Gyrate rhombicosidodecahedron

Gyrate rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gyrid
Elements
Faces	5+5+5+5 triangles, 5+5+5+5+10 squares, 1+1+5+5 pentagons
Edges	5+5+5+5+5+5+5+5+10+10+10+10+10+10+10+10
Vertices	5+5+5+5+10+10+10+10
Vertex figures	5+45 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	10 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 angles	3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	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 density	1
Related polytopes
Army	Gyrid
Regiment	Gyrid
Dual	Deltogyrate deltoidal hexecontahedron
Conjugate	Gyrate quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The gyrate rhombicosidodecahedron is one of the 92 Johnson solids (J₇₂). It consists of 5+5+5+5 triangles, 5+5+5+5+10 squares, and 1+1+5+5 pentagons. It can be constructed by rotating one of the pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "gyrid".

• Quickfur. "The Gyrate Rhombicosidodecahedron".

• Wikipedia Contributors. "Gyrate rhombicosidodecahedron".
• McCooey, David. "Gyrate Rhombicosidodecahedron"