Catalan Solids

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Triakis Icosahedron
Vertices:  32  (20[3] + 12[10])
Faces:60  (isosceles triangles)
Edges:90  (60 short + 30 long)
Symmetry:  Full Icosahedral  (Ih)
Dihedral Angle:  acos(−3*(8+5*sqrt(5))/61)    ≈160.612552209 degrees
Dual Solid:  Truncated Dodecahedron
(values below based on unit-edge-length Truncated Dodecahedron)
Short Edge (60):  5*(7+sqrt(5))/22    ≈2.09910635852267947646
Long Edge (30):  (5+sqrt(5))/2    ≈3.6180339887498948482
[3]-Vertex Radius (20):  5*(3*sqrt(3)+2*sqrt(15))/22    ≈2.9413907079821512843
[10]-Vertex Radius (12):  sqrt(5*(5+2*sqrt(5)))/2    ≈3.44095480117793384552
Edge-scribed Radius:  (5+3*sqrt(5))/4    ≈2.9270509831248422723
Inscribed Radius:  5*sqrt(61*(41+18*sqrt(5)))/122    ≈2.8852583129200411870
Volume:125*(19+9*sqrt(5))/44    ≈111.14946533380144110


References:[1]Eugène Catalan, Mémoire sur la Théorie des Polyèdres,
Journal de l'École polytechnique 41 (1865), 1-71, +7 plates.