# Trigyrate rhombicosidodecahedron

Trigyrate rhombicosidodecahedron Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymTagyrid
Elements
Faces1+1+3+3+6+6 triangles, 3+3+3+3+6+6+6 squares, 3+3+3+3 pentagons
Edges8×3+16×6
Vertices3+3+3+3+6+6+6+6+6+6+6+6
Vertex figures15+15 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
30 irregular tetragons, edge lengths 1, 2, 2, (1+5)/2
Measures (edge length 1)
Circumradius$\frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295$ Volume$\frac{60+29\sqrt5}{3} ≈ 41.61532$ Dihedral angles3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$ 3–5: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$ 4–4: $\arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°$ 4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ Central density1
Related polytopes
ArmyTagyrid
RegimentTagyrid
DualTrideltogyrate deltoidal hexecontahedron
ConjugateTrigyrate quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The trigyrate rhombicosidodecahedron is one of the 92 Johnson solids (J75). It consists of 1+1+3+3+6+6 triangles, 3+3+3+3+6+6+6 squares, and 3+3+3+3 pentagons. It can be constructed by rotating three mutually non-adjacent pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

## Vertex coordinates

A trigyrate rhombicosidodecahedron of edge length 1 has vertices given by:

• $\left(±\frac{5+\sqrt5}{4},\,0,\,\frac{3+\sqrt5}{4}\right),$ • $\left(\frac{5+\sqrt5}{4},\,0,\,-\frac{3+\sqrt5}{4}\right),$ • $\left(0,\,±\frac{3+\sqrt5}{4},\,-\frac{5+\sqrt5}{4}\right),$ • $\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$ • $\left(±\frac12,\,±\frac12,\,\frac{2+\sqrt5}{2}\right),$ • $\left(\frac12,\,±\frac12,\,-\frac{2+\sqrt5}{2}\right),$ • $\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),$ • $\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,-\frac12\right),$ • $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),$ • $\left(\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),$ • $\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),$ • $\left(±\frac12,\,±\frac{5+4\sqrt5}{10},\,\frac{10+3\sqrt5}{10}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,±\frac{5+2\sqrt5}{5},\,\frac{15+\sqrt5}{20}\right),$ • $\left(0,\,±\frac{15+13\sqrt5}{20},\,\frac{5+\sqrt5}{20}\right),$ • $\left(-\frac{10+3\sqrt5}{10},\,±\frac12,\,-\frac{5+4\sqrt5}{10}\right),$ • $\left(-\frac{15+\sqrt5}{20},\,±\frac{1+\sqrt5}{4},\,-\frac{5+2\sqrt5}{5}\right),$ • $\left(-\frac{5+\sqrt5}{20},\,0,\,-\frac{15+13\sqrt5}{20}\right).$ 