TRUNCATED DODECAHEDRON

In geometry, the 'truncated dodecahedron' is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
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Contents

Geometric relations

Area and volume

Cartesian coordinates

See also

References

External links

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:

Dodecahedron

Truncated dodecahedron

Icosidodecahedron

Truncated icosahedron

Icosahedron

It shares its vertex arrangement with three uniform star polyhedra:

U42

U48

U63

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:
:
:

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

★

★ dodecahedron

★ icosahedron

★ icosidodecahedron

★ truncated icosahedron

References

★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X (Section 3-9)

External links

★

★ The Uniform Polyhedra

★ Virtual Reality Polyhedra The Encyclopedia of Polyhedra