Small rhombicosidodecahedron

Small rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Srid
Coxeter diagram	x5o3x ()
Conway notation	eD
Stewart notation	E5
Elements
Faces	20 triangles, 30 squares, 12 pentagons
Edges	60+60
Vertices	60
Vertex figure	Isosceles trapezoid, edge lengths 1, √2, (1+√5)/2, √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	${\frac {60+29{\sqrt {5}}}{3}}\approx 41.61532$
Dihedral angles	4–3: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	5–4: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
Central density	1
Number of external pieces	62
Level of complexity	4
Related polytopes
Army	Srid
Regiment	Srid
Dual	Deltoidal hexecontahedron
Conjugate	Quasirhombicosidodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

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

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

along with all even permutations of

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

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 ${\sqrt {\frac {9a^{2}+5b^{2}+8ab+(3a^{2}+b^{2}+4ab){\sqrt {5}}}{8}}}$ and its volume is given by ${\frac {15a^{3}+45a^{2}b+15ab^{2}+5b^{2}}{4}}+(21a^{3}+45a^{2}b+45ab^{2}+5b^{3}){\frac {\sqrt {5}}{12}}$.

It has coordinates given by all even permutations of:

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

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

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:

• Gyrate rhombicosidodecahedron - One cap rotated
• Parabigyrate rhombicosidodecahedron - Two opposite caps rotated
• Metabigyrate rhombicosidodecahedron - Two non-opposite, non-adjacent caps rotated
• Trigyrate rhombicosidodecahedron - Three non-adjacent caps rotated
• Diminished rhombicosidodecahedron - One cap removed
• Paragyrate diminished rhombicosidodecahedron - One cap removed, the opposite cap rotated
• Metagyrate diminished rhombicosidodecahedron - One cap removed, another non-opposite, non-adjacent cap rotated
• Bigyrate diminished rhombicosidodecahedron - One cap removed, two more non-adjacent ones rotated
• Parabidiminished rhombicosidodecahedron - Two opposite caps removed
• Metabidiminished rhombicosidodecahedron - Two non-adjacent, non-opposite caps removed
• Gyrate bidiminished rhombicosidodecahedron - Two non-opposite caps removed, a third non-adjacent one rotated
• Tridiminished rhombicosidodecahedron - Three non-adjacent caps removed

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#39).

• Klitzing, Richard. "srid".
• Quickfur. "The Rhombicosidodecahedron".

• Wikipedia contributors. "Rhombicosidodecahedron".
• McCooey, David. "Rhombicosidodecahedron"

• Hi.gher.Space Wiki Contributors. "Rhodoperihedron".