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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
6-simplex
Multi-dimensional generalization of triangle
0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron
Simplex
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
Regular 5-polytope
geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets.
5-simplex
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
Encrypted messaging application
simplex-chat. "Release v6.5.6 · simplex-chat/simplex-chat". GitHub. Retrieved 2026-07-17. simplex-chat. "Release v7.0.0-beta.3 · simplex-chat/simplex-chat"
SimpleX_Chat
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
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
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
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
Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex 6-demicube, Truncated 6-demicube
List_of_mathematical_shapes
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
Polytope in 8-dimensional geometry
touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets. Since every 7-simplex has 7 6-simplex facets, each
4_21_polytope
Type of 7-polytope
7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces
7-simplex
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
Tiling of n-dimensional space
the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. The (2n−1)-simplex honeycombs
Simplicial_honeycomb
Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex 6-demicube, Truncated 6-demicube
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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
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
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
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
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
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
Convex regular 9-polytope
252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or
9-simplex
geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex. There are six unique
Stericated_5-simplexes
7-dimensional hypercube
infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces. Coxeter, Regular Polytopes, p. 12, Sec. 1.8 Configurations Coxeter
7-cube
Convex regular 9 dimensional polytope
cells, 4032 5-cell 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces. It has two constructed forms
9-orthoplex
Uniform 6-dimensional polytope
uniform 6-polytope. The simplest uniform polypeta are regular polypeta: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex
Uniform_6-polytope
cyclotruncated 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
Natural number
regular 6-simplex separately. In similar fashion, A 6 {\displaystyle \mathrm {A_{6}} } is associated with classical Chevalley Lie algebra A 6 {\displaystyle
63_(number)
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
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
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
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
Shape with six sides
circle, equal to 120°. The Schläfli symbol denotes this polygon as { 6 } {\displaystyle \{6\}} . However, the regular hexagon can also be considered as cutting
Hexagon
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
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
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
Polyhedron with four faces
three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a
Tetrahedron
Viral disease caused by herpes simplex viruses
Herpes simplex, often known simply as herpes, is a viral infection caused by the herpes simplex virus. Herpes infections are categorized by the area of
Herpes
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
the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex are located
Rectified_5-simplexes
cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex. There are six unique cantellations for the 8-simplex, including
Cantellated_8-simplexes
Uniform Polytope
the short branch leaves the rectified 6-simplex, . Removing the node on the end of the 2-length branch leaves the, 6-demicube, . Removing the node on the
2_31_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
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
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
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
Species of flowering plant
Ruellia simplex, the Mexican petunia, Mexican bluebell, Britton's wild petunia, or Purple showers, is a species of flowering plant in the family Acanthaceae
Ruellia_simplex
Species of virus
Herpes simplex virus 1 and 2 (HSV-1 and HSV-2) are two members of the human Herpesviridae family, a set of viruses that produce viral infections in the
Herpes_simplex_virus
Uniform 7-dimensional polytope
leaves the 6-simplex, . Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 311, . Every simplex facet touches
3_21_polytope
Geometric space with six dimensions
called a 6-polytope. The most studied are the regular polytopes, of which there are only three in six dimensions: the 6-simplex, 6-cube, and 6-orthoplex
Six-dimensional_space
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
Geometric object
9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.) The family starts uniquely as 6-polytopes. The triangular prism
Uniform_k_21_polytope
Shape with five sides
i = 1 5 d i 6 = 5 ( ( R 2 + L 2 ) 3 + 6 R 2 L 2 ( R 2 + L 2 ) ) , ∑ i = 1 5 d i 8 = 5 ( ( R 2 + L 2 ) 4 + 12 R 2 L 2 ( R 2 + L 2 ) 2 + 6 R 4 L 4 ) . {\displaystyle
Pentagon
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
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
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
Construction for n-dimensional noise functions
Simplex noise is the result of an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with fewer directional artifacts, in higher
Simplex_noise
Polyhedron with 12 faces
angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show
