Pentakis icosidodecahedron

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Pentakis icosidodecahedron
Pentakis icosidodecahedron.png
Type Geodesic polyhedron (2,0)
Faces 80 triangles
(20 equilateral; 60 isosceles)
Edges 120 (2 types)
Vertices 42 (2 types)
Vertex configuration (12) 35
(30) 36
Conway notation k5aD = dcD = uI
Symmetry group Icosahedral (Ih)
Dual polyhedron Chamfered dodecahedron
Properties convex
Net
Pentakis icosidodecahedron net.png

In geometry, the pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron (chamfered dodecahedron).

Contents

Construction

Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained even with the 12 order-5 vertices at a different distance from the center as the other 30.

It can also be topologically constructed from the icosahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices. If this is all that is done, all 80 triangles will be equilateral, but the 4 faces will be coplanar. If the divided edges are lengthened to make the new vertices the same distance from the center as the original vertices, the result is a pentakis icosidodecahedron.

Conway (u2)I(k5)aI
Image Icosahedron subdivided.png Conway polyhedron flat k5aI.png
Form2-frequency subdivided icosahedron Subdivided icosidodecahedron

Geodesic domes

The pentakis icosidodecahedron is a common geometry for geodesic domes derived from the icosahedron. Buckminster Fuller referred to it as the 2-frequency alternategeodesic subdivision of the icosahedron, because the edges are divided into 2 equal parts and then lengthed slightly to keep the new vertices on a geodesic great circle, creating a polyhedron with two distinct edge lengths and face shapes. An -frequency alternate geodesic subdivision divides each edge into equal parts, and lengthens them as necessary to keep the new vertices on the surface of the same sphere. It produces a polyhedron with triangular faces. The higher the frequency of subdivision, the larger the number of distinct edge lengths and face shapes that will be required, but the more spherical the shape. In the design of geodesic domes, the cost of more roundness is increased complexity.

3D model of a pentakis icosidodecahedron Pentakis icosidodecahedron.stl
3D model of a pentakis icosidodecahedron

It represents the exterior envelope of a vertex-centered orthogonal projection of the 600-cell, one of six convex regular 4-polytopes, into 3 dimensions.

See also

References