# Tortuous tunnel

Tortuous tunnel
Rank3
TypeQuasi-convex Stewart toroid
SpaceSpherical
Notation
Stewart notationM,
Q3Q3/S3S3
Elements
Faces6 squares, 6+6+6 triangles
Edges3+3+3+6+12+12
Vertices3+6+6
Measures (edge length 1)
Volume${\displaystyle \sqrt2 \approx 1.41421}$
Surface area${\displaystyle \frac{9\sqrt3}{2}+6 \approx 13.79423}$
Related polytopes
Convex hullTriangular orthobicupola
Abstract & topological properties
Flag count156
Euler characteristic0
SurfaceTorus
OrientableYes
Genus1
Properties
SymmetryA2×A1, order 12
ConvexNo

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:

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