Some Archimedean Solids in the Icosahedral Lattice

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This Demonstration considers a set of points of integral linear combinations , where the are six vertices of a regular icosahedron and the coefficients to are integers between and or and , for a total of or points. The coordinates of the are the six permutations of , , and (the golden ratio). The convex hull of is a triacontahedron. Certain choices of linear combinations give the vertices of a dodecahedron, an icosidodecahedron, a truncated dodecahedron, and a truncated icosahedon. Given one vertex on a solid, all the other vertices are points in that are at the same distance as the given one.

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To get a rhombicosidodecahedron or a great rhombicosidodecahedron, coefficients of absolute value 3 or 5 are needed.

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Contributed by: Izidor Hafner (April 2013)
Open content licensed under CC BY-NC-SA


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The set of all triplets with integer coefficients is called phi space.



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