In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
The 5-demicube honeycomb is a uniform space-filling tessellation in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation in Euclidean 7-space.
The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.
In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation. Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.
In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation. It is composed entirely of omnitruncated 5-simplex facets.
In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation. The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
In geometry, the quarter hypercubic honeycomb is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q{4,3...3,4} or Coxeter symbol qδ4 representing the regular form with three quarters of the vertices removed and containing the symmetry of Coxeter group for n ≥ 5, with = and for quarter n-cubic honeycombs = .
In four-dimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.
In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb. Its facets are 5-demicubes and runcinated 5-demicubes.
In six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb. Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.
In seven-dimensional Euclidean geometry, the quarter 8-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 8-demicubic honeycomb, and a quarter of the vertices of a 8-cube honeycomb. Its facets are 8-demicubes h{4,36}, pentic 8-cubes h6{4,36}, {3,3}×{32,1,1} and {31,1,1}×{31,1,1} duoprisms.
= 
