Geometric Reflections

Kaleidoscopes create wonderful geometric patterns.

They are both beautiful and thought provoking.

There is something pleasing to a mystic in such a land of mirrors. For a mystic is one who holds that two worlds are better than one. In the highest sense, indeed, all thought is reflection — Chesterton

In this talk, I outlined the mathematical theory of kaleidoscopes.

We introduced Coxeter geometries, and classified them in the plane.

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Hyperbolic Visualizations!

Thanks to Vi Hart, Andrea Hawksley, Elisabetta A. Matsumoto, and Henry Segerman for making these amazing things!

• exploration
• exploration
• How to see 3-manifolds by William P Thurston

CMESG Day 3

Today at CMESG, we met with our working groups again. My working group on “problem based learning” started to design lesson plans around our problems. I was in the team of people working on upper-division problems. In particular, we wanted to design a lesson around the problem:

Classify the Platonic solids and prove that there are only five.

This turned out to be much larger than we expected, and the problem sort of blew-up in our face. We were not sure where to get started. It was a neat instance of what commonly happens when people approach a new problem; it gets out of hand and they’re not sure how to proceed. We struggled to figure out how much graph theory and group theory to introduce. Where would people take the problem?

If I were to run a problem based learning session, I’d like to go through a three levels of testing before trying the session with my students.

• Try the problem on a non-mathematical friend. Is this interesting?
• Experiment with some math friends. What content might it have?
• Take the problem to a math club as a lesson plan. Where do people take it?

Once I knew that the problem was intrinsically interesting, could have some mathematical content, and wouldn’t go too wonky, I would write it up as a lesson plan to be used in a real class.

Science Rendezvous!

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This year, at Science Rendezvous, we shared symmetry and geometry. These areas of math are very beautiful and full of lovely patterns. In particular, we focused on how to connect geometry and symmetry using group theory. This approach was pioneered by Donald Coxeter, one of the most famous mathematicians of the twentieth century, and former professor at the University of Toronto. The big theme of our display was the notion of symmetry groups. This talk Symmetry and Groups by Professor Raymond Flood of Gresham College gives a great introduction to this connection.

Lukas brought his kaleidoscope, and I got it on video!

Three-Dimensional Kaleidoscope

My highschool student, Lukas Boelling, made this three-dimensional icosahedral/dodecahedral kaleidoscope with his dad, @eric_boelling. Lukas based his models off this excellent paper: Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes by Roe Goodman. Stay tuned for more models!

Homework #5 Question 4

Consider a solid ball of radius . Cut a cylindrical hole, through the center of the ball, such that the remaining body has height . Call this the donut . Use Cavalieri’s principle to calculate the volume of . Calculate the volumes of and .

Several students have asked what looks like. Here are some pictures that I found to illustrate the concept. The donut is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes and , so that it has total height .

Basic Combinatorics.

First we recall a little bit of terminology:

1.1. Sets and functions

A set is a collection of elements . We write a set by surrounding its list elements with curly braces. For example: , . We also use set constructor notation where is some statement about that can be true or false. For example: , . We write: , for the set of natural numbers, for the set of rational numbers, for the set of integers.

We write to mean that is in the set . We write . We write . We write for if . If then we say that and are disjoint sets.

We write for the set of ordered pairs of elements.

Definition 1 A function is injective (one to one) if: implies . A function is surjective (onto) if: for all there is such that . A function is bijective if: for all there is such that . The number of elements in a set is written .

1.2. Basic formulae

The basic facts of combinatorics are very simple.

1. If then there is no injective function from a set with elements to a set with elements. (This is called pigeon hole principle.)
2. If there is a bijective function from to then .
3. If and are disjoint then .
4. If and are disjoint then .
5. .
6. The number of element subsets of a set with elements is: . (Why is this an integer? Prove it.)
7. The number of functions from an -element set to a element set is .
8. If then the number of bijective functions from to (permutations of ) is .

There are a couple formulae that are handy to remember:

1. There are subset of .
2. Suppose you have objects of types 1, 2, , respectively. The number of ways of arranging all the objects is:
3. Suppose you have identical objects that you want to distribute among containers. The number of ways to do this is: . (Why?)

Dodecahedral Crafts

This evening I made some dodecahredral crafts to show my mentoring students. This week we’re proving that the dodecahderon actually exists, by constructing it explicitly. The model on the right illustrates our proof splendidly. For instructions, check out Laszlo Bardos’ site CutOutFoldUp.com. The dodecahedral calendar is from Marlies’ Crafts.

Singularities I.1

These are some notes that I’m writing up on Gromov’s papers on singularities of maps. This first post will look at some of the introductory material in: Singularities, Expanders and Topology of Maps. Pt 1. These notes will be a partial introduction to what is going on.

Of Waists and Spheres

These are some notes that I’m writing up on Gromov‘s waist inequality. We’ll look at some standard material about the Borsuk-Ulam theorem and finish with a nice application of the inequality.