Non-Regular Toroidal Solids

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Hexagonal Trapezohedron-Antiprism Toroid
Vertices:  24  (12[3] + 12[5])
Faces:24  (12 equilateral triangles + 12 mirror-symmetric pentagons)
Edges:48  (12 short + 12 medium + 24 long)
Symmetry:  6-fold Antiprismatic  (D6v)
Dihedral Angle 1 (12):  acos(sqrt(3*(38*sqrt(3)−63))/3)    ≈14.261802346 degrees
Dihedral Angle 2 (24):  acos(−sqrt(3)/3)    ≈125.264389683 degrees
Dihedral Angle 3 (12):  acos(−(2*sqrt(3)−1)/3)    ≈214.778108668 degrees
Dual Toroid:  Hexagonal Trapezohedron-Antiprism Toroid (itself)
(values below based on a Hexagonal Antiprism with edge length = 2)
Short Edge (12):  sqrt(6*(11*sqrt(3)−19))    ≈0.56156326406372182922
Medium Edge (12):  2*sqrt(211*sqrt(3)−365)    ≈1.3604710905169707931
Long Edge (24):  2
[5]-Vertex Radius (12):  sqrt(3+sqrt(3))    ≈2.1753277471610748738
[3]-Vertex Radius (12):  sqrt(3*(57*sqrt(3)−97))    ≈2.2761124959628021851
Volume:4*sqrt(6*(775643*sqrt(3)−1343453))    ≈2.8487495646855335448


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