3D Printer Models for MAT 232

This semester, I am teaching MAT 232 Multivariable Calculus. We often talk about level curves and use the saddle surface as a key example. Every time it comes up, I ask students to stare at the part of their hand where the thumb meets the palm. Of course, they stare at me like I am crazy! This region of the hand is a good model for a saddle surface. If you start looking around at biological examples, you’ll see saddle surfaces everywhere.

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.

Backing Up with TAR

My laptop is on its last legs. I manage to wear out laptops at an alarming rate. The Lenovo X220 has served me well, but the battery is flaking out and it shuts down everytime the wind blows. So, I looked up some options for backing up the disk. I found this brief guide to backing up.

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

IBL Geometry Materials

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

• 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!

MSLC Semina — Exploring Knights Tours

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.

UTM Math Club — The Diamond System

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.

Math Learning Center Orientation

Today I gave a little bit of an orientation to the Math Learning Center at MCS TA Professional Development day.

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.

Symmetry Groups at Science Unlimited

I gave a talk about symmetry groups at Science Unlimited 2019.
The slides are available here, for the curious.

Some Mathematical Reading

I just stumbled on this excellent list “Readings for Math Teachers” by Theron Hitchman. Lots of great stuff to read and ponder.

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

MSLC Summer Seminar

• 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

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.