# Small rhombicosidodecahedron

(Redirected from Srid)
Small rhombicosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymSrid
Coxeter diagramx5o3x ()
Conway notationeD
Stewart notationE5
Elements
Faces20 triangles, 30 squares, 12 pentagons
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge lengths 1, 2, (1+5)/2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle {\frac {60+29{\sqrt {5}}}{3}}\approx 41.61532}$
Dihedral angles4–3: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
5–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Central density1
Number of external pieces62
Level of complexity4
Related polytopes
ArmySrid
RegimentSrid
DualDeltoidal hexecontahedron
ConjugateQuasirhombicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits4
ConvexYes
NatureTame

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

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

• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$,

along with all even permutations of

• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$.

## Representations

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)
• xoF(Af)V(xB)F(xB)V(Af)Fox xFf(oV)F(Bx)A(Bx)F(oV)fFx&#xt (K2 axial, square-first)

## Semi-uniform variant

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 side length y, and 30 rectangles as faces.

With edges of length a (of pentagons) and b (of triangles), its circumradius is given by ${\displaystyle {\sqrt {\frac {9a^{2}+5b^{2}+8ab+(3a^{2}+b^{2}+4ab){\sqrt {5}}}{8}}}}$ and its volume is given by ${\displaystyle {\frac {15a^{3}+45a^{2}b+15ab^{2}+5b^{2}}{4}}+(21a^{3}+45a^{2}b+45ab^{2}+5b^{3}){\frac {\sqrt {5}}{12}}}$.

It has coordinates given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {b}{2}},\,\pm {\frac {a}{2}},\,\pm {\frac {a\varphi +b}{2}}\varphi \right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {a+b\varphi }{2}},\,\pm {\frac {a\varphi ^{2}+b}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {a\varphi }{2}},\,\pm {\frac {a\varphi +b}{2}},\,\pm {\frac {a+b}{2}}\varphi \right)}$.

where ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$.

The variant where the pentagons have edge lengths ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ times those of the triangles is the direct rectification of the icosidodecahedron.

## Related polyhedra

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: