartial Hackenbush game 3

H ACKENBUSH AND ITS VARIATIONS

4.2 Winning Nim

Then, the Nim sum of m and n is defined as

QUANTIFYING GENERALIZED H ACKENBUSH

4.4 Analogy to Hackenbush

Figure 4: The Hackenbush game consisting of 1, 3, 5, and 7 is equivalent

5 Quantifying generalized Hackenbush

H ACKENBUSH AND ITS VARIATIONS

1 Ice Cube Challenge!

Adapted from http://inquiryiselementary.blogspot.com/p/about.html

1.1 Background for Implementation

1.2 Background for Students

2.1 Background for Implementation

2.2 Background for Students

3.1 Background for Implementation

3.2 Background for Students

The Magical Number: e

2 Compound Interest and Natural Logarithms

3 Derivative of e x and Proving Eulers Identity

Alex Yu (left) working on his project, Making Music from Scratch,

Charlie Kip (left) and Keren Shao

Applications of Knot Theory to DNA

1 Introductory Knot Theory

A PPLICATIONS OF K NOT T HEORY TO DNA

Figure 2: The 3 Reidemeister Moves (Adapted from Wikimedia Commons)

Figure 4: Computing Linking Number (Adapted from the Wikimedia Commons)

A PPLICATIONS OF K NOT T HEORY TO DNA

S UPERCOILING AND T OPOISOMERASES

Figure 8: Alternate DNA Structures (Adapted from Biophysical Studies on DNA

3 Supercoiling and Topoisomerases

A PPLICATIONS OF K NOT T HEORY TO DNA

Figure 9: Supercoiling (Adapted from Stanford Biochem 201 Slides 1999)

Figure 10: Twist Equation

Figure 11: Writhe Equation