## Something important about spheres

**Kaleidotile** is another new geometry center program for the
Mac which can be downloaded from Jeff Week's Geometry
Games
website

**Apple Menu **

. . . **Math Apps**

. . . .-> **Kaliedotile **

Fool around with Kaleidotile for a while, then begin in earnest.

Focus on the first three options on the symmetry group panel: (2,3,3),
(2,3,4), and (2,3,5). These let you make all five platonic solids and 11
of the 13 archimedean solids (the polyhedra with regular polygons for
faces,
meeting the same way at each vertex). We will be using Kaleidotile to
explore
a specific property of spherical polyhedra.

Warm up(just to give something to think about while you play with
Kaleidotile):

**Q1** Assuming the names of the platonic solids are familiar, can you
decipher the system for the names of the Archimedean polyhedra? Hint: There
is a cub-octa-hedron and a icosa-dodeca-hedron; what is the
tetra-tetra-hedron?

But this question is what we are really interested in:

**Q2** For at least three polyhedra, count the number of vertices,
edges and faces. (If any polyhedra are particularly familiar, you can use
these). Hmmm.... see any numerical patterns? It's hard to give a hint
without
giving it away! In particular, look for a constant common to all spherical
polyhedra.

If you don't have Kaleidotile, here are some polyhedra to consider:

Hmmm

**Chaim Goodman-Strauss**
Dept. Mathematics
Univ. Arkansas
Fayetteville, AR 72701
*strauss@comp.uark.edu*
* 501-575-6332*