Tortuous tunnel

Tortuous tunnel
Rank3
TypeQuasi-convex Stewart toroid
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${\displaystyle {\sqrt {2}}\approx 1.41421}$
Surface area${\displaystyle {\frac {9{\sqrt {3}}}{2}}+6\approx 13.79423}$
Related polytopes
Convex hullTriangular orthobicupola
Abstract & topological properties
Flag count156
Euler characteristic0
SurfaceTorus
OrientableYes
Genus1
Properties
SymmetryA2×A1, order 12
Flag orbits13
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:

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