Truncated 24-cell honeycomb

Truncated 24-cell honeycomb(No image)TypeUniform 4-honeycombSchläfli symbolt{3,4,3,3}tr{3,3,4,3}t2r{4,3,3,4}t2r{4,3,31,1}t{31,1,1,1}Coxeter-Dynkin diagrams4-face typeTesseract Truncated 24-cell Cell typeCube Truncated octahedron Face typeSquareTriangleVertex figureTetrahedral pyramidCoxeter groups, [3,4,3,3], [4,3,31,1], [4,3,3,4], [31,1,1,1]PropertiesVertex transitive In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.

Alternate names Truncated icositetrachoric tetracomb Truncated icositetrachoric honeycomb Cantitruncated 16-cell honeycomb Bicantitruncated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter groupCoxeterdiagramFacetsVertex figureVertexfiguresymmetry(order)= [3,4,3,3]4: 1: , [3,3](24)= [3,3,4,3]3: 1: 1: , (6)= [4,3,3,4]2,2: 1: , (4)= [31,1,3,4]1,1: 2: 1: , [ ](2)= [31,1,1,1]1,1,1,1:1: [ ]+(1)

See also Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 16-cell honeycomb 24-cell honeycomb Rectified 24-cell honeycomb Snub 24-cell honeycomb 5-cell honeycomb Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb

References Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 99 o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99