User:Tomruen/3-3-3 prism

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
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}×{ }. 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.

Related figures
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.