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(5^{3})/2 |
Wythoff symbol | ^{5}⁄_{2} | 2 3 |
Coxeter diagram | |
Symmetry group | I_{h}, H_{3}, [5,3], (*532) |
References | U_{53}, C_{69}, W_{41} |
Properties | Regular nonconvex deltahedron |
(3^{5})/2 (Vertex figure) |
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 . 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)-D simplex faces of the core nD 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.
Images[edit]
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. |
It is a stellation of the icosahedron, counted by Wenninger as model [W41] and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter. |
× 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) |
As a snub[edit]
The great icosahedron can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry: . 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): . It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or , and is called a retrosnub octahedron.
Tetrahedral | Pyritohedral |
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Related polyhedra[edit]
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.
Name | Great stellated dodecahedron |
Truncated great stellated dodecahedron | Great icosidodecahedron |
Truncated great icosahedron |
Great icosahedron |
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Coxeter-Dynkin diagram |
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Picture |
References[edit]
- ^ Klitzing, Richard. "uniform polyhedra Great icosahedron".
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
- Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 0676126. (1st Edn University of Toronto (1938))
- H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp. 96–104
External links[edit]
Notable stellations of the icosahedron | |||||||||
Regular | Uniform duals | Regular compounds | Regular star | Others | |||||
(Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron | 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 polyhedra and compounds with icosahedral symmetry. |