Great icosahedron
| Great icosahedron | |
|---|---|
| |
| Type | Kepler–Poinsot polyhedron |
| Stellation core | icosahedron |
| Elements | F = 20, E = 30 V = 12 (χ = 2) |
| Faces by sides | 20{3} |
| Schläfli symbol | {3,5⁄2} |
| Face configuration | V(53)/2 |
| Wythoff symbol | 5⁄2 | 2 3 |
| Coxeter diagram | Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node.png |
| Symmetry group | Ih, H3, [5,3], (*532) |
| References | U53, C69, W41 |
| Properties | Regular nonconvex deltahedron |
| File:Great icosahedron vertfig.svg (35)/2 (Vertex figure) |
File:Great stellated dodecahedron.png Great stellated dodecahedron (dual polyhedron) |
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node.png. It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n–1)-dimensional simplex faces of the core n-polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.
Construction
[edit | edit source]The edge length of a great icosahedron is times that of the original icosahedron.
Images
[edit | edit source]| Transparent model | Density | Stellation diagram | Net |
|---|---|---|---|
| File:GreatIcosahedron.jpg A transparent model of the great icosahedron (See also Animation) |
File:Great icosahedron cutplane.png It has a density of 7, as shown in this cross-section. |
File:Great icosahedron stellation facets.svg It is a stellation of the icosahedron, counted by Wenninger as model [W41] and the 16th of 17 stellations of the icosahedron and the 7th of 59 stellations by Coxeter. |
File:Great icosahedron net.png × 12 Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines. |
| File:Great icosahedron tiling.svg This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow) |
Formulas
[edit | edit source]For a great icosahedron with edge length E (the edge of its dodecahedron core),
As a snub
[edit | edit source]The great icosahedron can be constructed as a uniform snub, with different colored faces and only tetrahedral symmetry: File:CDel node h.pngFile:CDel 3x.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node h.pngFile:CDel 3x.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node h.png. This construction can be called a retrosnub tetrahedron or retrosnub tetratetrahedron,[1] similar to the snub tetrahedron symmetry of the icosahedron, as a partial faceting of the truncated octahedron (or omnitruncated tetrahedron): File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png. It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, File:CDel node h.pngFile:CDel 3x.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node h.pngFile:CDel 3x.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel node.png, and is called a retrosnub octahedron.
Related polyhedra
[edit | edit source]
It shares the same vertex arrangement as the regular convex icosahedron. It also shares the same edge arrangement as the small stellated dodecahedron.
A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great stellated dodecahedron.
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two great dodecahedra inscribed within and sharing the edges of the icosahedron.
References
[edit | edit source]- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (1st Edn University of Toronto (1938))
- H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)., 3.6 6.2 Stellating the Platonic solids, pp. 96–104
External links
[edit | edit source]- Weisstein, Eric W., "Great icosahedron" ("Uniform polyhedron") at MathWorld.
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Uniform polyhedra and duals
