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
Symmetry	I_h
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.

[edit] 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:

Truncated dodecahedron

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

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

$A = 5 (\sqrt{3}+6\sqrt{5+2\sqrt{5}}) a^2 \approx 100.99076a^2$

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

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).

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)