Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with C4 Symmetry
(1 of 5)
Vertices:  12  (4[3] + 4[4] + 4[5])
Faces:14  (4 isosceles triangles + 4 obtuse triangles
    + 4 irregular tetragons + {1 + 1} squares)
Edges:24  (5 different lengths)
Symmetry:  4-fold Cyclic  (C4)
Dual Solid:  Simplest C4 (2 of 5)
(values below based on edge-scribed radius = 1)
Edge 1 (4):  sqrt(root[16th-order polynomial])    ≈0.86878460231836564274
Edge 2 (8):  sqrt(root[16th-order polynomial])    ≈0.98620758716519280269
Edge 3 (4):  sqrt(root[16th-order polynomial])    ≈1.3378035176665106939
Edge 4 (4):  sqrt(root[16th-order polynomial])    ≈1.4552265025133378539
Edge 5 (4):  sqrt(root[16th-order polynomial])    ≈1.5726494873601650138
Edge-scribed Radius:  1
Volume:sqrt(root[16th-order polynomial])    ≈3.4847651004012041302