These homework problems are meant to expand your understanding of what goes on during class. Any you turn in will be graded and returned to you. Answers may or may not be posted on the web, depending on demand.

1. Can a finite figure have more than one rotational symmetry? (That is, can a finite figure have rotational symmetries with different centers of rotation?)

2. Recall that a regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. Show that a regular n-sided polygon has precisely n rotational symmetries and precisely n reflection symmetries.

3. Is there a finite figure whose symmetry group is something other than C_N or D_N? If not, then we have classified the symmetry groups of finite figures. That is, we have listed all possible symmetry groups.

4. We saw in class that, on the square, both
   

   r5 m r-3 m r2 = r2 r2 m r-3 m r5 = r2.

   That is, we have written both sides of the (true) first equation backwards and obtained the (also true) second equation. The question is: is this always the case? That is, if you write both sides of a true equation backwards, do you obtain a true equation? You may answer this for the square or for an n-sided regular polygon.

5. (Hamoon's conjecture) As a follow-up to the previous question: if we re-arrange the exponents of the various r terms on each side of a (true) equation to obtain a new equation, must this new equation also hold? You may answer this for the square or for an n-sided regular polygon.

6. Prove that the identity of a group is unique.

7. Please make a model cube, tetrahedron, icosahedron, or dodecahedron. (You may make more than one, but please make at least one.) There are links on the class web page to paper models, or you may build frame models with, say, sticks or toothpicks.

   Some links for ideas and/or making such models:

   • Korthals Altes has some nice models for paper polyhedra, with PDF files
   • douglas zongker | polyhedra models, a beautiful page with PDF templates
   • georgehart.com, another beautiful page with virtual tours of polyhedra and some extremely complicated polyhedra
   • emit.org has the basic polyhedra, but nicely done
   • Fr. Magnus Wenninger, who wrote the book on building polyhedra, has a nice page with some of his exotic creations
   • The Geometry Junkyard has a page on Models of Polyhedra. Mostly good links to various people's creations.
   • A collection of polyhedra-related links from SpaceStation42.com

These problems are also available as a PDF file.