Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84 / QGSI 2.0; site qubix.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!sdcsvax!sdcrdcf!hplabs!oliveb!ios!qubix!lab From: lab@qubix.UUCP (Q-Bick) Newsgroups: net.math Subject: Re: Archemedian polyhedra: the EPCOT ball Message-ID: <1427@qubix.UUCP> Date: Fri, 12-Oct-84 17:01:38 EDT Article-I.D.: qubix.1427 Posted: Fri Oct 12 17:01:38 1984 Date-Received: Sun, 14-Oct-84 08:31:57 EDT References: <3732@decwrl.UUCP> <1395@qubix.UUCP> Organization: Quadratix ... Quartix Lines: 35 Took a look at my photographs (and at the cover of the EPCOT guide provided by Eastern Airlines) and reconstructed it, finding a dodecahedron a better and simpler basis than an icosahedron. Simply: start with a dodecahedron. Cover each face with a pentagonal pyramid, where each non-base face is an equilateral triangle. You now have 60 triangles. Divide each edge into 8 units and connect two edges of each triangle with a line parallel to the third edge. You now have 1+ (2+1)+(3+2)+(4+3)+(5+4)+(6+5)+(7+6)+(8+7)=64 subtriangles. Put a tetrahedron on each subtriangle. 3 exposed faces 64 tetrahedra 60 major triangles --------------- * ------------- * ---------- = 11520 tetrahedron major triangle dome faces on the dome For the purist looking for the icosahedron, its vertices are easy to find - wherever five triangles meet (fairly easy to find on the dome). The edges are totally hidden, as they pass through the ALTITUDES of the triangles (== apexes of the tetrahedra). However, the troughs are such a noticeable feature that you would probably never see the icosahedron, even if you were looking for it. (For that matter, the curvature of the sphere hides conceals the transition from one pentagon to the next, concealing even *their* existence.) BTW, the 30-face value in my former article was the number of skew equilateral rhombi (two non-coplanar equilateral triangles with a common edge) covering the dome. The acute ends of the rhombi are the apexes of the pentagonal pyramids - where 5 triangles meet on the dome. The major axes of the rhombi are the edges of the icosahedron; their minor axes, those of the dodecahedron. -- The Ice Floe of Larry Bickford {amd,decwrl,sun,idi,ittvax}!qubix!lab You can't settle the issue until you've settled how to settle the issue.