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.
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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