and 6 bitruncated 5-cell facets around a vertex. It can be constructed as five sets of parallel hyperplanes that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets. Alternate names Cyclotruncated pentachoric tetracomb Small truncated-pentachoric tetracomb Truncated 5-cell honeycomb The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cyclocantitruncated 5-cell honeycomb. Alaternate names Great cyclorhombated pentachoric tetracomb Great truncated-pentachoric tetracomb Cantellated 5-cell honeycomb The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncitruncated 5-cell honeycomb. Alternate names Cycloprismatorhombated pentachoric tetracomb Great prismatodispentachoric tetracomb Bitruncated 5-cell honeycomb The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncicantitruncated 5-cell honeycomb. Alternate names Great cycloprismated pentachoric tetracomb Grand prismatodispentachoric tetracomb Omnitruncated 5-cell honeycomb The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a cyclosteriruncicantitruncated 5-cell honeycomb. . It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets. Coxeter calls this Hinton's honeycomb after C. H. Hinton, who described it in his book The Fourth Dimension in 1906. The facets of all omnitruncated simplectic honeycombs are called permutohedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n). Alternate names Omnitruncated cyclopentachoric tetracomb Great-prismatodecachoric tetracomb A4* lattice The A lattice is the union of five A4 lattices, and is the dual to the omnitruncated 5-cell honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell. ∪ ∪ ∪ ∪ = dual of Alternated form This honeycomb can be alternated, creating omnisnub 5-cells with irregular 5-cells created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10. See also Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 16-cell honeycomb 24-cell honeycomb Truncated 24-cell honeycomb Snub 24-cell honeycomb Notes References Norman Johnson Uniform Polytopes, Manuscript (1991) 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (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