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geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes
5-simplex_honeycomb
Regular 5-polytope
The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplex with a petrie polygon cycle of 5 long edges. Its symmetry is isomorphic to
5-simplex
Geometric figure
the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells
5-cell_honeycomb
Type of uniform space-filling tessellation
uniform honeycombs in 5-space: 5-cube honeycomb 5-demicube honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb "The
5-demicubic_honeycomb
dimension n+1.[citation needed] 5-cubic honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb 5-demicubic honeycomb Six-dimensional space, 6-polytope
List_of_mathematical_shapes
cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex
Cyclotruncated 5-simplex honeycomb
Cyclotruncated_5-simplex_honeycomb
600-cell prism Grand antiprism prism 5-cubic honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb 5-demicubic honeycomb Six-dimensional space, 6-polytope
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Tiling of n-dimensional space
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde
Simplicial_honeycomb
origin and the six vertices of the 5-simplex. The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each
Stericated_5-simplexes
In geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry
Omnitruncated simplicial honeycomb
Omnitruncated_simplicial_honeycomb
cyclotruncated simplicial honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
Five dimensional space-filling tessellation
5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex
Omnitruncated 5-simplex honeycomb
Omnitruncated_5-simplex_honeycomb
geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified
6-simplex_honeycomb
7-homeycomb
geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified
7-simplex_honeycomb
Tiling of five-dimensional space
and uniform honeycombs in 5-space: 5-demicubic honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb de Bruijn,
5-cubic_honeycomb
geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified
8-simplex_honeycomb
Uniform 6-dimensional polytope
5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb C ~ 5 {\displaystyle {\tilde {C}}_{5}} There are 35 uniform honeycombs, including:
Uniform_6-polytope
omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-simplex facets. The facets
Omnitruncated 6-simplex honeycomb
Omnitruncated_6-simplex_honeycomb
uniform honeycombs in 8-space: 8-cubic honeycomb 8-demicubic honeycomb 8-simplex honeycomb Truncated 8-simplex honeycomb 521 honeycomb 251 honeycomb 152 honeycomb
Omnitruncated 8-simplex honeycomb
Omnitruncated_8-simplex_honeycomb
5-dimensional geometric object
semiregular 5-polytope, their elements are: The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube honeycomb, , vertex
5-polytope
Construction for n-dimensional noise functions
disphenoid honeycomb. Simplex noise is useful for computer graphics applications, where noise is usually computed over 2, 3, 4, or possibly 5 dimensions
Simplex_noise
omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets. The facets
Omnitruncated 7-simplex honeycomb
Omnitruncated_7-simplex_honeycomb
Five-dimensional geometric shape
{A}}_{4}} , family, all new, including: 4-simplex honeycomb Truncated 4-simplex honeycomb Omnitruncated 4-simplex honeycomb There are 9 uniquely ringed forms
Uniform_5-polytope
Type of uniform tessellation
vertices of two 8-demicube honeycombs (called a D82 or D8+ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A83 lattice):
5_21_honeycomb
Compact regular space-filling tessellation
order-5 5-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {3,3,3,5}, it has five 5-cells
Order-5_5-cell_honeycomb
5-simplex, 031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb
Rectified_5-simplexes
Seven-dimensional geometric object
forms Uniform 6-simplex honeycomb: {3[7]} Uniform Cyclotruncated 6-simplex honeycomb: t0,1{3[7]} Uniform Omnitruncated 6-simplex honeycomb: t0,1,2,3,4,5
Uniform_7-polytope
Type of geometric object
uniquely ringed forms 8-simplex honeycomb: {3[9]} C ~ 8 {\displaystyle {\tilde {C}}_{8}} 271 uniquely ringed forms Regular 8-cube honeycomb: {4,36,4}, B ~ 8
Uniform_9-polytope
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
3_31_honeycomb
The 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group. This honeycomb is highly
E9_honeycomb
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32
2_22_honeycomb
