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Rhombicosidodecahedron
Rhombicosidodecahedron
(Click here for rotating model)
Type Archimedean solid
Elements F = 62, E = 120, V = 60 (χ = 2)
Faces by sides 20{3}+30{4}+12{5}
Schläfli symbol r\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}
Wythoff symbol 3 5 | 2
Coxeter-Dynkin CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW ring.png
Symmetry Ih
or (*532)
References U27, C30, W14
Properties Semiregular convex
Rhombicosidodecahedron color
Colored faces
Rhombicosidodecahedron
3.4.5.4
(Vertex figure)
Deltoidalhexecontahedron.jpg
Deltoidal hexecontahedron
(dual polyhedron)
Rhombicosidodecahedron Net
Net

In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

It has 20 regular triangular faces, 30 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.

Contents

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.

Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolas: the gyrate, parabigyrate, metabigyrate and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolas, sometimes also rotating one or more of the other cupolas.

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are

(±1, ±1, ±φ3),
(±φ3, ±1, ±1),
(±1, ±φ3, ±1),
(±φ2, ±φ, ±2φ),
(±2φ, ±φ2, ±φ),
(±φ, ±2φ, ±φ2),
(±(2+φ), 0, ±φ2),
(±φ2, ±(2+φ), 0),
(0, ±φ2, ±(2+φ)),

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

Related polyhedra

The rhombicosidodecahedron shares its vertex arrangement with 3 nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).

It also shares its vertex arrangement with the uniform compounds of 6 or 12 pentagrammic prisms.

Small rhombicosidodecahedron.png
Rhombicosidodecahedron
Small dodecicosidodecahedron.png
Small dodecicosidodecahedron
Small rhombidodecahedron.png
Small rhombidodecahedron
Small stellated truncated dodecahedron.png
Small stellated truncated dodecahedron
UC36-6 pentagrammic prisms.png
Compound of six pentagrammic prisms
UC37-12 pentagrammic prisms.png
Compound of twelve pentagrammic prisms

See also

References

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

External links

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