```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 areunit radius, with four of them defining our regular tetrahedron (above), this rhombic dodecahedron has a volume of six.  Use your knowledge of geometry and algebrato 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 Urner4dsolutions.netNote:  '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 theirgeometric significance and appearance in computations.phi = (1 + sqrt(5))/2 and is known as the "golden mean"or "golden proportion" (pronounced fee, fie... or someuse letter the greek letter tau). phi/sqrt(2) = about 1.1441228.
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