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7-homeycomb
the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex
7-simplex_honeycomb
Seven-dimensional geometric object
6-simplex honeycomb: {3[7]} Uniform Cyclotruncated 6-simplex honeycomb: t0,1{3[7]} Uniform Omnitruncated 6-simplex honeycomb: t0,1,2,3,4,5,6,7{3[7]} C
Uniform_7-polytope
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
the 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
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
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
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
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
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
Tessellation in Euclidean geometry
in 7-space: 7-cube honeycomb 7-demicube honeycomb 7-simplex honeycomb Truncated 7-simplex honeycomb Omnitruncated 7-simplex honeycomb Coxeter, Regular and
Quarter_7-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
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
Rectified_7-simplexes
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
6-cubic honeycomb 6-simplex honeycomb 6-demicubic honeycomb 222 honeycomb Seven-dimensional space, uniform 7-polytope 7-simplex, Rectified 7-simplex, Truncated
List_of_mathematical_shapes
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
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
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
Polytope contained by 7-polytope facets
7-space: Regular and uniform tessellations include: A ~ 7 {\displaystyle {\tilde {A}}_{7}} 29 uniquely ringed forms, including: 7-simplex honeycomb:
Uniform_8-polytope
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
3_31_honeycomb
6-cubic honeycomb 6-simplex honeycomb 6-demicubic honeycomb 222 honeycomb Seven-dimensional space, uniform 7-polytope 7-simplex, Rectified 7-simplex, Truncated
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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 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
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
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
7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex. There are 14 unique sterication for the 7-simplex
Stericated_7-simplexes
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
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
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
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
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
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
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
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
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
Regular 5-polytope
different edge lengths. The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplex with a petrie polygon cycle of 5 long edges. Its symmetry
5-simplex
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
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
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
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
7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex. There are 8 unique runcinations of the 7-simplex
Runcinated_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
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
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
cell centers of the 7-orthoplex. The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices
Rectified_7-orthoplexes
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
Uniform polytope
2160 demihepteract facets) 152 honeycomb, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets) 162 honeycomb, tessellates hyperbolic 9-space
Uniform_1_k2_polytope
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
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
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
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
Four-dimensional analogue of the cube
4-polytopes and honeycombs, {p,3,3} with tetrahedral vertex figures, {3,3}. The tesseract is also in a sequence of regular 4-polytope and honeycombs, {4,3,p}
Tesseract
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
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
(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
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
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
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
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
Uniform 7-polytope
seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation
Truncated_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
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
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 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
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
6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex
Truncated_6-simplexes
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 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
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
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
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 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
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
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms"
Stericated_7-orthoplexes
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
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
Regular object in four dimensional geometry
pp. 292–293, Table I(ii); "24-cell". Coxeter 1973, p. 139, §7.9 The characteristic simplex. Coxeter 1973, p. 290, Table I(ii); "dihedral angles". Kim &
24-cell
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
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
Convex regular 8-polytope
opposites. It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb. Klitzing, Richard. "x3o3o3o3o3o3o4o - ek". Coxeter, Regular
8-orthoplex
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms"
Pentellated_7-orthoplexes
Convex uniform honeycombs: Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups)) Density
Goursat_tetrahedron
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) Klitzing, Richard. "7D uniform polytopes (polyexa) with
Runcinated_7-cubes
Shape with six sides
so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use
Hexagon
Geometric object with flat sides
polyhedron Honeycomb (geometry) Intersection of a polyhedron with a line List of regular polytopes Opetope Polytope de Montréal Coxeter 1973, pp. 141–144, §7-x
Polytope
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) Klitzing, Richard. "7D uniform polytopes (polyexa) with
Runcinated_7-orthoplexes
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
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
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)
Polyhedron with 8 triangles and 6 squares
the rectified cubic honeycomb (of alternating cuboctahedra and octahedra), the 24-cell honeycomb and the tesseractic honeycomb, respectively. Each tessellation
Cuboctahedron
Convex regular 8-polytope
tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately
8-simplex
10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are
Rectified_10-simplexes
Science that studies the disease called herpes
Herpes simplex research includes all medical research that attempts to prevent, treat, or cure herpes, as well as fundamental research about the nature
Herpes_simplex_research
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices. Each can
A7_polytope
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms"
Cantellated_7-cubes
Uniform 6-polytope
contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple
1_22_polytope
Uniform 6-polytope
6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces
6-simplex
6-dimensional hypercube
infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets. This configuration matrix represents the 6-cube. The rows and columns
6-cube
Uniform 8 dimensional polytope
5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex
8-demicube
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
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