Non-Regular Toroidal Solids

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Hexagonal Antiprism-Trapezohedron 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+2*sqrt(3))/3)    ≈32.060660106 degrees
Dihedral Angle 2 (12):  acos(−sqrt(3)/3)    ≈125.264389683 degrees
Dihedral Angle 3 (12):  acos(−(2*sqrt(3)−1)/3)    ≈145.221891332 degrees
Dihedral Angle 4 (12):  acos(−sqrt(3)/3)    ≈234.735610317 degrees
Dual Toroid:  Hexagonal Antiprism-Trapezohedron Toroid (itself)
(values below based on a Hexagonal Antiprism with edge length = 1)
Short Edge (12):  sqrt(3*sqrt(3)−5)    ≈0.44289098286895826186
Medium Edge (12):  sqrt(2*(sqrt(3)−1))/2    ≈0.60500033370605560912
Long Edge (24):  1
[3]-Vertex Radius (12):  sqrt(3*(sqrt(3)−1))/2    ≈0.74097105589669153429
[5]-Vertex Radius (12):  sqrt(3+sqrt(3))/2    ≈1.0876638735805374369
Volume:sqrt(6*(3*sqrt(3)−5))/2    ≈0.54242845985433675517


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