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Truncated dodecahedron
Truncated dodecahedron
(Click here for rotating model)
Type Archimedean solid
Elements F = 32, E = 90, V = 60 (χ = 2)
Faces by sides 20{3}+12{10}
Schläfli symbol t{5,3}
Wythoff symbol 2 3 | 5
Coxeter-Dynkin CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.png
Symmetry Ih
or (*532)
References U26, C29, W10
Properties Semiregular convex
Truncated dodecahedron color
Colored faces
Truncated dodecahedron
3.10.10
(Vertex figure)
Triakisicosahedron.jpg
Triakis icosahedron
(dual polyhedron)
Truncated dodecahedron Net
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

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 part of a truncation process between a dodecahedron and icosahedron:

Uniform polyhedron-53-t0.png
Dodecahedron
Uniform polyhedron-53-t01.png
Truncated dodecahedron
Uniform polyhedron-53-t1.png
Icosidodecahedron
Uniform polyhedron-53-t12.png
Truncated icosahedron
Uniform polyhedron-53-t2.png
Icosahedron

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

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

Cartesian coordinates

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

(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).

See also

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)

External links

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