Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with C8v Symmetry
(Octagonal Pyramid)
Vertices:  9  (8[3] + 1[8])
Faces:9  (8 isosceles triangles + 1 regular octagon)
Edges:16  (8 short + 8 long)
Symmetry:  8-fold Pyramidal  (C8v)
Dual Solid:  Octagonal Pyramid (itself)
(values below based on edge-scribed radius = 1)
Short Edge (8):  2*sqrt(2*(sqrt(2*(2−sqrt(2)))−1))    ≈0.81187289789667923758
Long Edge (8):  2*sqrt(2+sqrt(2)+2*sqrt(2+sqrt(2)))    ≈5.3328160262353855454
Edge-scribed Radius:  1
Volume:32*(4+2*sqrt(2)−sqrt(2*(10+7*sqrt(2))))/3    ≈5.5443526927817601854