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Homework -- Putting it all Together

Please answer these questions in a document named "homework8.html" in your public_html directory. Once again, you will need to include Kali pictures. If you find that you are having trouble writing mathematics in html, consider writing or typing the mathematics on a sheet of paper and scanning it in. (Until recently, there has been no good way to include symbols like "

" in html documents. Programers at the Geometry Center are now working on Java applets (WWW computer programs) to remedy this situation.)

Use the costs presented in this chapter to prove that the seventeen plane group symbols listed in our table are all the orbifold symbols that cost exactly two dollars. (Unlike the case for the spherical symmetry groups, all symbols that cost $2 do correspond to real plane symmetry groups.)
Explain in your own words why the fact that there are exactly seventeen $2 orbifolds implies that there are only seventeen different symmetries a wallpaper pattern can have.
On the Explanation of Costs page is a detailed computation of the orbifold Euler characteristic of the orbifold of a brick. Perform a similar computation of the orbifold Euler characteristic of the soccer ball shown at the bottom of that page.
Kali allows you to draw several plane symmetry groups that aren't included in the list on the costs page. Why aren't the dihedral, cyclic, and frieze groups included in the list of crystallographic groups you compiled above? What symbols from the orbifold notation would you use to describe these groups? (Conway describes the frieze groups as patterns on the equator of the celestial sphere, and introduces infinite order kaleidoscopic and gyration points.)
You may wish to refer to the paper by Professor Schattschneider mentioned in last section's homework.
For the past three weeks, we have discussed ways of determining the orbifold of a symmetric pattern. If our answer to question one is really a proof that there are no more or less than seventeen wallpaper patterns, it must also be true that every orbifold described in the table determines some wallpaper group! This is, in fact, true. Here are a few questions to get you used to the idea of converting orbifolds back into wallpaper patterns.
- Give an example of a symmetrical pattern that has a square as its orbifold.
- Design a pattern whose orbifold has one ninety degree kaleidoscopic corner and an order four gyration point. Give a convincing argument for why your pattern has this orbifold.

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