Sceptre

Sceptre
Rank3
Notation
Stewart notationS*
Elements
Faces
Edges11×10+8×5
Vertices3×10+8×5
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
Flag orbits60
ConvexNo

The sceptre or S* is a regular faced polyhedron. It is formed by blending four pentagonal cupolae, three pentagonal antiprisms, and a dodecahedron.

It is meant to be excavated from a great rhombicosidodecahedron, thus its important feature is the height between its two decagonal faces. Other polyhedra of the same height can be used in its place, for example, a pentagonal rotunda can replace a pentagonal cupola and a pentagonal antiprism.

Vertex coordinates

The vertex coordinates of a sceptre, with the decagonal face nearest the dodecahedron centered at the origin and with edge length 1, are:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,{\sqrt {25+10{\sqrt {5}}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,{\sqrt {25+10{\sqrt {5}}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,{\sqrt {25+10{\sqrt {5}}}}\right)}$.

Bibliography

• Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.
• Jim McNeill. "K5 Galaxy"