Truncated icosidodecahedron


In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron and Small Rhombicosidodecahedron, and less narrowly beating the Truncated Icosahedron ; it also has by far the greatest volume when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry, the truncated icosidodecahedron is a zonohedron.

Names

The name great rhombicosidodecahedron refers to the relationship with the rhombicosidodecahedron.
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.

Area and volume

The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

Cartesian coordinates

for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:
where φ = is the golden ratio.

Dissection

The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares whose height to base ratio is the golden ratio. The rest of its space can be dissected into 12 nonuniform pentagonal cupolas below the decagons and 20 nonuniform triangular cupolas below the hexagons.

Orthogonal projections

The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.
Centered byVertexEdge
4-6		Edge
4-10		Edge
6-10		Face
square		Face
hexagon		Face
decagon
Solid
Wireframe
Projective
symmetry		+
Dual
image

Spherical tilings and Schlegel diagrams

The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Schlegel diagrams are similar, with a perspective projection and straight edges.

Geometric variations

Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.

Truncated icosidodecahedral graph

In the mathematical field of graph theory, a truncated icosidodecahedral graph is the graph of vertices and edges of the truncated icosidodecahedron, one of the Archimedean solids. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph.

3-fold symmetry
2-fold symmetry

Related polyhedra and tilings

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.