Truncated dodecahedron

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Truncated dodecahedron
Truncated dodecahedron
(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
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:

A = 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right) a^2 \approx 100.99076a^2
V = \frac{5}{12} \left(99+47\sqrt{5}\right) a^3 \approx 85.0396646a^3

Cartesian coordinates[edit]

The following Cartesian coordinates define the vertices of a truncated dodecahedron with edge length 2(τ−1), centered at the origin:[1]

(0, ±1/τ, ±(2+τ))
(±(2+τ), 0, ±1/τ)
(±1/τ, ±(2+τ), 0)
(±1/τ, ±τ, ±2τ)
(±2τ, ±1/τ, ±τ)
(±τ, ±2τ, ±1/τ)
(±τ, ±2, ±τ2)
(±τ2, ±τ, ±2)
(±2, ±τ2, ±τ)

where τ = (1 + √5) / 2 is the golden ratio (also written φ).

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]
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 tiling[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.

Uniform tiling 532-t01.png Truncated dodecahedron stereographic projection decagon.png
Decagon-centered
Truncated dodecahedron stereographic projection triangle.png
Triangle-centered
Orthographic projection Stereographic projections

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:

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg POV-Ray-Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

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.

Dimensional family of truncated polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Truncated
figures
Spherical triangular prism.png
3.4.4
Uniform tiling 332-t01-1-.png
3.6.6
Uniform tiling 432-t01.png
3.8.8
Uniform tiling 532-t01.png
3.10.10
Uniform tiling 63-t01.png
3.12.12
Uniform tiling 73-t01.png
3.14.14
Uniform tiling 83-t01.png
3.16.16
H2 tiling 23i-3.png
3.∞.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
t{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
t{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
t{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
t{∞,3}
Uniform dual figures
Triakis
figures
Triangular dipyramid.png
V3.4.4
Triakistetrahedron.jpg
V3.6.6
Triakisoctahedron.jpg
V3.8.8
Triakisicosahedron.jpg
V3.10.10
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
V3.12.12
Ord7 triakis triang til.png
V3.14.14
Ord8 triakis triang til.png
V3.16.16
Ord-infin triakis triang til.png
V3.∞.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png

Notes[edit]

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]