Higher Genus Toroidal Solids

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Octagonal Dodecahedron with 8 overarching faces
Vertices:  32  ({(2 * 2) * 8}[3])
Faces:12  ({2 * 2} convex octagons + {(2 * 2) * 2} nonconvex octagons)
Edges:48 (12 different lengths)
Symmetry:  2-fold Rotoreflection  (S4)
(values below based on integer coordinates)
Edge 1 (4):  30
Edge 2 (2):  8*sqrt(34)    ≈46.647615158762403767
Edge 3 (4):  24*sqrt(5)    ≈53.665631459994952714
Edge 4 (4):  30*sqrt(5)    ≈67.082039324993690892
Edge 5 (4):  15*sqrt(41)    ≈96.046863561492730297
Edge 6 (4):  105
Edge 7 (4):  8*sqrt(241)    ≈124.19339757008018983
Edge 8 (4):  138
Edge 9 (4):  144
Edge 10 (4):  66*sqrt(5)    ≈147.58048651498611996
Edge 11 (4):  7*sqrt(561)    ≈165.79806995257815727
Edge 12 (6):  264
Volume:9075136  [EXACT]
Planes
Red: x + 132 = 0
x - 132 = 0
y + 132 = 0
y - 132 = 0
Blue: x + 2z + 276 = 0
x - 2z - 276 = 0
y + 2z - 276 = 0
y - 2z + 276 = 0
Green: 3x - 5y + 4z - 528 = 0
3x - 5y - 4z + 528 = 0
5x + 3y + 4z + 528 = 0
5x + 3y - 4z - 528 = 0


References:[1]Minimal surfaces for planar octagons and nonagons (Ivan Neretin)