Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with O Symmetry
(Uniform #12) (Snub Cube)
(1 of 4)
Vertices:  24  (24[5])
Faces:38  (32 equilateral triangles + 6 squares)
Edges:60
Symmetry:  Chiral Octahedral  (O)
Square-Triangle Angle:  acos(−sqrt(11−2*cbrt(17+3*sqrt(33))    
    −2*cbrt(17−3*sqrt(33)))/3)    
≈142.983430074 degrees
Triangle-Triangle Angle:  acos(−sqrt(3*(11+4*cbrt(17+3*sqrt(33))    
    +4*cbrt(17−3*sqrt(33))))/9)    
≈153.234587713 degrees
Dual Solid:  Pentagonal Icositetrahedron
(values below based on midscribed radius = 1)
Edge Length:  2*sqrt(3*(5−cbrt(19+3*sqrt(33))    
    −cbrt(19−3*sqrt(33))))/3    
≈0.801781129201327280647
Circumscribed Radius:  sqrt(3*(8−cbrt(19+3*sqrt(33))    
    −cbrt(19−3*sqrt(33))))/3    
≈1.0773640261238718692
Midscribed Radius:  1
Square Center Radius:  sqrt(3*(cbrt(19+3*sqrt(33))    
    +cbrt(19−3*sqrt(33))−2))/3    
≈0.91612594942734873530
Triangle Center Radius:  sqrt(4+cbrt(19+3*sqrt(33))    
    +cbrt(19−3*sqrt(33)))/3    
≈0.97284578346453266494
Volume:8*sqrt(3*(cbrt(1124803+344589*sqrt(33))    
    −cbrt(344589*sqrt(33)−1124803)−44))/9    
≈4.0664527252304884030