Non-Regular Toroidal Solids

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Cube-Octahedron Toroid
Vertices:  12  (6[3] + 6[5])
Faces:12  (6 equilateral triangles + 6 mirror-symmetric pentagons)
Edges:24  (6 short + 6 medium + 12 long)
Symmetry:  3-fold Antiprismatic  (D3v)
Dihedral Angle 1 (6):  acos(5*sqrt(3)/9)    ≈15.793169048 degrees
Dihedral Angle 2 (12):  acos(0)    90 degrees
Dihedral Angle 3 (6):  acos(−1/3)    ≈250.528779366 degrees
Dual Toroid:  Cube-Octahedron Toroid (itself)
(values below based on a Cube with edge length = 4)
Short Edge (6):  1
Medium Edge (6):  4
Long Edge (12):  3*sqrt(2)    ≈4.2426406871192851464
[5]-Vertex Radius (6):  3
[3]-Vertex Radius (6):  2*sqrt(3)    ≈3.4641016151377545871
Volume:19  [EXACT]


References:[1]Paul Gailiunas, Some Self-reciprocal Polyhedra.
[2]T. Bakos, Octahedra Inscribed in a Cube,
The Mathematical Gazette 43 (1959), 17-20.