to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such
List_of_regular_polytopes
uniform honeycombs in 5-space: 5-cube honeycomb 5-demicube honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb Coxeter
Quarter_5-cubic_honeycomb
Polytope contained by 7-polytope facets
including: 7-simplex honeycomb: {3[8]} C ~ 7 {\displaystyle {\tilde {C}}_{7}} 135 uniquely ringed forms, including: Regular 7-cube honeycomb: {4,34,4} =
Uniform_8-polytope
Group of irregular uniform polytopes
express one honeycomb, 1111, , represents a lower symmetry form of the 16-cell honeycomb, and 01111, for the rectified 16-cell honeycomb. The 5-dimensional
Gosset–Elte_figures
Four-dimensional geometrical object
forms a uniform honeycomb which Coxeter calls Hinton's honeycomb. Omnitruncated 5-cell Omnitruncated pentachoron Omnitruncated 4-simplex Great prismatodecachoron
Runcinated_5-cell
Type of geometrical object
family, generated by end-ringed Coxeter diagrams are: 621 honeycomb: 261 honeycomb: 162 honeycomb: Richeson, D.; Euler's Gem: The Polyhedron Formula and
Uniform_10-polytope
Tiling of hyperbolic 3-space by uniform polyhedra
fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure
1_52_honeycomb
Four-dimensional analogue of the tetrahedron
hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α4 polytope), the simplest possible convex 4-polytope, and is
5-cell
order-6 hexagonal tiling honeycomb has a half-symmetry construction: . It also has an index-6 subgroup, [6,3*,6], with a non-simplex fundamental domain. This
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Tessellation in Euclidean geometry
uniform honeycombs in 7-space: 7-cube honeycomb 7-demicube honeycomb 7-simplex honeycomb Truncated 7-simplex honeycomb Omnitruncated 7-simplex honeycomb Coxeter
Quarter_7-cubic_honeycomb
along three orthogonal axes. The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions. The half-symmetry uniform construction
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
uniform honeycombs in 8-space: 8-cube honeycomb 8-demicube honeycomb 8-simplex honeycomb Truncated 8-simplex honeycomb Omnitruncated 8-simplex honeycomb Coxeter
Quarter_8-cubic_honeycomb
Regular paracompact honeycomb
field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact
Hexagonal_tiling_honeycomb
Space-filling tessellation
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently,
Bitruncated_cubic_honeycomb
Class of eight-dimensional polytopes
geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications
Stericated_8-simplexes
Polytope whose facets are all simplices
4-polytope 4-simplex, 16-cell, 600-cell Dual convex uniform honeycombs: Disphenoid tetrahedral honeycomb Dual of cantitruncated cubic honeycomb Dual of omnitruncated
Simplicial_polytope
Regular tiling of a two-dimensional space
≈ 0.907 {\textstyle {\frac {\pi }{2{\sqrt {3}}}}\approx 0.907} . The honeycomb theorem states that hexagonal tiling is the best way to divide a surface
Hexagonal_tiling
Tessellation of convex uniform polyhedron cells
Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored. These nonsimplectic
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Uniform polytope
but can be extended backwards to include the 5-demicube (demipenteract) in 5 dimensions, and the 4-simplex (5-cell) in 4 dimensions. Each polytope is constructed
Uniform_1_k2_polytope
Uniform 6-polytope
vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb. Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed
Pentellated_6-simplexes
Four-dimensional shape
the cells of the dual of the uniform rectified 5-simplex, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral prism vertex figure
Tetrahedral_bipyramid
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
Runcinated_5-simplexes
Type of geometric object
Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex. The rectified 9-simplex is the vertex figure of the 10-demicube
Rectified_9-simplexes
5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex. There are unique 4 degrees of cantellation for the 5-simplex
Cantellated_5-simplexes
uniform honeycombs in 5-space: 6-cube honeycomb 6-demicube honeycomb 6-simplex honeycomb Truncated 6-simplex honeycomb Omnitruncated 6-simplex honeycomb Coxeter
Quarter_6-cubic_honeycomb
In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets. It is also named
1_33_honeycomb
Four-dimensional analogue of the cube
characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct construction of the tesseract possible. The characteristic 5-cell
Tesseract
Feature of a polyhedron, polytope, etc.
facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For (boundary complexes of)
Facet_(geometry)
Type of 7-polytope
geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex. There are 20 unique
Hexicated_7-simplexes
five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees of truncation
Truncated_5-simplexes
Convex uniform 7-polytope in seven-dimensional geometry
the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located
Rectified_7-simplexes
(Jonathan Bowers) The rectified 8-simplex is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb. The Cartesian coordinates
Rectified_8-simplexes
Regular polytope whose 2D form is a pentagon
{ } Pentagon, {5} Dodecahedron, {5, 3} (12 pentagonal faces) 120-cell, {5, 3, 3} (120 dodecahedral cells) Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates
Pentagonal_polytope
geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex. There are 14 unique sterication
Stericated_7-simplexes
Motor vehicle
The Mercedes Simplex was an automobile produced from 1902 to 1909 by the Daimler Motoren Gesellschaft (DMG, Daimler Motor Society, a predecessor of Daimler-Benz
Mercedes_Simplex
5-dimensional regular honeycomb
is the order-5 5-cell honeycomb, {3,3,3,5}. It is related to the order-4 120-cell honeycomb, {5,3,3,4}, and order-5 120-cell honeycomb, {5,3,3,5}. It
120-cell_honeycomb
Quasiregular space-filling tesselation
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Eight-dimensional geometric tessellation
8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube
2_51_honeycomb
Geometric object
figure (17280 7-simplex and 2160 7-orthoplex facets) 5 21 honeycomb: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex
Uniform_k_21_polytope
Isogonal polytope with uniform facets
more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be
Uniform_polytope
Polytope with highest degree of symmetry
5 dimensions). 1-simplex to 4-simplex The simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex
Regular_polytope
three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular
Alternated hexagonal tiling honeycomb
Alternated_hexagonal_tiling_honeycomb
Four-dimensional geometric object with flat sides
Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related
4-polytope
5-dimensional hypercube
tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic
5-cube
Uniform polytope
5-cell: 201, (5 tetrahedra cells) Pentacross: 211, (32 5-cell (201) facets) 221, (72 5-simplex and 27 5-orthoplex (211) facets) 231, (576 6-simplex and
Uniform_2_k1_polytope
6-dimensional geometric object
construction. The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified
6-polytope
8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex
Truncated_8-simplexes
Regular tiling of the plane
tiling) List of uniform tilings Simplectic honeycomb Tilings of regular polygons Triangular tiling honeycomb Tilings and patterns, p.102-107 "The Lattice
Triangular_tiling
Uniform polytope
and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134
1_32_polytope
geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex. There are 8 unique runcinations
Runcinated_7-simplexes
Regular object in four dimensional geometry
its predecessor, enclosing more content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity
24-cell
is a nonuniform triangular prism. Cantellated pentachoron Cantellated 4-simplex (small) prismatodispentachoron Rectified dispentachoron Small rhombated
Cantellated_5-cell
Four-dimensional analog of the icosahedron
a sequence of 4-polytope and honeycombs with icosahedron vertex figures: The regular complex polygons 3{5}3, and 5{3}5, , in C 2 {\displaystyle \mathbb
600-cell
of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex. The rectified 10-simplex is the vertex
Rectified_10-simplexes
Solid with eight equal triangular faces
polyhedra construction, and it can tile with different polyhedra to create a honeycomb. The vertices and edges of a regular octahedron give rise to a graph,
Regular_octahedron
Abstract regular 4-polytope
contained within a flat 3-dimensional subspace. 5-simplex 57-cell Icosahedral honeycomb - regular hyperbolic honeycomb with same Schläfli type, {3,5,3}. (The 11-cell
11-cell
The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets. The tritruncated 6-simplex is the intersection
Truncated_6-simplexes
runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled
Runcinated_8-simplexes
Convex uniform honeycombs: Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups)) Density
Goursat_tetrahedron
bitruncation. The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each
Truncated_5-cell
Simplex formed from a right-angled path
Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions
Schläfli_orthoscheme
6-dimensional hypercube
(part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets. This configuration matrix represents the 6-cube. The
6-cube
geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex. There are 8 unique sterications
Stericated_6-simplexes
Geometric object with flat sides
and noncompact hyperbolic tilings, such as the icosahedral honeycomb {3,5,3}, and order-5 pentagonal tiling {5,5}. In 2 dimensions, all regular polygons
Polytope
Four-dimensional analog of the octahedron
characteristic simplex. Coxeter 1973, p. 290, Table I(ii); "dihedral angles". Coxeter 1970, p. 45, Table 2: Reflexible honeycombs and their groups; Honeycomb [3,3
16-cell
Convex regular 10-polytope
462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle
10-simplex
a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex. There are 8 unique
Runcinated_6-simplexes
Planar surface that forms part of the boundary of a solid object
2-face. The facets of a 4D polytope or 3-honeycomb are its 3-faces or cells. The facets of a 5D polytope or 4-honeycomb are its 4-faces. In related terminology
Face_(geometry)
5 SIMPLEX-HONEYCOMB
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