Non-Regular Toroidal Solids

(box: x-ray)  (F: faces)  (slider: perspective)  (image: L=rotate R=zoom)  (M: metrics)

Pentagonal Antiprism-Trapezohedron Toroid
Vertices:  20  (10[3] + 10[5])
Faces:20  (10 equilateral triangles + 10 mirror-symmetric pentagons)
Edges:40  (10 short + 10 medium + 20 long)
Symmetry:  5-fold Antiprismatic  (D5v)
Dihedral Angle 1 (10):  acos(sqrt(15*(5+2*sqrt(5)))/15)    ≈37.377368141 degrees
Dihedral Angle 2 (10):  acos(−sqrt(5)/5)    ≈116.565051177 degrees
Dihedral Angle 3 (10):  acos(−sqrt(5)/3)    ≈138.189685104 degrees
Dihedral Angle 4 (10):  acos(−sqrt(5)/5)    ≈243.434948823 degrees
Dual Toroid:  Pentagonal Antiprism-Trapezohedron Toroid (itself)
(values below based on a Pentagonal Antiprism with edge length = 1)
Short Edge (10):  (3−sqrt(5))/2    ≈0.38196601125010515180
Medium Edge (10):  (sqrt(5)−1)/2    ≈0.61803398874989484820
Long Edge (20):  1
[3]-Vertex Radius (10):  (sqrt(15)−sqrt(3))/4    ≈0.53523313465963489791
[5]-Vertex Radius (10):  sqrt(2*(5+sqrt(5)))/4    ≈0.95105651629515357212
Volume:5*(sqrt(5)−1)/12    ≈0.51502832395824570684


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