S5S5

S5S5
Rank	3
Type	Acrohedron
Notation
Coxeter diagram	xox5oxo&#xt
Stewart notation	S5S5
Elements
Faces	10+10 triangles, 2 pentagons
Edges	5+10+20
Vertices	5+10
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	No

S₅S₅ or double-ess-five is a non-convex regular faced polyhedron. It can be formed as an outer-blend of two pentagonal antiprisms.

Vertex coordinates

Vertex coordinates for S₅S₅ with unit side length can be given as

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)$,
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0\right)$,
• $\left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)$.

Related polyhedra

The prize substitute is the member of the cult of the double-ess-five with the fewest faces.

See also: Cult of the double-ess-five

The relative positioning of its pentagonal faces makes S₅S₅ useful for tunnelling Stewart toroids. In many cases S₅S₅ can be substituted for another polyhedron with the same configuration of pentagonal faces.

Bibliography

• Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.