# Great icosahedron

Great icosahedron
Rank3
TypeRegular
Notation
Bowers style acronymGike
Coxeter diagramo5/2o3x ()
Schläfli symbol${\displaystyle \{3,5/2\}}$
Elements
Faces20 triangles
Edges30
Vertices12
Vertex figurePentagram, edge length 1
Petrie polygons6 skew decagrams
Holes12 pentagrams
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Edge radius${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Inradius${\displaystyle {\frac {3{\sqrt {3}}-{\sqrt {15}}}{12}}\approx 0.11026}$
Volume${\displaystyle 5{\frac {3-{\sqrt {5}}}{12}}\approx 0.31831}$
Dihedral angle${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
Central density7
Number of external pieces180
Level of complexity9
Related polytopes
ArmyIke, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentSissid
DualGreat stellated dodecahedron
Petrie dualPetrial great icosahedron
φ 2 Small stellated dodecahedron
ConjugateIcosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count120
Euler characteristic2
Schläfli type{3,5}
OrientableYes
Genus0
SkeletonIcosahedral graph
Properties
SymmetryH3, order 120
Flag orbits1
ConvexNo
NatureTame

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

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
Skew icosahedron
8
8
11

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[1] 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:

### 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} ${\displaystyle 1}$
Grand hecatonicosachoron {5,3,5/2} ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$