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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
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

Colored faces

3.10.10
(Vertex figure)

Triakis icosahedron
(dual polyhedron)

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.

Contents

  • Geometric relations 1
  • Area and volume 2
  • Cartesian coordinates 3
  • Orthogonal projections 4
  • Spherical tilings and Schlegel diagrams 5
  • Vertex arrangement 6
  • Related polyhedra and tilings 7
  • Truncated dodecahedral graph 8
  • See also 9
  • Notes 10
  • References 11
  • External links 12

Geometric relations

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

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

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

(0, ±1/ϕ, ±(2+ϕ))
(±(2+ϕ), 0, ±1/ϕ)
(±1/ϕ, ±(2+ϕ), 0)
(±1/ϕ, ±ϕ, ±2ϕ)
(±2ϕ, ±1/ϕ, ±ϕ)
(±ϕ, ±2ϕ, ±1/ϕ)
(±ϕ, ±2, ±ϕ2)
(±ϕ2, ±ϕ, ±2)
(±2, ±ϕ2, ±ϕ)

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
Image
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
image

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

Related polyhedra and tilings

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

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
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.

*n32 symmetry mutation of truncated spherical tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Truncated
figures
Config. 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 3.∞.∞
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞

Truncated dodecahedral graph

Truncated dodecahedral graph
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]


Circular

See also


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

  • (Section 3-9)  
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids.  

External links

  • Eric W. Weisstein, Truncated dodecahedron (Archimedean solid) at MathWorld
  • Richard Klitzing, 3D convex uniform polyhedra, o3x5x - tid
  • Editable printable net of a truncated dodecahedron with interactive 3D view
  • The Uniform Polyhedra
  • Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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