Trigyrate rhombicosidodecahedron

Trigyrate rhombicosidodecahedron
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Tagyrid
Elements
Faces	1+1+3+3+6+6 triangles, 3+3+3+3+6+6+6 squares, 3+3+3+3 pentagons
Edges	8×3+16×6
Vertices	3+3+3+3+6+6+6+6+6+6+6+6
Vertex figures	15+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
Volume
Dihedral angles	3–4:
	3–5:
	4–4:
	4–5:
Central density	1
Related polytopes
Army	Tagyrid
Regiment	Tagyrid
Dual	Trideltogyrate deltoidal hexecontahedron
Conjugate	Trigyrate quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

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[edit | edit source]

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

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