Truncated dodecahedron

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Truncated dodecahedron
Truncateddodecahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 32, E = 90, V = 60 (χ = 2)
Faces by sides 20{3}+12{10}
Conway notation tD
Schläfli symbols t{5,3}
t0,1{5,3}
Wythoff symbol 2 3 | 5
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry group Ih, H3, [5,3], (*532), order 120
Rotation group I, [5,3]+, (532), order 60
Dihedral Angle 10-10: 116.57°
3-10: 142.62°
References U26, C29, W10
Properties Semiregular convex
Truncated dodecahedron.png
Colored faces
Truncated dodecahedron vertfig.png
3.10.10
(Vertex figure)
Triakisicosahedron.jpg
Triakis icosahedron
(dual polyhedron)
Truncated dodecahedron flat.png
Net

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

Geometric relations[edit]

This polyhedron can be formed from a 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[edit]

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

Cartesian coordinates[edit]

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[edit]

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
Image Dodecahedron t01 v.png Dodecahedron t01 e3x.png Dodecahedron t01 exx.png Dodecahedron t01 A2.png Dodecahedron t01 H3.png
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
image
Dual dodecahedron t12 v.png Dual dodecahedron t12 e3x.png Dual dodecahedron t12 exx.png Dual dodecahedron t12 A2.png Dual dodecahedron t12 H3.png

Spherical tilings and Schlegel diagrams[edit]

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
Uniform tiling 532-t01.png Truncated dodecahedron stereographic projection decagon.png
Decagon-centered
Truncated dodecahedron stereographic projection triangle.png
Triangle-centered
Truncated dodecahedron ortho-color.png Truncated dodecahedron schlegel.png Truncated dodecahedron schlegel-tricenter.png

Vertex arrangement[edit]

It shares its vertex arrangement with three nonconvex uniform polyhedra:

Truncated dodecahedron.png
Truncated dodecahedron
Great icosicosidodecahedron.png
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron.png
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron.png
Great dodecicosahedron

Related polyhedra and tilings[edit]

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[edit]

Truncated dodecahedral graph
Truncated dodecahedral graph.png
5-fold symmetry schlegel diagram
Vertices 60
Edges 90
Automorphisms 120
Chromatic number 2
Properties Cubic, Hamiltonian, regular, zero-symmetric

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]

Truncated Dodecahedral Graph.svg
Circular

See also[edit]

Notes[edit]

  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[edit]

  • 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. 

External links[edit]