Gyrate bidiminished rhombicosidodecahedron

Gyrate bidiminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gybadrid
Elements
Faces	4×1+3×2 triangles, 4×1+8×2 squares, 4×1+3×2 pentagons, 2 decagons
Edges	8×1+41×2
Vertices	4×1+23×2
Vertex figures	5+15 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	20 scalene triangles, edge lengths √2, (1+√5)/2, √(5+√5)/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	$5\frac{11+5\sqrt5}{3} ≈ 36.96723$
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: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	4–10: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$
	5–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
Central density	1
Related polytopes
Army	Gybadrid
Regiment	Gybadrid
Dual	Gyrate bistellated deltoidal hexecontahedron
Conjugate	Gyrate bireplenished quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A1×I×I, order 2
Convex	Yes
Nature	Tame

The gyrate bidiminished rhombicosidodecahedron is one of the 92 Johnson solids (J₈₂). It consists of 4×1+3×2 triangles, 4×1+8×2 squares, 4×1+3×2 pentagons, and 2 decagons. It can be constructed by removing two non-opposite pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating a third cap by 36°.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "gybadrid".

• Quickfur. "The Gyrate Bidiminished Rhombicosidodecahedron".

• Wikipedia Contributors. "Gyrate bidiminished rhombicosidodecahedron".
• McCooey, David. "Bigyrate Diminished Rhombicosidodecahedron"