assignments
due Mon Sep 24
Assignment 1
The short version: Write coherently, and maybe even elegantly and inspiringly, about some mathematics that you now understand but did not a month ago.
You are welcome to interpret that as broadly as you like. We'll have some class time on Friday to talk about it. Here are a few more specific things that you might try (but note that this is not a list of questions that should all be answered).
1. Answer two or three of the questions from the SET handout.
2. Can you find a 6 by 6 row-complete Latin square (aka cheese-tasting schedule) using the mod arithmetic method? Other even orders? Can you show that 3 by 3 or 5 by 5 ones don't exist?
3. How about orthogonal Latin squares? Can you use the method we saw in class to construct a pair of 9 by 9 orthogonal squares? Can you explain why it works for all odd numbers? What about triples of mutually orthogonal squares (i.e. a set of three squares, any pair of which are orthogonal). What patterns are there in any of the sets you've found (whether in or out of class)? Here's a link to a pair of orthogonal Latin squares of order 10:
What patterns can you find in these?
4. A tetrahedron is a triangular-based pyramid. How many "parts" does it have. Consider the sequence "point, line, triangle, tetrahedron". We might call the next one a 4-tetrahedron. How many parts does a 4-tetrahedron have? Do 4-tetrahedrons exist?
5. What happens when you cut a Mobius strip along its length in various ways. Can you explain why, and predict what will happen in more complicated cases? What if there are more twists in the strip?
6. Play with planar diagrams (polygons with an even number of sides and whose sides are identified in pairs). How many are there with 4 sides? Can you explain the surface that corresponds to each? Investigate those with 6 sides.
due Fri Oct 19
Assignment 2
A much smaller assignment than the last one. Play with one of the applets and comment on what you find. Examples of more specific options (remember, you only need to do one):
- Explore the Mandelbrot Set. How small a near-copy can you find? What are its coordinates? What are some of differences between the small one(s) and the main picture?
- Explain how to solve the Chaos game.
- Upload your own pictures into the box-counting dimension program. Find some dimensions.
