Truncated_dodecahedron

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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.

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:

It shares its vertex arrangement with three nonconvex uniform polyhedra:

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:

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

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

Cartesian coordinates

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

(0, ±1/?, ±(2+?))

(±(2+?), 0, ±1/?)

(±1/?, ±(2+?), 0)

(±1/?, ±?, ±2?)

(±2?, ±1/?, ±?)

(±?, ±2?, ±1/?)

(±?, ±2, ±?²)

(±?², ±?, ±2)

(±2, ±?², ±?)

where ? = (1+√5)/2 is the golden ratio (also written ?).

See also

• Spinning truncated dodecahedron

• icosahedron

• icosidodecahedron

• truncated icosahedron

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Truncated_dodecahedron". The list of authors you can find on this page.