Grand antiprism
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| Grand antiprism | |
|---|---|
 (Schlegel diagram wireframe) |
|
| Type | Uniform polychoron |
| Cells | 100+200 (3.3.3)  20 (3.3.3.5)  |
| Faces | 20 {5} 700 {3} |
| Edges | 500 |
| Vertices | 100 |
| Vertex figure | 12 (3.3.3) 2 (3.3.3.5) (Dissected regular icosahedron) |
| Symmetry group | Ionic diminished Coxeter group [[10,2+,10]], of order 400 |
| Schläfli symbol | s{5}.s{5} (extended) |
| Properties | convex |
In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform polychoron, discovered in 1965 by Conway and Guy[1].
The vertex figure of the grand antiprism is a dissected regular icosahedron: a regular icosahedron in which a patch of 8 triangles is replaced by a pair of trapezoids, edge lengths φ, 1, 1, 1 (where φ is the golden ratio), joined together along their edge of length φ, to give a tetradecahedron whose faces are the 2 trapezoids and the 12 remaining equilateral triangles.
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[edit] Structure
The 20 pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism.
The 300 tetrahedra join the two rings to each other, and are laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder.
This structure is analogous to that of the 3-dimensional antiprisms. However, the grand antiprism is the only uniform analogue of the antiprism in 4 dimensions (except 16-cell which is regular analogue of digonal antiprism).
[edit] Construction
The grand antiprism can be constructed by diminishing the 600-cell: subtracting 20 pyramids whose bases are three-dimensional pentagonal antiprisms. Conversely, the two rings of pentagonal antiprisms in the grand antiprism may be triangulated by 10 tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of each, yielding 150 tetrahedra per ring. These combined with the 300 tetrahedra that join the two rings together yield the 600 tetrahedra of the 600-cell.
This relationship is analogous to how a pentagonal antiprism can be constructed from an icosahedron by removing two opposite vertices, thereby removing 5 triangles from the opposite 'poles' of the icosahedron, leaving the 10 equatorial triangles and two pentagons on the top and bottom.
(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids.)
[edit] Projections
These are two perspective projections, projecting the polytope into a hypersphere, and applying a stereographic projection into 3-space.
 Wireframe, viewed down one of the pentagonal antirprism columns. |
 with transparent triangular faces |
 Orthographic projection Centered on hyperplane of an antiprism in one of the two rings. |
[edit] Alternate names
- Pentagonal double antiprismoid Norman W. Johnson
- Gap (Jonathan Bowers: for grand antiprism)
[edit] See also
[edit] References
- ^ J.H. Conway and M. J. T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965. (Michael Guy is son of Richard K. Guy)
[edit] External links
- Grand antiprism, Section 5 of George Olshevsky's catalogue of uniform polychora.
- In the Belly of the Grand Antiprism (middle section, describing the analogy with the icosahedron and the pentagonal antiprism)
