Tortuous tunnel

Tortuous tunnel
(3D model)
Rank	3
Type	Quasi-convex Stewart toroid
Space	Spherical
Notation
Stewart notation	M, Q3Q3/S3S3
Elements
Faces	6 squares, 6+6+6 triangles
Edges	3+3+3+6+12+12
Vertices	3+6+6
Measures (edge length 1)
Volume	$\sqrt2 \approx 1.41421$
Surface area	$\frac{9\sqrt3}{2}+6 \approx 13.79423$
Related polytopes
Convex hull	Triangular orthobicupola
Abstract & topological properties
Flag count	156
Euler characteristic	0
Surface	Torus
Orientable	Yes
Genus	1
Properties
Symmetry	A2×A1, order 12
Convex	No

The tortuous tunnel or M is a quasi-convex Stewart toroid. It can be made by excavating 2 octahedra from a triangular orthobicupola. It has the fewest vertices of any known quasi-convex Stewart toroid, with 15.

Vertex coordinates

A tortuous tunnel with edge length 1 has the following vertex coordinates:

• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,\pm\frac{\sqrt6}{3}\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,\pm\frac{\sqrt6}{3}\right)$,
• $\left(\pm\frac12,\,\pm\frac{\sqrt3}{2},\,0\right)$,
• $\left(\pm1,\,0,\,0\right)$,
• $\left(\pm\frac12,\,\frac{\sqrt3}{6},\,0\right)$,
• $\left(0,\,-\frac{\sqrt3}{3},\,0\right)$.

Gallery

• 

  A view of the tunnel.

External links

• Doskey, Alex. "Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1".
• McNeil, Jim. "Simple Stewart toroids".

Bibliography

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