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Projects for Self Schoolers
Posted:
Nov 12, 2008 2:47 PM


So if you're in that distinct ethnic minority of being focussed on spatial geometry (socalled "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. Visavis 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 spacefilling rhombic dodecahedron, the encasement for each ball in a densepacking 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.



