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}
Schläfli symbol	t{5,3}
Wythoff symbol	2 3 | 5
Coxeter–Dynkin
Symmetry	Ih, [5,3], (*532)
Dihedral Angle	decagon-decagon: 116.57 decagon-triangle: 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.

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:

Picture	Dodecahedron	Truncated dodecahedron	Icosidodecahedron	Truncated icosahedron	Icosahedron
Coxeter-Dynkin

It shares its vertex arrangement with three nonconvex uniform polyhedra:

Truncated dodecahedron

Great icosicosidodecahedron

Great ditrigonal dodecicosidodecahedron

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).

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]

See also

• Spinning truncated dodecahedron
• Icosahedron
• Icosidodecahedron
• Truncated icosahedron

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

• 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