Metagyrate diminished rhombicosidodecahedron

Metagyrate diminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Magydrid
Elements
Faces	3×1+6×2 triangles, 3×1+11×2 squares, 3×1+4×2 pentagons, 1 decagon
Edges	7×1+49×2
Vertices	3×1+26×2
Vertex figures	5+30 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	10 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	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	${\frac {115+54{\sqrt {5}}}{6}}\approx 39.29128$
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: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	4–10: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
	5–10: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	52
Level of complexity	210
Related polytopes
Army	Magydrid
Regiment	Magydrid
Dual	Metadeltogyrate stellated deltoidal hexecontahedron
Conjugate	Metagyrate replenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count	420
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A1×I×I, order 2
Convex	Yes
Nature	Tame

The metagyrate diminished rhombicosidodecahedron is one of the 92 Johnson solids (J₇₈). It consists of 3×1+6×2 triangles, 3×1+11×2 squares, 3×1+4×2 pentagons, and 1 decagon. It can be constructed by removing one of the pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating another non-opposite cap by 36°.

Vertex coordinates

A metagyrate diminished rhombicosidodecahedron of edge length 1 has vertices given by:

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

External links

• Klitzing, Richard. "magydrid".
• Quickfur. "The Metagyrate Diminished Rhombicosidodecahedron".

• Wikipedia contributors. "Metagyrate diminished rhombicosidodecahedron".
• McCooey, David. "Metagyrate Diminished Rhombicosidodecahedron"