PROJECT 3 OPERATING INSTRUCTIONS

Before I go into details, here is a philosophic description of what the point of the project is – my goals and expectations for you. I think that will help you interpret the more specific directions appropriately. What I am after is for you to have the experience of digging into some geometrical topic more deeply than we have been able to, using the thinking and analyzing skills that you have been honing in the course of the quarter. Then I want you to use the communication skills we have also worked on to tell me about whatever it is you have found. If I succeed in helping you find a good topic, you will be able to challenge yourself at the right level and have a lot of fun doing some good learning.

So the first thing I need to provide is clues about finding places to dig in. I will list several that seem to me to have lots of Good Stuff available with no more mathematical hardware needed than what we have developed this quarter. If, on the other hand, you have spotted something else that attracts you, by all means run it past me and I will see if it can be made to work.

Topics that leap to mind:

1) Symmetries – in particular, the so-called ribbon symmetries. Those are the ones that hold for repeated patterns printed on an infinitely long ribbon. They are cool and have, as I recall, some interesting applications. Certainly some interesting histories.

NOTE: This seems a good place to break in and be a little more specific. What I am saying here is illustrated with ribbon symmetries, but has parallels in many of the possible areas. When you start looking in books and on-line, you will find a lot to look at. You will find some interesting things about how these patterns turn up in different cultures. All of that is fine, and it is perfectly appropriate to turn in a certain amount of it as part of your report. The danger that lurks is that of turning it into a show-and-tell of pretty designs or a straight cultural report. One way to legitimate it is to include with the pretty ones a description of which symmetries are represented, including which are less than obvious. Alternatively if you go the cultural route, you could figure out and describe which symmetries turn up on one culture but not another.

2) Symmetries in the plane. Chapter 22 gets you started on that and we will at least hit that a glancing blow in class. Actually, I gave a partial introduction to it when we were doing symmetries of figures. There is some really neat stuff on relating the symmetries – as I recall, anything can be produced by repeated reflections, but I have to reconstruct that every time I get to it!

3) Ruler and compass constructions. These you may have run into, but if so, I can just about guarantee that you got it as a bunch of procedures for accomplishing particular tasks. The fun of it is in figuring out the geometry behind it. Our book has a nice clear introduction on pp. 480 – 482 and some good exercises – but to my taste they should be pushing you harder to justify steps. If you opt for this topic I will be delighted to help you find some reasonable-level challenges.

4) Maurits Escher the mathematical artist. Here again the risk is of succumbing to the “Wow!!” factor, but in fact he does marvelous things with tessellations and with symmetries. Something along the lines of choosing a couple of tessellations and analyzing their symmetries, or, for those of you who have done more advanced math courses, looking into the way he uses projections and the like in conjunction with his tessellations could be a lot of fun. And a certain amount of “Wow!” is totally acceptable!

For the moment, those are all that I have up my sleeve. If I think of more I will notify you, and if you think of something, you tell me. This should get you started.

Oh, yes, the product: You should turn in something careful and neat – also grammatical and spell-checked. If you have unearthed a really challenging problem and spend your time solving it and writing up the solution, you might get by with less than six pages, but I’d say the general aim should be 6 – 10 pages (double-spaced.)

The paper is due on the day we would have had our final – Monday, March 16. On the other hand, if you turn it in during our last class I will read it over the week-end, and that way if I feel that you need to do more about it you will have more time to do so before the grading deadline of Friday the 20^th.