# Mathematical explorations

Mathematical discovery and creative problem solving

## Course information

Code: PMU199F

Instructor: Marco Gualtieri, office hours by appointment.

Class schedule: R3-5, BA 2179, starting September 12, 2013

Evaluation: Participation/presentation 30%, Assignments 70%

This course is an exploration of great ideas in mathematics. Some of the topics we will study are centuries-old, while others are still under active development today. Among other things, we will learn about ancient Indian prosody, the concept of infinity, the nature of space and spacetime, the meaning of higher-dimensional polyhedra, and how to create and understand fractals. Students will read and critique assigned texts and videos, write micro-essays, carry out experiments, develop mathematical intuition and skills, and engage in class discussion.

## Assignments

### 1: What is a set?

Read this short introduction to sets (PDF)

Assignment 1: Due Sept. 19

### 2: Graphs and permutations

Assignment 2: Due Sept. 26

### 3: Counting subsets

Assignment 3: Due Oct. 10 (Should be October 3!)

### 4: Levels of Infinity

Assignment 4: Due Oct 10

### 5: Higher Dimensions

Assignment 5: Due Oct 17

### 6: Polyhedra and polytopes

Assignment 6: Due Oct 24

### 7: Chaos part I

Assignment 7: Due Oct 31

Sage worksheet you should use to get started

Screencast 1 introducing the assignment (Youtube link)

Screencast 2 explaining how to submit/share worksheet (Youtube link).

### 8: Chaos Part II

Assignment 8: Due Nov 7

Screencast (Youtube link)

### 9: Chaos Part III (final part)

Assignment 9: Due Nov 14

### 10: Probability

Assignment 10: Due Nov 21

## Essay in a straitjacket

The micro-essays within the assignments should be between 300 and 500 words. They must be preceded by an outline. The mini-essay should have a simple structure and a clear idea, and should avoid embellishments and flowery language. Stick to clear factual statements and ensure that each sentence carries actual content and does not duplicate previous statements. Evaluation will be by two main criteria: Does the essay make a clear statement? Does it make an interesting statement?

## General resources

- A mathematical column by Keith Devlin
- Vi Hart’s series on Fibonacci
- Knuth’s “The art of computer programming”: Look at part 7.2.1.7 of Volume 4B, also available from here. A piece of language which Knuth uses is “binary n-tuple”, which refers to an ordered list of n objects, where each object can be of only two types. So for example \((a, b, a, a, b)\) is a binary 5-tuple, where each object in the list is either \(a\) or \(b\).
- Aristotle’s Physics, Volume VI. For a more modern perspective, Part 3.1 and Part 3.3 by Nick Huggett. For a less modern perspective, see Thomas Aquinas, part 11, #860-863.
- Documentary on Benoit Mandelbrot, from Nova (2008)
- David Dewey’s introduction to the Mandelbrot set – vintage WWW page, but very well explained. See also the zoomable juliamap software (google labs). For more information about Julia sets, see this page and applet.
- The concept of infinity. Watch the video, but with a critical eye.
- Carl Sagan on the 4th dimension. Based partially on the classic
work
*Flatland*(PDF) with LaTeX source available on github. - One of my presentations on polyhedra.
- The topology of surfaces.

## Writing assistance

## Accessibility Needs

If you require accommodations for a disability, contact Accessibility Services.