# Cake pan

Cake pan
Rank3
TypeQuasi-convex Stewart toroid
SpaceSpherical
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
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 the fewest faces of any known Stewart toroid with 21 faces.

## Vertex coordinates

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

• ${\displaystyle \left(\pm\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,\frac{\sqrt{6}}{3}\pm\frac{1}{2}\right)}$,
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,\frac{\sqrt{6}}{3}\pm\frac{1}{2}\right)}$,
• ${\displaystyle \left(\pm\frac{1}{2},\,\pm\frac{\sqrt{3}}{2},\,\pm\frac{1}{2}\right)}$,
• ${\displaystyle \left(0,\,\pm1,\,\pm\frac{1}{2}\right)}$.