## P5.js Test-Run

Posted in Computers, geometry by pgadey on 2020/04/19

I played around with P5.js today. It is really nice software! There is an in-browser P5.js editor, so you can start playing around instantaneously. function setup() { angleMode(RADIANS); // Change the mode to RADIANS createCanvas(200, 130); }

 function draw() { background(256); v1 = [1,0]; v2 = [cos(PI/3),sin(PI/3)]; n=0; m=0; while (n <= 5){ while (m <= 5){ p = [20+(20*v1*n)+(20*v2*m), 20+(20*v1*n)+(20*v2*m)] if ( (n*m) % 2 == 0){ circle(p,p,20); fill(200, 0, 0); } else { circle(p,p,20); fill(0, 0, 200); } m += 1; } m = 0; n += 1; } } 

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## Bruce and Katharine Cornwell Films

Posted in Math by pgadey on 2020/04/08 Bruce and Katharine Cornwell produced beautiful animations of mathematics back in the 1960s. I watched the first couple minutes of their film “Possibly So, Pythagoras” (1963) and learned half a dozen pleasant things.

Check out their films, available on Vimeo here.

You can read the memorial from Reed College here.

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## IBL Geometry Materials

Posted in Teaching and Learning by pgadey on 2019/10/31

Turns out that JIBLM has a lot of resources for Geometry!

Tonnes of excellent material to work with!

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## Geometric Reflections

Posted in Math by pgadey on 2019/07/25 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. ## Hyperbolic Visualizations!

Posted in Math by pgadey on 2019/07/15 Thanks to Vi Hart, Andrea Hawksley, Elisabetta A. Matsumoto, and Henry Segerman for making these amazing things!

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## CMESG Day 3

Posted in Teaching and Learning by pgadey on 2019/06/03

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.

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## Science Rendezvous!

Posted in Math by pgadey on 2019/05/12

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

Posted in Math by pgadey on 2019/05/05

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!

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## Homework #5 Question 4

Posted in Math by pgadey on 2018/07/20

Consider a solid ball of radius $R$. Cut a cylindrical hole, through the center of the ball, such that the remaining body has height $h$. Call this the donut $D(R,h)$. Use Cavalieri’s principle to calculate the volume of $D(R,h)$. Calculate the volumes of $D(25,6)$ and $D(50,6)$. Several students have asked what $D(R,h)$ looks like. Here are some pictures that I found to illustrate the concept. The donut $D(R,h)$ 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 $z=h/2$ and $z=-h/2$, so that it has total height $h$.  Tagged with: ,

## Basic Combinatorics.

Posted in Uncategorized by pgadey on 2014/07/27

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: ${X = \{1,2,3\}}$, ${Y = \{\heartsuit, \clubsuit, \star\}}$. We also use set constructor notation ${Y = \{x : P(x)\}}$ where ${P(x)}$ is some statement about ${x}$ that can be true or false. For example: ${X = \{n : n\text{ is even}\}}$, ${Z = \{n : n\text{ is prime}\}}$. We write: ${\{\} = \emptyset}$, ${{\mathbb N}}$ for the set of natural numbers, ${{\mathbb Q}}$ for the set of rational numbers, ${{\mathbb Z}}$ for the set of integers.

We write ${x \in X}$ to mean that ${x}$ is in the set ${X}$. We write ${X \cup Y = \{x : x \in X \text{ or } x \in Y\}}$. We write ${X \cap Y = \{x : x \in X \text{ and } x \in Y\}}$. We write ${X \sqcup Y}$ for ${X \cup Y}$ if ${X \cap Y = \emptyset}$. If ${X \cap Y = \emptyset}$ then we say that ${X}$ and ${Y}$ are disjoint sets.

We write ${X \times Y = \{(x,y) : x \in X,\ y \in Y\}}$ for the set of ordered pairs of elements.

Definition 1 A function ${f : X \rightarrow Y}$ is injective (one to one) if: ${x \neq y}$ implies ${f(x) \neq f(y)}$. A function is surjective (onto) if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. A function is bijective if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. The number of elements in a set ${X}$ is written ${|X|}$.

1.2. Basic formulae

The basic facts of combinatorics are very simple.

1. If ${k < n}$ then there is no injective function from a set with ${n}$ elements to a set with ${k}$ elements. (This is called pigeon hole principle.)
2. If there is a bijective function from ${X}$ to ${Y}$ then ${|X| = |Y|}$.
3. If ${X}$ and ${Y}$ are disjoint then ${|X \cup Y| = |X| + |Y| - |X \cap Y|}$.
4. If ${X}$ and ${Y}$ are disjoint then ${|X \sqcup Y| = |X| + |Y|}$.
5. ${|X \times Y| = |X| \cdot |Y|}$.
6. The number of ${k}$ element subsets of a set with ${n}$ elements is: ${\binom{n}{k} = \frac{ n! }{ (n-k)! k! }}$. (Why is this an integer? Prove it.)
7. The number of functions from an ${n}$-element set to a ${k}$ element set is ${k^n}$.
8. If ${|X| = n}$ then the number of bijective functions from ${X}$ to ${X}$ (permutations of ${X}$) is ${n!}$.

There are a couple formulae that are handy to remember:

1. There are ${2^n}$ subset of ${\{1, \dots, n\}}$.
2. Suppose you have ${n_1, \dots, n_k}$ objects of types 1, 2, ${\dots}$, ${k}$ respectively. The number of ways of arranging all the objects is: ${{ (n_1 + \dots + n_k)! }/{ n_1! n_2! \dots n_k! }}$
3. Suppose you have ${n}$ identical objects that you want to distribute among ${k}$ containers. The number of ways to do this is: ${\binom{n+k-1}{k-1}}$. (Why?)

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