Cake pan

Cake pan
(3D model)
Rank	3
Type	Quasi-convex Stewart toroid
Space	Spherical
Notation
Stewart notation	Q3P6/P3Q3
Elements
Faces	3+3+3+3+3 squares, 3+3 triangles
Edges	3+3+3+3+3+3+3+6+6+6
Vertices	3+3+6+6
Measures (edge length 1)
Volume	$\frac{5\sqrt{3}}{4} \approx 2.16506$
Surface area	$\frac{3\sqrt{3}}{2}+15 \approx 17.59808$
Related polytopes
Convex hull	Elongated triangular cupola
Abstract & topological properties
Flag count	156
Euler characteristic	0
Surface	Torus
Orientable	Yes
Genus	1
Properties
Symmetry	A2×I, order 6
Convex	No

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:

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

Gallery

• 

  A view of the tunnel.

External links

• McNeil, Jim. "Simple Stewart toroids".

Bibliography

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