# S5S5

S5S5
Rank3
TypeAcrohedron
Notation
Coxeter diagramxox5oxo&#xt
Stewart notationS5S5
Elements
Faces10+10 triangles, 2 pentagons
Edges5+10+20
Vertices5+10
Abstract & topological properties
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexNo

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

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

## Related polyhedra

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.