# Oriental hat

Oriental hat
Rank3
TypeQuasi-convex Stewart toroid
Notation
Stewart notationR5/S5Q5
Elements
Faces5 pentagons, 5 squares, 5+5+5+5+5 triangles
Edges60
Vertices10+5+5+5
Measures (edge length 1)
Central density0
Related polytopes
Convex hullPentagonal rotunda
Abstract & topological properties
Flag count240
Euler characteristic0
OrientableYes
Genus1
Properties
SymmetryH2×I, order 10
Flag orbits24
ConvexNo

The oriental hat is a quasi-convex Stewart toroid. It can be made by excavating a pentagonal rotunda by a pentagonal antiprism and a pentagonal cupola.

## Vertex coordinates

The vertex coordinates for a tunnelled pentagonal rotunda with edge length 1 can be given as:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}$,
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0\right)}$.

## Related polytopes

The oriental hat can be elongated in two ways to form quasi-convex Stewart toroids, both with the elongated pentagonal rotunda as their convex hull.[1] It cannot be gyroelongated.

12 tunnelled pentagonal rotundae are used in the construction of the Webb toroid.