Great icosahedron
Great icosahedron | |
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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 |
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Symmetry group | Ih, H3, [5,3], (*532) |
References | U53, C69, W41 |
Properties | nonconvex deltahedron
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(35)/2 (Vertex figure) |
Great stellated dodecahedron (dual polyhedron) |
In
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
The edge length of a great icosahedron is times that of the original icosahedron.
Images
Transparent model | Density | Stellation diagram | Net |
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A transparent model of the great icosahedron (See also Animation) |
It has a density of 7, as shown in this cross-section. |
Coxeter .
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× 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. |
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
For a great icosahedron with edge length E,
As a snub
The great icosahedron can be constructed as a uniform
Tetrahedral | Pyritohedral |
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Related polyhedra
It shares the same
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.
Name | Great stellated dodecahedron |
Truncated great stellated dodecahedron
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Great icosidodecahedron |
Truncated great icosahedron |
Great icosahedron |
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Coxeter-Dynkin
diagram |
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Picture |
References
- ^ Klitzing, Richard. "uniform polyhedra Great icosahedron".
- ISBN 0-521-09859-9.
- MR 0676126. (1st Edn University of Toronto (1938))
- ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp. 96–104
External links
- Weisstein, Eric W., "Great icosahedron" ("Uniform polyhedron") at MathWorld.
- Uniform polyhedra and duals
Notable stellations of the icosahedron | |||||||||
Regular | Uniform duals | Regular compounds
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Regular star | Others | |||||
(Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron
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Great triambic icosahedron | Compound of five octahedra | Compound of five tetrahedra | Compound of ten tetrahedra | Great icosahedron | Excavated dodecahedron | Final stellation |
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The stellation process on the icosahedron creates a number of related compounds with icosahedral symmetry .
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