Metabigyrate rhombicosidodecahedron

Metabigyrate rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Mabgyrid
Elements
Faces	2+2+2+2+4+4+4 triangles, 1+1+2+2+4+4+4+4+4+4 squares, 2+2+2+2+4 pentagons
Edges	8×2+26×4
Vertices	4×2+13×4
Vertex figures	10+30 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	20 irregular tetragons, edge lengths 1, √2, √2, (1+√5)/2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	${\frac {60+29{\sqrt {5}}}{3}}\approx 41.61532$
Dihedral angles	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	3–5: $\arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }$
	4–4: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
Central density	1
Number of external pieces	62
Level of complexity	120
Related polytopes
Army	Mabgyrid
Regiment	Mabgyrid
Dual	Metabideltogyrate deltoidal hexecontahedron
Conjugate	Metabigyrate quasirhombicosidodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	K2×I, order 4
Convex	Yes
Nature	Tame

The metabigyrate rhombicosidodecahedron is one of the 92 Johnson solids (J₇₄). It consists of 2+2+2+2+4+4+4 triangles, 1+1+2+2+4+4+4+4+4+4 squares, and 2+2+2+2+4 pentagons. It can be constructed by rotating two non-opposite pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "mabgyrid".
• Quickfur. "The Metabigyrate Rhombicosidodecahedron".

• Wikipedia contributors. "Metabigyrate rhombicosidodecahedron".
• McCooey, David. "Metabigyrate Rhombicosidodecahedron"