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. __TOC__

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 uniform star 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::$A\; =\; 5\; (sqrt\{3\}+6sqrt\{5+2sqrt\{5)\; a^2\; approx\; 100.99076a^2$:$V\; =\; frac\{5\}\{12\}\; (99+47sqrt\{5\})\; a^3\; approx\; 85.0396646a^3$

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).

ee also

*
*icosahedron
*icosidodecahedron
*truncated icosahedron

References

* (Section 3-9)

External links

*
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
* [http://www.lifeisastoryproblem.org/explore/net_trunc_dodecahedron.pdf Printable net of a Truncated Dodecahedron] [http://www.lifeisastoryproblem.org Life is a Story Problem.org]