Small ditrigonary icosidodecahedral antiprism

The small ditrigonary icosidodecahedral antiprism or sidtidap, is a nonconvex uniform polychoron that consists of 2 small ditrigonary icosidodecahedra, 12 pentagrammic antiprisms, and 40 tetrahedra. Each vertex joins 1 small ditrigonary icosidodecahedron, 3 pentagrammic antiprisms, and 4 tetrahedra.

Small ditrigonary icosidodecahedral antiprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sidtidap
Coxeter diagram	β2β5o3o ()
Elements
Cells	40 tetrahedra, 12 pentagrammic antiprisms, 2 small ditrigonary icosidodecahedra
Faces	40+120 triangles, 24 pentagrams
Edges	60+120
Vertices	40
Vertex figure	Triangular cupola, edge lengths (√5–1)/2 (3 base edges), 1 (remaining edges)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106$
Hypervolume	$\frac{\sqrt{110+85\sqrt5}}{8} ≈ 2.16530$
Dichoral angles	Tet–3–stap: $\arccos\left(-\sqrt{\frac{5+3\sqrt5}{20}}\right) ≈ 139.91776^\circ$
	Sidtid–3–tet: $\arccos\left(-\frac{\sqrt{7-3\sqrt5}}{2}\right) ≈ 105.66967^\circ$
	Sidtid–5/2–stap: 90°
Height	$\sqrt{\frac{\sqrt5-1}{2}} ≈ 0.78615$
Number of external pieces	174
Level of complexity	32
Related polytopes
Army	Semi-uniform Dope
Regiment	Sidtidap
Dual	Small triambic icosahedral antitegum
Conjugate	None
Abstract & topological properties
Euler characteristic	–10
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	No
Nature	Tame

It can be obtained as a holosnub dodecahedral prism.

Cross-sections

Vertex coordinates

The vertices of a small ditrigonary icosidodecahedral antiprism of edge length 1 are given by:

• $\left(±\frac12,\,±\frac12,\,±\frac12,\,±\sqrt{\frac{\sqrt5-1}{8}}\right),$

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

• $\left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0,\,±\sqrt{\frac{\sqrt5-1}{8}}\right).$

Related polychora

The regiment of the small ditrigonary icosidodecahedral antiprism also includes the ditrigonary dodecadodecahedral antiprism and the great ditrigonary icosidodecahedral antiprism.

The small icosicosidodecahedral alterprism is a partial Stott expansion of this polychoron.

External links

• Bowers, Jonathan. "Category 20: Miscellaneous" (#966).

• Klitzing, Richard. "sidtidap".