Date: Nov 12, 2008 2:47 PM
Author: Kirby Urner
Subject: Projects for Self Schoolers

So if you're in that distinct ethnic minority of being 
focussed on spatial geometry (so-called "solid"), not
just the flat stuff, you might want to pick up a few
clues for some interesting research projects, from this
distillation of relatively recent results.

First, if you know what a tetrahedron is, a regular one,
then consider that your "water cup" for pouring into
other shapes, measuring their volume. Vis-a-vis this
approach, a class of polyhedra called the Waterman
Polyhedra all have whole number volumes. Use the
Internet to find out more.

Second, consider the space-filling rhombic dodecahedron,
the encasement for each ball in a dense-packing we call
the CCP and/or FCC (other things). Given those balls are
unit radius, with four of them defining our regular
tetrahedron (above), this rhombic dodecahedron has a
volume of six. Use your knowledge of geometry and algebra
to verify this claim.

Third, consider the rhombic triacontahedron, yes a quasi-
spherical shape of 30 diamond faces. Inscribed about
a sphere, such that its 30 face centers kiss the sphere's
surface, we define its radius as equal to that of the
encased sphere's.

Verify that if this sphere has a radius of phi/sqrt(2),
that this shape has a volume of 7.5, relative to our
basic 'water cup' (above). This is not such an easy
problem, answer tomorrow (you don't have to look).

Kirby Urner
4dsolutions.net

Note: 'Connections: The Geometric Bridge Between Art
and Science' (Jay Kappraff, NJIT), is a good source of
information on both phi and sqrt(2), in terms of their
geometric significance and appearance in computations.
phi = (1 + sqrt(5))/2 and is known as the "golden mean"
or "golden proportion" (pronounced fee, fie... or some
use letter the greek letter tau). phi/sqrt(2) = about
1.1441228.