# Tridiminished rhombicosidodecahedron

Tridiminished rhombicosidodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymTedrid
Elements
Faces
Edges7×3+9×6
Vertices3+3+3+6×6
Vertex figures15 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
30 scalene triangles, edge lengths 2, (1+5)/2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle {\frac {105+46{\sqrt {5}}}{6}}\approx 34.64319}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
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 pieces32
Level of complexity50
Related polytopes
ArmyTedrid
RegimentTedrid
DualTristellated deltoidal hexecontahedron
ConjugateTrireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count300
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
Flag orbits50
ConvexYes
NatureTame

The tridiminished rhombicosidodecahedron (OBSA: tedrid) is one of the 92 Johnson solids (J83). It consists of 1+1+3 triangles, 3+3+3+6 squares, 3+3+3 pentagons, and 3 decagons. It can be constructed by removing three pentagonal cupolaic caps of the small rhombicosidodecahedron.

## Vertex coordinates

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

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