I got interested in getting some 3D printed models of saddle surfaces to hand around the class. I found a great project 3D Printed Models for Multivariable Calculus put together by John Zweck. The STL files for the models are freely available, and I asked Reinhard Grassmann of the Continuum Robotics Lab if he could 3D print some models of saddle surfaces and the paraboloid for me.

They arrived yesterday and they turned out GREAT! You can clearly see the level curves in one model, and the coordinate grid in another. They feel great to hold and are durable enough to hand around to a class of students.

]]>Here are the relevant commands for reference:

`$ sudo tar -cvpzf backup.tar.gz --exclude=/backup.tar.gz --one-file-system /`

$ sudo tar -xvpzf /path/to/backup.tar.gz -C /restore/location --numeric-owner

- Euclidean Geometry: An Introduction to Mathematical Work by TJ Hitchman
- Hilbert Geometry: A Guided Inquiry Approach by David Clark
- Modern Geometry I by Nathaniel Miller
- Modern Geometry II by Nathaniel Miller
- Euclidean and Non-Euclidean Geometries by Charles Coppin

Tonnes of excellent material to work with!

]]>At the MSLC Seminar we had an “improvisational seminar” this week. We started off chit-chatting about various problems, and a theme emerged. One participant posed the following problem:

The game of Knight Placement is played on an 8×8 chessboard. Two players alternately take turns placing knights on the board. A move consists of adding a knight to the board, such that no knight is under attack. A player loses if they’re unable to place a knight. Who wins under optimal play?

I followed this question up with:

Suppose that the Queen of Chess has a garrison of twenty-five knights. The knights are kept on a 5×5 chessboard. One fine morning, the Queen shows up and orders the knights to all switch places, or be severely punished. Can every knight switch places simultaneously?

This got us thinking about knights tours. In a knight’s tour, a knight travels to every cell of a chessboard by visiting each square exactly once. Notice that if the 5×5 board has a knight’s tour, then the garrison can re-arrange themselves by each stepping along the tour.

We found a couple small boards with and without closed knight’s tours. Wikipedia turned our attention to this paper:

Allen J. Schwenk (1991). “*Which Rectangular Chessboards Have a Knight’s Tour?*” Mathematics Magazine: 325–332. (link)

Working through that paper might make a good session at MSLC Seminar. If anyone knows the history / providence of the puzzles above, I would be hear about them.

]]>I gave a string workshop at the UTM Math Club. It was very experimental. I wanted to highlight the algorithmic aspect of string figures. We were going to do the first bit of inoli’s Diamonds System.

The plan was: distribute sting, teach Half-Second Star, teach Osage Two Diamonds, and then ask people to vary the figure in hopes of produce any number of diamonds other than two. The following issues arose: not everyone was able to form Osage Two Diamonds, and there were rather lost with the variation part. The game of small changes is hard to get started. Once I showed them a variation that produced Osage One Diamond, they were able to puzzle out n=3 and n=4 diamonds by experimentation.

The string that I distributed was not excellent quality. That made things even harder. Asking people to make their own loops, took a long time. Perhaps 15~20 minutes. It is probably wise to prepare a bunch of loops ahead of time. That seems like the best practice.

One neat thing happened during the talk. A young couple wandered in to Math Club to see what we were going. They were walking past, and noticed everyone playing with string and decided to come and check it out. As soon as they say down, they both started to play with the string. The girl asked the her boyfriend if she could show him a magic trick. She performed an excellent rendition of Loop Through Neck. Her patter was fine, and she even misdirected well with strong eye contact and did a couple moved to establish fairness and rhythm. It was lovely to watch. Don’t you just love seeing string “in the wild”?

The talk seemed more mathematical or algorithmic this time, despite me note introducing String Figure Calculus. I think that the presentation:

“Here is an algorithm. It outputs n=2. Vary it to get something other than two!”

works well for computational/mathematically minded audiences. It seems like a math problem. One angle, that I did not explore, but might be neat to explore in a future talk, is the formalization process:

“Here is an algorithm. How can we code it?”

One Math Club member started heading in that direction. He wanted to vary systematically.

After the talk, and exploration, I gave everyone a copy of inoli’s Diamonds System article.

They were left muttering “… conflated DNA looms???”

Professional development for TAs is where people get started on their teaching careers. These mini-workshops for incoming TAs are a valuable opportunity to share our hard won insights in to teaching and learning with people who are at the front lines. Teaching assistants interact directly with students, and are often the part of a course that students related to best. Almost all of out teaching assistants are themselves students at UTM. They have the freshest perspective on how these courses are taught.

My contribution to the program for TA Professional Development was communication strategies for use in one-on-one interaction with students. I wanted to get across two ideas: “asking is more important than telling” and “students don’t know”. I tried to bundle these together in a communication exercise.

The teaching assistants were all given a simple picture, and asked to describe the picture “mathematically” to their neighbour. The task is difficult because the person describing the picture could not directly describe the subject.

]]>I gave a talk about symmetry groups at Science Unlimited 2019.

The slides are available here, for the curious.

An absolutely spot-on quote from one of the articles:

]]>The teaching of mathematics, like mathematics itself, is an endless journey of study. I believe that teaching mathematics can be as intellectually demanding as doing mathematics. If our society could come to see teaching as a job that is emotionally, physically, and intellectually demanding, we would then be able to give teachers the respect they deserve, attract more talented people to the profession, and speed up the pace of pedagogical innovation through the study of teaching. — Adventures in Teaching, Darryl Yong

- May 30th “Derivation and applications of the gamma function” by David Salwinski
- June 6th “An Extension of Heron’s Formula” by Zohreh Shahbazi
- June 13th “What is Homology?” by Parker Glynn-Adey
- June 20th “Exploring Mathematics Learning Support Across Canadian Universities” by Rubina Shaik and Shrijan Rajkarnikar
- June 27th “Liouville numbers and irrationality measure” by David Salwinski
- July 4th “Representation theory” by Lisa Jeffery
- July 11th “Geodesics on Surfaces of Revolution” by Amanda Petcu
- July 18th “An (informal) Introduction to Model Theory and Skolem’s Paradox.” by Yasin Mobassir
- July 25th “Geometric Reflections” by Parker Glynn-Adey
- August 1st “The Inscribed Square Problem” by Amanda Petcu

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.