# Cake pan

Cake pan
Rank3
TypeQuasi-convex Stewart toroid
Notation
Stewart notationQ3P6/P3Q3
Elements
Faces3+3+3+3+3 squares, 3+3 triangles
Edges3+3+3+3+3+3+3+6+6+6
Vertices3+3+6+6
Measures (edge length 1)
Volume${\displaystyle {\frac {5{\sqrt {3}}}{4}}\approx 2.16506}$
Surface area${\displaystyle {\frac {3{\sqrt {3}}}{2}}+15\approx 17.59808}$
Related polytopes
Convex hullElongated triangular cupola
Abstract & topological properties
Flag count156
Euler characteristic0
SurfaceTorus
OrientableYes
Genus1
Properties
SymmetryA2×I, order 6
Flag orbits26
ConvexNo

The cake pan is a quasi-convex Stewart toroid. It is a tunnelling of a elongated triangular cupola by a triangular cupola and a triangular prism.

It has 21 faces, the fewest faces of any known Stewart toroid.

## Vertex coordinates

A cake pan with edge length 1 has the following vertex coordinates:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {\sqrt {6}}{3}}\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {6}}{3}}\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1}{2}}\right)}$.