| Great icosahedron | |
|---|---|
| Type | Kepler–Poinsot polyhedron |
| Stellation core | icosahedron |
| Elements | F = 20, E = 30V = 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 | |
| Symmetry group | Ih, H3, [5,3], (*532) |
| References | U53, C69, W41 |
| Properties | Regular nonconvex deltahedron |
| (35)/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)-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 7 + 3 5 2 {\displaystyle {\frac {7+3{\sqrt {5}}}{2}}} times that of the original icosahedron.
Images
| Transparent model | Density | Stellation diagram | Net |
|---|---|---|---|
| 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. | × 12Net (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 (the edge of its dodecahedron core),
Inradius = E ( 3 3 − 15 ) 4 {\displaystyle {\text{Inradius}}={\frac {{\text{E}}(3{\sqrt {3}}-{\sqrt {15}})}{4}}}
Midradius = E ( 5 − 1 ) 4 {\displaystyle {\text{Midradius}}={\frac {{\text{E}}({\sqrt {5}}-1)}{4}}}
Circumradius = E 2 ( 5 − 5 ) 4 {\displaystyle {\text{Circumradius}}={\frac {{\text{E}}{\sqrt {2(5-{\sqrt {5}})}}}{4}}}
Surface Area = 3 3 ( 5 + 4 5 ) E 2 {\displaystyle {\text{Surface Area}}=3{\sqrt {3}}(5+4{\sqrt {5}}){\text{E}}^{2}}
Volume = 25 + 9 5 4 E 3 {\displaystyle {\text{Volume}}={\tfrac {25+9{\sqrt {5}}}{4}}{\text{E}}^{3}}
As a snub
The great icosahedron can be constructed as 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 |
|---|---|
Related polyhedra
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 | Greatstellateddodecahedron | Truncated great stellated dodecahedron | Greaticosidodecahedron | Truncatedgreaticosahedron | Greaticosahedron |
|---|---|---|---|---|---|
| Coxeter-Dynkindiagram | |||||
| Picture |
- 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
- Weisstein, Eric W., "Great icosahedron" ("Uniform polyhedron") at MathWorld.
- Uniform polyhedra and duals
| 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 |
|---|---|---|---|---|---|---|---|---|---|
| The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. | |||||||||
References
Klitzing, Richard. "uniform polyhedra Great icosahedron". https://bendwavy.org/klitzing/dimensions/../incmats/gike.htm ↩