RHOMBICOSIDODECAHEDRON

The 'rhombicosidodecahedron', or 'small rhombicosidodecahedron', is an Archimedean solid. It has 20 regular triangular faces, 30 regular square faces, 12 regular pentagonal faces, 60 vertices and 120 edges.
The name ''rhombicosidodecahedron'' 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.
It can also called a ''cantellated dodecahedron'' or a ''cantellated icosahedron'' from truncation operations of the uniform polyhedron.
__TOC__

Contents

Geometric relations

Area and volume

Cartesian coordinates

Vertex arrangement

See also

References

External links

Geometric relations

If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosadodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.
The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of 6 or 12 pentagrammic prisms.
The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.

Area and volume

The area ''A'' and the volume ''V'' of a rhombicosidodecahedron of edge length ''a'' are:
:
A & = left { 30 + sqrt{ 30 left [ 10 + 3sqrt{5} + sqrt{15 (5 + 2sqrt{5})}
ight ] }
ight } a^2 \
& pprox 59.3059828a^2 \
V & = rac{1}{3} (60+29sqrt{5})a^3 pprox 41.6153238a^3 \
end{align}

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2, centered at the origin, are
: (±1, ±1, ±τ³),
: (±τ³, ±1, ±1),
: (±1, ±τ³, ±1),
: (±τ², ±τ, ±2τ),
: (±2τ, ±τ², ±τ),
: (±τ, ±2τ, ±τ²),
: (±(2+τ), 0, ±τ²),
: (±τ², ±(2+τ), 0),
: (0, ±τ², ±(2+τ)),
where τ = (1+√5)/2 is the golden ratio.

Vertex arrangement

The rhombicosidodecahedron shares its vertex arrangement with 3 nonconvex uniform polyhedrons:

Small dodecicosidodecahedron

Small rhombidodecahedron

Small stellated truncated dodecahedron

See also

★

★ dodecahedron

★ icosahedron

★ icosidodecahedron

★ rhombicuboctahedron

★ truncated icosidodecahedron (great rhombicosidodecahedron)

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