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In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex. There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations .
A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex.It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).
The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5-simplex vertices and edges. This compound has [[3,3,3,3]] symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ .
A 20/28 hp plate-frame Simplex built in 1941, running at the Moseley Railway Trust Simplex 40 hp armoured 600 mm (1 ft 11 + 5 ⁄ 8 in) gauge, built for the British Army for World War I Two models of Simplex locomotive at Alan Keef's works, 1999 Red Rum at Leighton Buzzard Narrow Gauge Railway
An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex is the vertex figure of the A 5 lattice, . It and has a doubled symmetry from its symmetric Coxeter diagram.
Rob Roy: how to build a simple 3 1/2 in. gauge 0-6-0 tank locomotive based on the dockyard engines of the old Caledonian Railway. Hemel Hempstead: Model and Allied Publications, 1972 (112 p). Hemel Hempstead: Model and Allied Publications, 1972 (112 p).
The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).
The birectified 5-simplex, 0 22, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k 22 series. The birectified 5-simplex is the vertex figure for the third, the 1 22. The fourth figure is a Euclidean honeycomb, 2 22, and the final is a noncompact hyperbolic honeycomb, 3 22.