Tridiminished rhombicosidodecahedron

Tridiminished rhombicosidodecahedron
	(3D model); (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Tedrid
Elements
Faces	1+1+3 triangles, 3+3+3+6 squares, 3+3+3 pentagons, 3 decagons
Edges	3+3+3+3+3+3+3+6+6+6+6+6+6+6+6+6
Vertices	3+3+3+6+6+6+6+6+6
Vertex figures	15 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
Volume
Dihedral angles	3–4:
	4–5:
	4–10:
	5–10:
Central density	1
Number of external pieces	32
Level of complexity	50
Related polytopes
Army	Tedrid
Regiment	Tedrid
Dual	Tristellated deltoidal hexecontahedron
Conjugate	Trireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count	300
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

The tridiminished rhombicosidodecahedron 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[edit | edit source]

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

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

Tridiminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Tedrid
Elements
Faces	1+1+3 triangles, 3+3+3+6 squares, 3+3+3 pentagons, 3 decagons
Edges	3+3+3+3+3+3+3+6+6+6+6+6+6+6+6+6
Vertices	3+3+3+6+6+6+6+6+6
Vertex figures	15 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	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	${\frac {105+46{\sqrt {5}}}{6}}\approx 34.64319$
Dihedral angles	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	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	32
Level of complexity	50
Related polytopes
Army	Tedrid
Regiment	Tedrid
Dual	Tristellated deltoidal hexecontahedron
Conjugate	Trireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count	300
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A₂×I, order 6
Convex	Yes
Nature	Tame