Grand antiprism

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Grand antiprism
Grand antiprism.png
(Schlegel diagram wireframe)
Type Uniform polychoron
Uniform index 47
Cells 100+200 (3.3.3) Tetrahedron.png
20 (3.3.3.5)Pentagonal antiprism.png
Faces 20 {5}
700 {3}
Edges 500
Vertices 100
Vertex figure 12 (3.3.3)
2 (3.3.3.5)
Grand antiprism verf.png
Symmetry group Ionic diminished Coxeter group [[10,2+,10]], of order 400
Schläfli symbol s{5}.s{5} (extended)
Properties convex
Grand antiprism net.png
A net showing two disjoint rings of 10 antiprisms. 200 tetrahedra (yellow) are in face contact with the antiprisms and 100 tetrahedra (red) contact only other tetrahedra.

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][2]

Alternate names[edit]

  • Pentagonal double antiprismoid Norman W. Johnson
  • Gap (Jonathan Bowers: for grand antiprism)

Structure[edit]

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. These can be further divided into three sets. 100 face mate to one ring, 100 face mate to the other ring, and 100 are centered at the exact midpoint of the duocylinder and edge mate to both rings. This latter set forms a flat torus and can be "unrolled" into a flat 10X10 square array of tetrahedra that meet only at their edges and vertices. See figure below.

100 tetrahedrons in a 10x10 array forming the clifford torus boundary in the 600 cell and grand antiprism.

In addition the 300 tetrahedra can be partitioned into 10 disjoint Boerdijk–Coxeter helices of 30 cells each that close back on each other. The two pentagonal antiprism tubes, plus the 10 BC helices, form an irregular discrete Hopf fibration of the grand antiprism that Hopf maps to the faces of a pentagonal antiprism. The two tubes map to the two pentagonal faces and the 10 BC helices map to the 10 triangular faces.

The structure of the grand antiprism is analogous to that of the 3-dimensional antiprisms. However, the grand antiprism is the only convex uniform analogue of the antiprism in 4 dimensions (although the 16-cell may be regarded as a regular analogue of the digonal antiprism). The only nonconvex uniform 4-dimensional antiprism analogue uses pentagrammic crossed-antiprisms instead of pentagonal antiprisms, and is called the pentagrammic double antiprismoid.

Vertex figure[edit]

The vertex figure of the grand antiprism is a dissected regular icosahedron: a regular icosahedron with two vertices removed. In their place 8 triangles are 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.

Tetrahedron vertfig.png
12 (3.3.3)
Pentagonal antiprism vertfig.png
2 (3.3.3.5)
Dissected regular icosahedron.png
Dissected regular icosahedron

Construction[edit]

The regular 600-cell can be decomposed with the symmetry of the grand antiprism, with each of the 20 blue pentagonal antiprisms being divided into 15 regular tetrahedra.

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.

Computationally speaking, this diminishing may be realized by removing two rings of vertices from the 600-cell, each ring having 10 vertices, and each lying in mutually orthogonal planes, and taking the convex hull of the remaining vertices.

(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids. Equivalently, this may be realized as taking the convex hull of the vertices remaining after 24 vertices, corresponding to those of an inscribed 24-cell, are removed from the 600-cell.)

Projections[edit]

These are two perspective projections, projecting the polytope into a hypersphere, and applying a stereographic projection into 3-space.

Stereographic grand antiprism.png
Wireframe, viewed down one of the pentagonal antiprism columns.
Stereographic grand antiprism faces.png
with transparent triangular faces
Ortho solid 963-uniform polychoron grand antiprism.png
Orthographic projection
Centered on hyperplane of an antiprism in one of the two rings.

See also[edit]

Notes[edit]

  1. ^ 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)
  2. ^ Conway, 2008, p.402-403 The Grand Antiprism

References[edit]

External links[edit]