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.