Great stellated dodecahedron

Great stellated dodecahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Gissid
Coxeter diagram	x5/2o3o ()
Schläfli symbol	$\{5/2,3\}$
Elements
Faces	12 pentagrams
Edges	30
Vertices	20
Vertex figure	Triangle, edge length (√5–1)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{15}-\sqrt3}{4} ≈ 0.53523$
Edge radius	$\frac{3-\sqrt5}{4} ≈ 0.19098$
Inradius	$\sqrt{\frac{25-11\sqrt5}{40}} ≈ 0.10041$
Volume	$\frac{7\sqrt5-15}{4} ≈ 0.16312$
Dihedral angle	$\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495^\circ$
Central density	7
Number of pieces	60
Level of complexity	3
Related polytopes
Army	Doe
Regiment	Gissid
Dual	Great icosahedron
Petrie dual	Petrial great stellated dodecahedron
Conjugate	Dodecahedron
Convex core	Dodecahedron
Abstract properties
Flag count	120
Euler characteristic	2
Schläfli type	{5,3}
Topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great stellated dodecahedron, or gissid, is one of the four Kepler–Poinsot solids. It has 12 pentagrams as faces, joining 3 to a vertex.

It is the last stellation of the dodecahedron, from which its name is derived. It is also the only Kepler-Poinsot solid to share its vertices with the dodecahedron as opposed to the icosahedron. It has the smallest circumradius of any uniform polyhedron.

Vertex coordinates

The vertices of a great stellated dodecahedron of edge length 1, centered at the origin, are all sign changes of

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

along with all even permutations and all sign changes of

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

The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

In vertex figures

The great stellated dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.

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

Related polyhedra

o3o5/2o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Great icosahedron	gike	{3,5/2}	x3o5/2o ()
Truncated great icosahedron	tiggy	t{3,5/2}	x3x5/2o ()
Great icosidodecahedron	gid	r{3,5/2}	o3x5/2o ()
Truncated great stellated dodecahedron (degenerate, ike+2gad)		t{5/2,3}	o3x5/2x ()
Great stellated dodecahedron	gissid	{5/2,3}	o3o5/2x ()
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom)	sicdatrid	rr{3,5/2}	x3o5/2x ()
Truncated great icosidodecahedron (degenerate, ri+12(10/2))		tr{3,5/2}	x3x5/2x ()
Great snub icosidodecahedron	gosid	sr{3,5/2}	s3s5/2s ()

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#9).

• Klitzing, Richard. "gissid".

• Nan Ma. "Great stellated dodecahedron {5/2, 3}".

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