Great icosahedron
Great icosahedron | |
---|---|
Rank | 3 |
Type | Regular |
Space | Spherical |
Notation | |
Bowers style acronym | Gike |
Coxeter diagram | o5/2o3x () |
Schläfli symbol | |
Elements | |
Faces | 20 triangles |
Edges | 30 |
Vertices | 12 |
Vertex figure | Pentagram, edge length 1 |
Measures (edge length 1) | |
Circumradius | |
Edge radius | |
Inradius | |
Volume | |
Dihedral angle | |
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 | H_{3}, 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[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 | {3,3,5/2} | ||
Grand hecatonicosachoron | {5,3,5/2} |
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.
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[edit | edit source]
- 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"