Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

Truncated dodecahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 90, V = 60 (χ = 2)
Faces by sides20{3}+12{10}
Conway notationtD
Schläfli symbolst{5,3}
t0,1{5,3}
Wythoff symbol5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle10-10: 116.57°
3-10: 142.62°
ReferencesU26, C29, W10
PropertiesSemiregular convex

Colored faces
3.10.10
(Vertex figure)

Triakis icosahedron
(dual polyhedron)
Net
3D model of a truncated dodecahedron

Geometric relations

This polyhedron can be formed from a regular dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.

It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.

Area and volume

The area A and the volume V of a truncated dodecahedron of edge length a are:

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin,[1] are all even permutations of:

(0, ±1/φ, ±(2 + φ))
(±1/φ, ±φ, ±2φ)
(±φ, ±2, ±(φ + 1))

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

Orthogonal projections

The truncated dodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
3-10		 Edge
10-10		 Face
Triangle		 Face
Decagon
Solid
Wireframe
Projective
symmetry		 [2] [2] [2] [6] [10]
Dual

Spherical tilings and Schlegel diagrams

The truncated dodecahedron 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
Triangle-centered

Vertex arrangement

It shares its vertex arrangement with three nonconvex uniform polyhedra:


Truncated dodecahedron
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron

It is part of a truncation process between a dodecahedron and icosahedron:

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

Truncated dodecahedral graph
Truncated dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

In the mathematical field of graph theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.[2]


Circular

Notes
  1. Weisstein, Eric W. "Icosahedral group". MathWorld.
  2. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

References
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
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