# Metagyrate diminished rhombicosidodecahedron

Metagyrate diminished rhombicosidodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymMagydrid
Elements
Faces3×1+6×2 triangles, 3×1+11×2 squares, 3×1+4×2 pentagons, 1 decagon
Edges7×1+49×2
Vertices3×1+26×2
Vertex figures5+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${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle {\frac {115+54{\sqrt {5}}}{6}}\approx 39.29128}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }}$
4–4: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–10: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
5–10: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces52
Level of complexity210
Related polytopes
ArmyMagydrid
RegimentMagydrid
ConjugateMetagyrate replenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count420
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA1×I×I, order 2
ConvexYes
NatureTame

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:

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