In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron[1] is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
If the bipyramids, the gyroelongated bipyramids, and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape.
Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles. Buckminster Fuller used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
^Conway, Symmetries of things, p.284
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In geometry, a disdyakistriacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and...
polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. More formally, the disdyakis dodecahedron is the Kleetope...
In geometry, the medial disdyakistriacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron. It...
={\tfrac {1+{\sqrt {5}}}{2}}} is the golden ratio. The great disdyakistriacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron...
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and edge centers to degree-four points. The 62 vertices of the disdyakistriacontahedron fall in three sets centered on the origin: Twelve vertices are...
rotation group and the full icosahedral group are given by: In the disdyakistriacontahedron one full face is a fundamental domain; other solids with the same...
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dodecahedron by a rhombic pyramid, and the disdyakistriacontahedron is the Kleetope of the rhombic triacontahedron. In fact, the base polyhedron of a Kleetope...
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diagram: . There are 120 triangles, visible in the faces of the disdyakistriacontahedron, and in the alternately colored triangles on a sphere: The dihedral...
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