Small rhombicosidodecahedral prism

Small rhombicosidodecahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sriddip
Coxeter diagram	x x5o3x ()
Elements
Cells	20 triangular prisms, 30 cubes, 12 pentagonal prisms, 2 small rhombicosidodecahedra
Faces	40 triangles, 60+60+60 squares, 24 pentagons
Edges	60+120+120
Vertices	120
Vertex figure	Isosceles trapezoidal pyramid, edge lengths 1, √2, (1+√5)/2, √2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825$
Hypervolume	$\frac{60+29\sqrt5}{3} ≈ 41.61532$
Dichoral angles	Trip–4–cube: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	Cube–4–pip: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	Srid–5–pip: 90°
	Srid–3–trip: 90°
	Srid–4–cube: 90°
Height	1
Central density	1
Number of external pieces	64
Level of complexity	16
Related polytopes
Army	Sriddip
Regiment	Sriddip
Dual	Deltoidal hexecontahedral tegum
Conjugate	Quasirhombicosidodecahedral prism
Abstract & topological properties
Flag count	3840
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

The small rhombicosidodecahedral prism or sriddip is a prismatic uniform polychoron that consists of 2 small rhombicosidodecahedra, 12 pentagonal prisms, 20 triangular prisms, and 30 cubes. Each vertex joins 1 small rhombicosidodecahedron, 1 pentagonal prism, 1 triangular prism, and 2 cubes. As the name suggests, it is a prism based on the small rhombicosidodecahedron. As such it is also a convex segmentochoron (designated K-4.111 on Richard Klitzing's list).

The small rhombicosidodecahedral prism can be vertex-inscribed into the small ditetrahedronary hexacosihecatonicosachoron.

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, srid atop srid

• 

  Net

Vertex coordinates

The vertices of a small rhombicosidodecahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

• $\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac12\right),$

along with all even permutations of the first three coordinates of:

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

Representations

The small rhombicosidodecahedral prism has the following Coxeter diagrams:

• x x5o3x (full symmetry)
• xx5oo3xx&#x (bases considered separately)

Related polychora

The pentagonal cupolic prism is a segmentochoron that can be obtained as a cap of the small rhombicosidodecahedral prism.

The regiment of the small rhombicosidodecahedral prism also includes the small dodecicosidodecahedral prism and the small rhombidodecahedral prism.

External links

• Bowers, Jonathan. "Category 19: Prisms" (#926).

• Klitzing, Richard. "Sriddip".

• Wikipedia Contributors. "Rhombicosidodecahedral prism".