Diminished rhombicosidodecahedron

Diminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Dirid
Elements
Faces	5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, 1 decagon
Edges	5+5+5+5+5+5+5+10+10+10+10+10+10+10
Vertices	5+5+5+10+10+10+10
Vertex figures	45 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	10 scalene triangles, edge lengths √2, (1+√5)/2, √(5+√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295$
Volume	$\frac{115+54\sqrt5}{6} ≈ 39.29128$
Dihedral angles	3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	4–10: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$
	5–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
Central density	1
Related polytopes
Army	Dirid
Regiment	Dirid
Dual	Stellated deltoidal hexecontahedron
Conjugate	Replenished quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The diminished rhombicosidodecahedron is one of the 92 Johnson solids (J₇₆). It consists of 5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, and 1 decagon. It can be constructed by removing one of the pentagonal cupolaic caps of the small rhombicosidodecahedron.

Vertex coordinates

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

• $\left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),$
• $\left(0,\,±\frac{3+\sqrt5}{4},\,-\frac{5+\sqrt5}{4}\right),$
• $\left(0,\,-\frac{3+\sqrt5}{4},\,\frac{5+\sqrt5}{4}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),$
• $\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,-\frac12\right),$
• $\left(±\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right).$

External links

• Klitzing, Richard. "dirid".

• Quickfur. "The Diminished Rhombicosidodecahedron".

• Wikipedia Contributors. "Diminished rhombicosidodecahedron".
• McCooey, David. "Diminished Rhombicosidodecahedron"