From:  (robert a. moeser)
Date:           Sat, 22 Jun 1996 07:55:14 -0500
Newsgroups:     sci.math
Subject:        snub cube and dodecahedron (repost?)


i am making accurate computer models of the Platonic and Archimedean
solids. to the extent possible, i would like to use "nice" constants
in generating them.

the basic metaphor in my approach is to lay out, in the plane, a figure
which when "folded" will yield the desired polyhedron.

describing the polygons and their relationships is straightforward, and
for 16 out of 18 of the models the remaining facts necessary, namely
the dihedral angles, or Pi - the fold angles, is also a "nice" number,
for example arcsin(sqrt(6)/3) and such like.

never mind precisely how i managed to "discover" the nice way of expressing
some of those angles!

but for the "snub cube", which requires two fold angles,
37.01657 degrees (0.64606 radians) and 26.76541 degrees (0.46714 radians)
i cannot come up with nice neat expressions for those constants.

the "snub dodecahedron" is the same - the angles in this case are
15.82463 (0.27619 radians) and 27.07008 (0.47246 radians).

in addition, although i understand how the constraints work to define the
snub cube and snub dodecahedron, i am still looking for a simple way
to create the geometry. all the Platonic solids and the other Archimedean
solids are easy to make by simply expressed "carving" operations.

(i practice with broccoli stalks.)

so i also ask if anyone has seen a recipe for creating these two
perplexing solids by applying simple operations on some original solid.

for example, a cuboctahedron is "simply" a cube with its corners
carved off at angles from the octahdron.

-- rob