Dodecahedron
Solid with twenty equal triangular faces
8.660 a 2 , V = 5 φ 2 6 a 3 ≈ 2.182 a 3 . {\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}
Regular_icosahedron
eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex. There are unique 3 degrees of rectifications
Rectified_8-simplexes
Algorithm for linear programming
Dantzig's simplex algorithm (or simplex method) is an algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and
Simplex_algorithm
Shape with three equal sides
Finetti diagram in the mathematical modellilng of population genetics, and a simplex plot in game theory and convex optimization.[citation needed] In the study
Equilateral_triangle
5-dimensional hypercube
2 80 4 6 4 4 4 80 3 3 8 12 6 40 2 16 32 24 8 10 ] {\displaystyle {\begin{bmatrix}{\begin{matrix}32&5&10&10&5\\2&80&4&6&4\\4&4&80&3&3\\8&12&6
5-cube
hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined
Rectified_7-orthoplexes
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
Uniform polytope in 8 dimensional geometry
ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, . Petrie polygon projections are 12, 18, or 30-sided based on the
2_41_polytope
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices. Each can
A6_polytope
Uniform 7-polytope
columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole
7-demicube
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
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
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
List_of_F4_polytopes
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
Geometric object with flat sides
is self-dual. Some common self-dual polytopes include: Every regular n-simplex, in any number of dimensions, with Schläfli symbol {3n}. These include
Polytope
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
Regular 7- polytope
[4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest
7-orthoplex
8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex. There are two unique pentellations of the 8-simplex. Including
Pentellated_8-simplexes
Polytope contained by 7-polytope facets
are exactly three such convex regular 8-polytopes: {3,3,3,3,3,3,3} - 8-simplex {4,3,3,3,3,3,3} - 8-cube {3,3,3,3,3,3,4} - 8-orthoplex There are no nonconvex
Uniform_8-polytope
Equiangular and equilateral polygon
Hexagons – {6/2}, {6/3} Octagons – {8/2}, {8/4} Enneagon – {9/3} Decagons – {10/2}, {10/4}, and {10/5} Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6} Depending
Regular_polygon
5-dimensional geometric object
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
geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication (6th order truncation) of the regular 8-simplex. Acronym: supane (Jonathan
Hexicated_8-simplexes
Species of flowering plant
Firmiana simplex, commonly known as the Chinese parasol tree, or wutong (Chinese: 梧桐; pinyin: wútóng; jap.: godō), is a tree in the family Malvaceae,
Firmiana_simplex
8-dimensional hypercube
family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets. The 8-cube is 8th in an infinite series of hypercubes: Coxeter
8-cube
Convex polytope, the n-dimensional analogue of a square and a cube
(n−1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n−1)-simplex's facets
Hypercube
American fire protection company
SimplexGrinnell, a subsidiary of Johnson Controls, is an American company specializing in active fire protection systems, communication systems and testing
SimplexGrinnell
Plane figure bounded by line segments
section, the coordinates of the centroid of a solid simple polygon are C x = 1 6 A ∑ i = 0 n − 1 ( x i + x i + 1 ) ( x i y i + 1 − x i + 1 y i ) , {\displaystyle
Polygon
Encephalitis associated with herpes simplex virus
Herpes simplex encephalitis (HSE), or simply herpes encephalitis, is a rare form of encephalitis caused by the herpes simplex virus. It is estimated to
Herpes_simplex_encephalitis
Type of geometric object
are exactly three such convex regular 9-polytopes: {3,3,3,3,3,3,3,3} - 9-simplex {4,3,3,3,3,3,3,3} - 9-cube {3,3,3,3,3,3,3,4} - 9-orthoplex There are no
Uniform_9-polytope
Defunct American motor vehicle manufacturer
Crane-Simplex was the common name of the Simplex Crane Model 5 luxury automobile, produced by the Simplex Automobile Company in New Brunswick, New Jersey
Crane-Simplex
Shape with four equal sides and angles
square. The answer to the puzzle is n ( n + 1 ) ( 2 n + 1 ) / 6 {\displaystyle n(n+1)(2n+1)/6} , a square pyramidal number. For n = 1 , 2 , 3 , … {\displaystyle
Square
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Rectified_8-cubes
Herpes simplex virus infection of the lip
A cold sore is a type of herpes infection caused by the herpes simplex virus that affects primarily the lip. Symptoms typically include a burning pain
Cold_sore
9-dimensional hypercube
infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets. Klitzing, Richard. "o3o3o3o3o3o3o3o4x - enne". H.S.M. Coxeter:
9-cube
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
Figurate number
{\displaystyle T_{0}=0} (see empty sum), the first few terms are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210
Triangular_number
Four-dimensional geometric object with flat sides
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
4-polytope
Solid with eight equal triangular faces
, r i = 6 6 a ≈ 0.408 a , r m = 1 2 a = 0.5 a . {\displaystyle r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707a,\qquad r_{i}={\frac {\sqrt {6}}{6}}a\approx
Regular_octahedron
Five-dimensional geometric shape
Wythoff construction operations upon regular 5-simplex (hexateron). The A5 family has symmetry of order 720 (6 factorial). 7 of the 19 figures, with symmetrically
Uniform_5-polytope
Infection by herpes simplex viruses of the genitals
Genital herpes is a herpes infection of the genitals caused by the herpes simplex virus (HSV). Most people either have no or mild symptoms and thus do not
Genital_herpes
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