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:


  Chaim Goodman-Strauss
  Dept. Mathematics
  Univ. Arkansas
  Fayetteville, AR 72701