Great icosahedron

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Great icosahedron
Great icosahedron.png
Bowers style acronymGike
Coxeter diagramo5/2o3x (CDel node.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schläfli symbol
Faces20 triangles
Vertex figurePentagram, edge length 1
Great icosahedron vertfig.png
Measures (edge length 1)
Edge radius
Dihedral angle
Central density7
Number of pieces180
Level of complexity9
Related polytopes
DualGreat stellated dodecahedron
Petrie dualPetrial great icosahedron
Convex coreIcosahedron
Abstract properties
Flag count120
Euler characteristic2
Schläfli type{3,5}
Topological properties
SymmetryH3, order 120

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[edit | edit source]

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

Variations[edit | edit source]

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[edit | edit source]

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

Name Picture Schläfli symbol Edge length
Grand hexacosichoron
Grand hecatonicosachoron
Schlegel wireframe 600-cell vertex-centered.png

Related polyhedra[edit | edit source]

Two uniform polyhedron compounds are composed of great icosahedra:

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 (CDel node 1.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node.png)
Great icosahedron.png
Truncated great icosahedron tiggy t{3,5/2} x3x5/2o (CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great truncated icosahedron.png
Great icosidodecahedron gid r{3,5/2} o3x5/2o (CDel node.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great icosidodecahedron.png
Truncated great stellated dodecahedron (degenerate, ike+2gad) t{5/2,3} o3x5/2x (CDel node.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Small complex icosidodecahedron.png
Great stellated dodecahedron gissid {5/2,3} o3o5/2x (CDel node.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Great stellated dodecahedron.png
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom) sicdatrid rr{3,5/2} x3o5/2x (CDel node 1.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Compound of small ditrigonal icosidodecahedron and the compound of five cubes.png
Truncated great icosidodecahedron (degenerate, ri+12(10/2)) tr{3,5/2} x3x5/2x (CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Great snub icosidodecahedron gosid sr{3,5/2} s3s5/2s (CDel node h.pngCDel 3.pngCDel node h.pngCDel 5-2.pngCDel node h.png)
Great snub icosidodecahedron.png

External links[edit | edit source]