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User:Tomruen/3-3-3 prism

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3-3-3 prism

Orthogonal projections in regular enneagon
Type p-q-r prism
Schläfli symbol {3}×{3}×{3} = {3}3
Coxeter diagram or
5-faces 9 {3}×{3}×{ }
4-faces 9 {3}×{3}
27 {3}×{4}
Cells 54 {3}×{ }
27 {4}×{ }
Faces 81 {4}
27 {3}
Edges 81
Vertices 27
Vertex figure 5-simplex
Symmetry [3[3,2,3,2,3]], order 64 =1296
Dual 3-3-3 pyramid
Properties convex, vertex-uniform, facet-transitive

In the geometry of 6 dimensions, the 3-3-3 prism or triangular triaprism is a four-dimensional convex uniform polytope. It can be constructed as the Cartesian product of three triangles and is the simplest of an infinite family of six-dimensional polytopes constructed as Cartesian products of three polygons.

Elements

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It has 27 vertices, 81 edges, 108 faces (81 squares, and 27 triangles), 54 triangular prism,{3}×{ }, 27 square prisms, { }×{ }×{ }, and 9 3-3 duoprisms, {3}×{3} ,27 3-4 duoprisms, {3}×{4}, and 18, 3-3 duoprism prisms, {3}×{3}×{ }.[1] It has Coxeter diagram , and Coxeter notation symmetry [3[3,2,3,2,3]], order 1296. The symmetry of each triangle is [3], dihedral order 6. All three triangles combined have symmetry order 63 = 216. The extended symmetry [3], order 6 comes from permuting the three planes of triangles.

Its vertex figure is a 6-simplex with 3 orthogonal longer edges.

Projections

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Orthogonal projections in regular enneagon
Upper left, edge colored by triangle.
Others show 3 sets of 9 triangles.
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The 3-3-3 prism shares vertices with a generalized cube, a complex polyhedron, 3{4}2{3}2, or , with 27 vertices, 27 3-edges, and 9 faces.

The 3-3-3 prism is the vertex figure of the birectified 222 honeycomb, 0222, {32,2,2}, or in 6-dimensions.

The 4-4-4 prism is the same as a 6-cube.

References

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  1. ^ Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o x3o x3o - trittip
  • Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
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