- Following the example given
in the notes, use the costs of orbifold
features 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 at most seventeen different types of symmetry in wallpaper patterns.

- 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.
- Design and display a symmetrical pattern that has a square as its
orbifold.
- Design and display 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.

- Design and display a symmetrical pattern that has a square as its
orbifold.

Author: Heidi Burgiel

Comments to:
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Created: Dec 7 1995 ---
Last modified: Jun 11 1996

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