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6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex
6-simplex_honeycomb
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
honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb 5-demicubic honeycomb Six-dimensional space, 6-polytope and uniform 6-polytope 6-simplex,
List_of_mathematical_shapes
Seven-dimensional geometric object
Cyclotruncated 6-simplex honeycomb: t0,1{3[7]} Uniform Omnitruncated 6-simplex honeycomb: t0,1,2,3,4,5,6,7{3[7]} C ~ 6 {\displaystyle {\tilde {C}}_{6}} , [4,34
Uniform_7-polytope
the 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
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
Geometric figure
geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed
5-cell_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
honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb 5-demicubic honeycomb Six-dimensional space, 6-polytope and uniform 6-polytope 6-simplex,
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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
Uniform 6-dimensional polytope
honeycombs include: A ~ 5 {\displaystyle {\tilde {A}}_{5}} There are 12 unique uniform honeycombs, including: 5-simplex honeycomb Truncated 5-simplex
Uniform_6-polytope
five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared
5-simplex_honeycomb
Uniform 6-polytope
6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex
Pentellated_6-simplexes
Five dimensional space-filling tessellation
geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely
Omnitruncated 5-simplex honeycomb
Omnitruncated_5-simplex_honeycomb
Construction for n-dimensional noise functions
case of the function is an orientation of the tetragonal disphenoid honeycomb. Simplex noise is useful for computer graphics applications, where noise is
Simplex_noise
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
cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated
Cyclotruncated 5-simplex honeycomb
Cyclotruncated_5-simplex_honeycomb
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
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
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
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
field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_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
6-dimensional geometric object
The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and
6-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
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 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 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
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
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
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
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
Group of irregular uniform polytopes
polytopes except for the 142 which has 3 types of 6-faces. The set of figures extend into honeycombs of (2,2,2), (3,3,1), and (5,4,1) families in 6,7,8
Gosset–Elte_figures
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
the 5-simplex honeycomb. Steritruncated hexateron Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers) The coordinates can be made in 6-space
Stericated_5-simplexes
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
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
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
in 5-space: 6-cube honeycomb 6-demicube honeycomb 6-simplex honeycomb Truncated 6-simplex honeycomb Omnitruncated 6-simplex honeycomb Coxeter, Regular and
Quarter_6-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
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
6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex. There are 8 unique sterications for the 6-simplex
Stericated_6-simplexes
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
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
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
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
geometry, 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
Feature of geometric structures
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)
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
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
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
Geometric object
facets) 6 21 honeycomb: 621, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets) Each polytope is constructed from (n − 1)-simplex and (n − 1)-orthoplex
Uniform_k_21_polytope
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
Type of geometric object
nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex. These polytopes are part of a family
Rectified_9-simplexes
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
6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex. There are unique 4 degrees of cantellation for the 6-simplex
Cantellated_6-simplexes
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
Uniform polytope
ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032, Seen in a configuration matrix, the element counts can be derived
1_32_polytope
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
Uniform polytope
starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5 dimensions, and the 4-simplex (5-cell) in 4 dimensions
Uniform_1_k2_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
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
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
six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation
Truncated_6-simplexes
Uniform polytope
(201) facets) 221, (72 5-simplex and 27 5-orthoplex (211) facets) 231, (576 6-simplex and 56 221 facets) 241, (17280 7-simplex and 240 231 facets) 251
Uniform_2_k1_polytope
and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent
Rectified_6-orthoplexes
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
(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
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
7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex. There are unique 6 degrees of cantellation for the 7-simplex,
Cantellated_7-simplexes
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
polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices. Each can be visualized as symmetric orthographic projections
A7_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
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
Four-dimensional analogue of the cube
The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct
Tesseract
6-dimensional hypercube
demihypercubes), which has 12 5-demicube and 32 5-simplex facets. This configuration matrix represents the 6-cube. The rows and columns correspond to vertices
6-cube
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
Regular polytope whose 2D form is a pentagon
faces) 120-cell, {5, 3, 3} (120 dodecahedral cells) Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ 120-cell facets) The
Pentagonal_polytope
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
Four-dimensional shape
as well, it exists as the cells of the dual to the rectified 24-cell honeycomb. Triangular bipyramid - A lower dimensional analogy of the tetrahedral
Tetrahedral_bipyramid
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
Runcinated_5-simplexes
Convex uniform honeycombs: Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups)) Density
Goursat_tetrahedron
Uniform 7-polytope
7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex
Truncated_7-simplexes
5-dimensional geometric object
elements are: The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified
5-polytope
Polytope with highest degree of symmetry
dimensions). 1-simplex to 4-simplex The simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named
Regular_polytope
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
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
the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are
Rectified_6-simplexes
polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices. Each can be visualized as symmetric orthographic projections
A8_polytope
Regular object in four dimensional geometry
Clifford torus, showed how the honeycombs correspond to Hopf fibrations, and made a particular study of the 24-cell's 4 rings of 6 octahedral cells with illustrations
24-cell
Regular 6 dimensional polytope
5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil
6-orthoplex
truncating the vertices of the 6-cube at 1 / ( 2 + 2 ) {\displaystyle 1/({\sqrt {2}}+2)} of the edge length. A regular 5-simplex replaces each original vertex
Truncated_6-cubes
Polyhedron with 12 faces
of five Federov polyhedra or parallelohedra, generated to create its honeycomb. Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital
Dodecahedron
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
Abstract regular 4-polytope
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
Polytope constructed from alternation of a hypercube
formed. The 2n facets become 2n (n − 1)-demicubes, and 2n - 1 (n − 1)-simplex facets are formed in place of the deleted vertices. They have been named
Demihypercube
a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex. There are 35 unique
Heptellated_8-simplexes
Uniform 6-polytope
6-dimensional space, 222, . The rectified 122 polytope (also called 0221) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb
1_22_polytope
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "6D uniform polytopes (polypeta) with acronyms"
Stericated_6-orthoplexes
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
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6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
6 SIMPLEX-HONEYCOMB
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