TRUNCATED ICOSIDODECAHEDRON

The 'Truncated icosidodecahedron' is an Archimedean solid. It has 30 regular square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.

Contents

Other names

Area and volume

Cartesian coordinates

See also

References

External links

Other names

Alternate interchangeable names include:

★ ''Great rhombicosidodecahedron''

★ ''Rhombitruncated icosidodecahedron''

★ ''Omnitruncated icosidodecahedron''
The name ''truncated icosidodecahedron'', originally given by Johannes Kepler, is somewhat misleading. If you truncate an icosidodecahedron by cutting the corners off, you do ''not'' get this uniform figure: instead of squares you get golden rectangles. However, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular.

Icosidodecahedron

The alternative name ''great rhombicosidodecahedron'' (as well as rhombitruncated icosidodecahedron) refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. Compare to small rhombicosidodecahedron.
One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See uniform great rhombicosidodecahedron.

Area and volume

The surface area ''A'' and the volume ''V'' of the truncated icosidodecahedron of edge length ''a'' are:
:
A & = 30 left [ 1 + sqrt{ 2 left ( 4 + sqrt{5} + sqrt{15+6sqrt{6}}
ight ) }
ight ] a^2 \
& pprox 175.031045a^2 \
V & = ( 95 + 50sqrt{5} ) a^3 pprox 206.803399a^3. \
end{align}

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2τ-2, centered at the origin, are all the even permutations of
: (±1/τ, ±1/τ, ±(3+τ)),
: (±2/τ, ±τ, ±(1+2τ)),
: (±1/τ, ±τ², ±(-1+3τ)),
: (±(-1+2τ), ±2, ±(2+τ)) and
: (±τ, ±3, ±2τ),
where τ = (1+√5)/2 is the golden ratio.

See also

★

★ dodecahedron

★ great truncated icosidodecahedron

★ icosahedron

★ truncated cuboctahedron

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