Great icosahedron

Great icosahedron
(3D model) (OFF file)
Rank	3
Type	Regular
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
Petrie polygons	6 skew decagrams
Holes	12 pentagrams
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779$
Edge radius	${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
Inradius	${\frac {3{\sqrt {3}}-{\sqrt {15}}}{12}}\approx 0.11026$
Volume	$5{\frac {3-{\sqrt {5}}}{12}}\approx 0.31831$
Dihedral angle	$\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$
Central density	7
Number of external pieces	180
Level of complexity	9
Related polytopes
Army	Ike, edge length ${\frac {{\sqrt {5}}-1}{2}}$
Regiment	Sissid
Dual	Great stellated dodecahedron
Petrie dual	Petrial great icosahedron
φ 2	Small stellated dodecahedron
Conjugate	Icosahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	120
Euler characteristic	2
Schläfli type	{3,5}
Orientable	Yes
Genus	0
Skeleton	Icosahedral graph
Properties
Symmetry	H3, order 120
Flag orbits	1
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.

Great icosahedra appear as cells in only one of the regular star polychora, namely the great faceted hexacosichoron.

Vertex coordinates

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

Related polytopes

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.

Alternative realizations

Symmetric realizations of {3,5}. Click on a node to be taken to the page for that realization.

• : Faithful realization
• : Non-faithful realization
• : Blend with the icosahedron.
• : Blend with the great icosahedron.
• : Blend with the hemiicosahedron.

The great icosahedron is conjugate to the icosahedron. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {3,5}. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The icosahedron and the great icosahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with the hemiicosahedron.

Faithful symmetric realizations of {3,5}

Dimension	Components	Name
3	Icosahedron	Icosahedron
3	Great icosahedron	Great icosahedron
6	Icosahedron Great icosahedron	Skew icosahedron
8	Icosahedron Hemiicosahedron
8	Great icosahedron Hemiicosahedron
11	Icosahedron Great icosahedron Hemiicosahedron

There are also realizations that are faithful but not symmetric. The particular case of 3-dimensional realizations with regular faces are called isomorphs. They have been investigated by Jim McNeill and others. For example, one of the pyramids of the icosahedron can be inverted, producing an irregular polyhedron that is concave but with no intersections.

Compounds

Two uniform polyhedron compounds are composed of great icosahedra:

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

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}	${\frac {1+{\sqrt {5}}}{2}}$

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"
• Hartley, Michael. "{3,5}*120".