Great icosahedron

Great icosahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Gike
Coxeter diagram	o5/2o3x ()
Schläfli symbol	$\{3,5/2\}$
Elements
Faces	20 triangles
Edges	30
Vertices	12
Vertex figure	Pentagram, edge length 1
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt5}{8}} ≈ 0.58779$
Edge radius	$\frac{\sqrt5-1}{4} ≈ 0.30902$
Inradius	$\frac{3\sqrt3-\sqrt{15}}{12} ≈ 0.11026$
Volume	$5\frac{3-\sqrt5}{12} ≈ 0.31831$
Dihedral angle	$\arccos\left(\frac{\sqrt5}{3}\right) ≈ 41.81031^\circ$
Central density	7
Number of pieces	180
Level of complexity	9
Related polytopes
Army	Ike
Regiment	Sissid
Dual	Great stellated dodecahedron
Petrie dual	Petrial great icosahedron
Conjugate	Icosahedron
Convex core	Icosahedron
Abstract properties
Flag count	120
Euler characteristic	2
Schläfli type	{3,5}
Topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great icosahedron, or gike, is one of the four Kepler–Poinsot solids. It has 20 triangles as faces, joining 5 to a vertex in a pentagrammic fashion.

It has the same edges as the small stellated dodecahedron, and the same vertices as the convex icosahedron. It is also one of the stellations of the icosahedron, and the only Kepler-Poinsot solid to be a stellation of the icosahedron as opposed to the dodecahedron.

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the small stellated dodecahedron.

Variations

The great icosahedron can also be considered to be a kind of retrosnub tetrahedron, by analogy with the snub cube and snub dodecahedron. It is the result of alternating the vertices of a degenerate uniform polyhedron with 8 degenerate hexagrams and 6 doubled-up squares and then adjusting edge lengths to be equal. It can be represented as s3/2s3/2s or s3/2s4o, with chiral tetrahedral and pyritohedral symmetry respectively, the conjugate of the icosahedron being viewed as a snub tetrahedron.

In vertex figures

The great icosahedron appears as a vertex figure of two Schläfli–Hess polychora.

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

Related polyhedra

Two uniform polyhedron compounds are composed of great icosahedra:

• Small retrosnub disoctahedron (2)
• Small retrosnub icosicosahedron (5)

The great icosahedron can be constructed by joining pentagrammic pyramids to the bases of a pentagrammic retroprism, conjugate to the icosahedron's view as a pentagonal antiprism augmented with pentagonal pyramids.

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" (#8).

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

• Klitzing, Richard. "Gike".

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

• Wikipedia Contributors. "Great icosahedron".
• McCooey, David. "Great Icosahedron"