Small rhombicosidodecahedron

From Polytope Wiki
Jump to navigation Jump to search
Small rhombicosidodecahedron
Small rhombicosidodecahedron.png
Bowers style acronymSrid
Coxeter diagramx5o3x (CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png)
Faces20 triangles, 30 squares, 12 pentagons
Vertex figureIsosceles trapezoid, edge lengths 1, 2, (1+5)/2, 2
Small rhombicosidodecahedron vertfig.png
Measures (edge length 1)
Dihedral angles4–3:
Central density1
Number of external pieces62
Level of complexity4
Related polytopes
DualDeltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SymmetryH3, order 120

The small rhombicosidodecahedron, or srid, also commonly known as simply the rhombicosidodecahedron, is one of the 13 Archimedean solids. It consists of 20 triangles, 30 squares, and 12 pentagons, with 1 triangle, 2 squares, and 1 pentagon meeting at each vertex. It can be obtained by cantellation of the dodecahedron or icosahedron, or equivalently by expanding either polyhedron's faces outward.

Vertex coordinates[edit | edit source]

A small rhombicosidodecahedron of edge length 1 has vertex coordinates given by all permutations of

along with all even permutations of

Representations[edit | edit source]

A small rhombicosidodecahedron has the following Coxeter diagrams:

  • x5o3x (full symmetry)
  • oxxFofxx5xxfoFxxo&#xt (H2 axial, pentagon-first)
  • xx(oF)fVxF(Vx)fo3of(Vx)FxVf(oF)xx&#xt (A2 symmetry, triangle-first)

Semi-uniform variant[edit | edit source]

The small rhombicosidodecahedron has a semi-uniform variant of the form x5o3y that maintains its full symmetry. This variant has 12 pentagons of side length x, 20 triangles of size length y, and 30 rectangles as faces.

With edges of length a (of pentagons) and b (of triangles), its circumradius is given by and its volume is given by .

Related polyhedra[edit | edit source]

The small rhombicosidodecahedron is the colonel of a three-member regiment that also includes the small dodecicosidodecahedron and the small rhombidodecahedron.

It is possible to cut off a pentagonal cupola cap from the rhombicosidodecahedron to diminish it, or to gyrate any such cap by 36° (so squares connect to other squares, and triangles connect to pentagons). The various combinations of diminishings and gyrations lead to a total of 12 Johnson solids:

o5o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Dodecahedron doe {5,3} x5o3o
Uniform polyhedron-53-t0.png
Truncated dodecahedron tid t{5,3} x5x3o
Uniform polyhedron-53-t01.png
Icosidodecahedron id r{5,3} o5x3o
Uniform polyhedron-53-t1.png
Truncated icosahedron ti t{3,5} o5x3x
Uniform polyhedron-53-t12.png
Icosahedron ike {3,5} o5o3x
Uniform polyhedron-53-t2.png
Small rhombicosidodecahedron srid rr{5,3} x5o3x
Uniform polyhedron-53-t02.png
Great rhombicosidodecahedron grid tr{5,3} x5x3x
Uniform polyhedron-53-t012.png
Snub dodecahedron snid sr{5,3} s5s3s
Uniform polyhedron-53-s012.png

External links[edit | edit source]