Proofs and Visualizations

Combine your intuition with advanced mathematical techniques to visualize and explore topics from number theory to hyperbolic geometry, and many beautiful places in between.
Teacher: Zandra Vinegar

There exist numbers that cannot be computed, geometric figures that have infinite perimeter but contain finite area, and ways of defining space so that unique, straight lines intersect at multiple points. I will rigorously prove each of these claims in class.

If you find these ideas bizarre, you’re certainly not alone. For hundreds of years, mathematicians called many beautiful examples and even entire branches of mathematics “pathological,” -- meaning so unbelievable that they should be ignored by "respectable" mathematicians -- and they refused to investigate further. However, the beauty and applicability of these topics to physics and computer science eventually rooted them at the heart of modern mathematics. So, if these kinds of ideas shake your world up a bit, then come to this class prepared to be shaken!

At its foundation, Proofs and Visualizations will focus on learning mathematical research techniques that will allow students to gain intuition for advanced areas of mathematics by exploring and experimenting. We will then hone this intuition into interesting and rigorous conclusions (a.k.a. proofs!). Our proofs, though, will look nothing like the typical, cut-and-dried "two-column proof" from high school geometry; rather, they will be more like those of a professional mathematician. If you've never written any proofs at all before, don’t worry. They are easy to learn and methods of proof will be covered in class.

Starting with a quick introduction to number theory, we will look at a new area of mathematics each week: knot theory, graph theory, fractals, non-Euclidean geometry, and more! While it will frequently be tempting to simply sit back and enjoy the beauty of these systems, we will dive deeper and explore their patterns and paradoxes rigorously.

For the application...

Prerequisites

None.

Relevant experience

Any courses, activities or programs in mathematics.

Application Question (Core-specific free response)

Start by watching this short video on youtube. There are 15 pendulums in the video. Call the shortest pendulum A, then B, C… the longest one is O. Please use these names in your solutions. You can see that the shorter a pendulum is, the faster it “ticks.”

A)
At 0:42, 0:47, and 0:58; the pendulums are in clear groups: 4 groups with 3-4 pendulums in each, then 3 groups with 5 pendulums in each, and finally 2 groups with 7-8 pendulums in each. At each of these times, which pendulums are in groups together? You can figure this out either by watching the video extremely carefully, or by reading the description of how the system was built and then using math/physics (be sure to show your work!). Make a list of as many “groups” as you can and the times when they exist. What's the temporal pattern?

Screenshots of the three times are shown below.

B)
In one paragraph, either describe one way in which you could investigate this system further OR design and describe another physical system (using pendulums, springs, etc.) with interesting and/or beautiful behavior. (You don’t need to build it, just find or come up with the idea!)