Small stellated dodecahedron

Small stellated dodecahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Sissid
Coxeter diagram	x5/2o5o ()
Schläfli symbol	$\{5/2,5\}$ $\{5,5\mid 3\}$
Elements
Faces	12 pentagrams
Edges	30
Vertices	12
Vertex figure	Pentagon, edge length (√5–1)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt5}{8}} ≈ 0.58779$
Edge radius	$\frac{\sqrt5-1}{4} ≈ 0.30902$
Inradius	$\sqrt{\frac{5-\sqrt5}{40}} ≈ 0.26287$
Volume	$\frac{3\sqrt5-5}{4} ≈ 0.42705$
Dihedral angle	$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$
Central density	3
Number of pieces	60
Level of complexity	3
Related polytopes
Army	Ike
Regiment	Sissid
Dual	Great dodecahedron
Petrie dual	Petrial small stellated dodecahedron
Conjugate	Great dodecahedron
Convex core	Dodecahedron
Abstract properties
Flag count	120
Euler characteristic	-6
Schläfli type	{5,5}
Topological properties
Surface	Bring's surface
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small stellated dodecahedron, or sissid, is one of the four Kepler-Poinsot solids. It has 12 pentagrams as faces, joining 5 to a vertex.

It is the first stellation of a dodecahedron, from which its name is derived.

Vertex coordinates

The vertices of a small stellated dodecahedron of edge length 1, centered at the origin, are all cyclic permutations of

• $\left(0,\,±\frac{1}{2},\,±\frac{\sqrt{5}-1}{4}\right).$

In vertex figures

The small stellated dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name	Picture	Schläfli symbol	Edge length
Great faceted hexacosichoron		{3,5/2,5}	$1$
Great hecatonicosachoron		{5,5/2,5}	$\frac{1+\sqrt{5}}{2}$

Related polyhedra

The small stellated dodecahedron is the colonel of a two-member regiment that also includes the great icosahedron.

Two uniform polyhedron compounds are composed of small stellated dodecahedra:

• Pentagrammatic snub pseudodisoctahedron (2)
• Pentagrammatic snub pseudicosicosahedron (5)

o5o5/2o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Great dodecahedron	gad	{5,5/2}	x5o5/2o ()
Truncated great dodecahedron	tigid	t{5,5/2}	x5x5/2o ()
Dodecadodecahedron	did	r{5,5/2}	o5x5/2o ()
Truncated small stellated dodecahedron (degenerate, triple cover of doe)		t{5/2,5}	o5x5/2x ()
Small stellated dodecahedron	sissid	{5/2,5}	o5o5/2x ()
Rhombidodecadodecahedron	raded	rr{5,5/2}	x5o5/2x ()
Truncated dodecadodecahedron (degenerate, sird+12(10/2))		tr{5,5/2}	x5x5/2x ()
Snub dodecadodecahedron	siddid	sr{5,5/2}	s5s5/2s ()

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#7).

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#1 under sissid).

• Klitzing, Richard. "Sissid".

• Nan Ma. "Small stellated dodecahedron {5/2, 5}".

• Wikipedia Contributors. "Small stellated dodecahedron".
• McCooey, David. "Small Stellated Dodecahedron"

• Hartley, Michael. "{5,5}*120".

Bibliography

• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.