The following is the outline of a talk I gave at the
1997
Art and Math conference at S.U.N.Y. Albany, N.Y. This page is not fully
self-contained as the talk also included slides and models.
A
Color-Matching Dissection of the Rhombic Enneacontahedron
A path of rhombi connect any two opposite edges; they share one direction.
Every pair of paths crosses exactly once, so every pair of edge directions
appears in one rhombus.
Three
Dimensions:
Start with a zonohedron, a polyhedron with
rhombic faces,
opposite sides parallel and equal
Dissect polyhedron into rhombohedra
One rhombohedron for each triple of edge directions
n direction zonohedron has n-choose-3 pieces, i.e., n(n-1)(n-2)/6
It is easy to find a dissection:
start with a rhombus at any vertex (uses 3directions)
add n-3 layers
Pieces can be shuffled to make other solutions
An equator of faces share one direction, called a zone
A path of rhombohedra connect any two opposite faces; they share two directions