Small ditrigonary icosidodecahedron

Small ditrigonary icosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sidtid
Coxeter diagram	x5/2o3o3*a ()
Elements
Faces	20 triangles, 12 pentagrams
Edges	60
Vertices	20
Vertex figure	Ditrigon, edge lengths 1 and (√5–1)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} \approx 0.86603$
Volume	$\frac{4\sqrt5}{3} \approx 2.98142$
Dihedral angle	$\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 142.62263^\circ$
Central density	2
Number of external pieces	72
Level of complexity	5
Related polytopes
Army	Doe, edge length $\frac{\sqrt5-1}{2}$
Regiment	Sidtid
Dual	Small triambic icosahedron
Conjugate	Great ditrigonary icosidodecahedron
Convex core	Truncated icosahedron
Abstract & topological properties
Flag count	240
Euler characteristic	-8
Orientable	Yes
Genus	5
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small ditrigonary icosidodecahedron, or sidtid, is a quasiregular uniform polyhedron. It consists of 20 equilateral triangles and 12 pentagrams, with three of each joining at a vertex.

It can be constructed as a holosnub dodecahedron. The pentagrammic faces lie in the same planes as the pentagons of the convex hull dodecahedron, and the triangles are the dodecahedron's vertex figures.

This polyhedron is the vertex figure of the small ditrigonary hexacosihecatonicosachoron.

Vertex coordinates

A small ditrigonary icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

• $\left(±\frac12,\,±\frac12,\,±\frac12\right),$

and even permutations of

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

The first set of vertices correspond to those of an inscribed unit cube. This relates to the fact that a uniform compound of 5 cubes has the same vertices and edges as this polyhedron.

Representations

A small ditrigonary icosidodecahedron has the following Coxeter diagrams:

• x5/2o3o3*a
• ß5o3o (as holosnub)

Related polyhedra

The small ditrigonary icosidodecahedron is the colonel of a three-member regiment that also includes the ditrigonary dodecadodecahedron and the great ditrigonary icosidodecahedron. This regiment also contains the rhombihedron, the uniform compound of 5 cubes. The pentagrammic cuploid and pentagonal cuploid are contained within the edge structure.

o5/2o3o3*a truncations

Name	OBSA	CD diagram	Picture
Small ditrigonary icosidodecahedron	sidtid	x5/2o3o3*a ()
(degenerate, double cover of id)		x5/2x3o3*a ()
(degenerate, double cover of ike)		o5/2o3x3*a ()
Small icosicosidodecahedron	siid	x5/2o3x3*a ()
(degenerate, double cover of ti)		x5/2x3x3*a ()
Small snub icosicosidodecahedron	seside	s5/2s3s3*a ()

External links

• Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#33).

• Klitzing, Richard. "sidtid".

• Wikipedia Contributors. "Small ditrigonal icosidodecahedron".
• McCooey, David. "Small Ditrigonal Icosidodecahedron"