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.

Truncated icosidodecahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 62, E = 180, V = 120 (χ = 2)
Faces by sides30{4}+20{6}+12{10}
Conway notationbD or taD
Schläfli symbolstr{5,3} or
t0,1,2{5,3}
Wythoff symbol2 3 5 |
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle6-10: 142.62°
4-10: 148.28°
4-6: 159.095°
ReferencesU28, C31, W16
PropertiesSemiregular convex zonohedron

Colored faces
4.6.10
(Vertex figure)

Disdyakis triacontahedron
(dual polyhedron)
Net
3D model of a truncated icosidodecahedron

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 (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and Small Rhombicosidodecahedron (89.23%), and less narrowly beating the Truncated Icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) 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 (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.

Names

The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of an icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid.

Alternate interchangeable names are:
Icosidodecahedron and its truncation

The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
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

Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:[4]

(±1/φ, ±1/φ, ±(3 + φ)),
(±2/φ, ±φ, ±(1 + 2φ)),
(±1/φ, ±φ2, ±(−1 + 3φ)),
(±(2φ − 1), ±2, ±(2 + φ)) and
(±φ, ±3, ±2φ),

where φ = 1 + √5/2 is the golden ratio.

Dissection
A toroidal polyhedron can be formed from the truncated icosidodecahedron, with pentagonal faces excavated by pentagonal rotundae, and a final excavated central rhombicosidodecahedron.

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.

Orthogonal projections
Centered by Vertex Edge
4-6		 Edge
4-10		 Edge
6-10		 Face
square		 Face
hexagon		 Face
decagon
Solid
Wireframe
Projective
symmetry		 [2]+ [2] [2] [2] [2] [6] [10]
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.

Orthographic projection Stereographic projections
Decagon-centered Hexagon-centered Square-centered

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
Truncated icosidodecahedral graph
5-fold symmetry
Vertices120
Edges180
Radius15
Diameter15
Girth4
Automorphisms120 (A5×2)
Chromatic number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

In the mathematical field of graph theory, a truncated icosidodecahedral graph (or great rhombicosidodecahedral 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.[5]

Schlegel diagram graphs

3-fold symmetry
2-fold symmetry
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.[6]

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Notes
  1. Wenninger, (Model 16, p. 30)
  2. Williamson (Section 3-9, p. 94)
  3. Cromwell (p. 82)
  4. Weisstein, Eric W. "Icosahedral group". MathWorld.
  5. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
  6. Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan

References
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