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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
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
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
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
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
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
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
Geometric space with five dimensions
three regular polytopes exist. In five dimensions, they are: The 5-simplex of the simplex family, {3,3,3,3}, with 6 vertices, 15 edges, 20 faces (each an
Five-dimensional_space
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
5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex 5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube
List_of_mathematical_shapes
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 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
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
Rectified_5-simplexes
Tiling of n-dimensional space
graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter
Simplicial_honeycomb
Convex regular 9-polytope
cells, 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)
9-simplex
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
uniform 5-polytope 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex 5-demicube
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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
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
Uniform 6-polytope
six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of
Pentellated_6-simplexes
Convex regular 8-polytope
8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces
8-simplex
Defunct American motor vehicle manufacturer
The Simplex Automobile Company was formed in 1907 to take over the manufacturer of the S & M Simplex. The Simplex was an American luxury Brass Era automobile
Simplex_Automobile_Company
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
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
Convex regular 9 dimensional polytope
tetrahedron 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
9-orthoplex
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
geometry, the cyclotruncated simplicial honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
Five-dimensional geometric shape
all convex: {3,3,3,3} - 5-simplex {4,3,3,3} - 5-cube {3,3,3,4} - 5-orthoplex There are no nonconvex regular polytopes in 5 dimensions or above. Unsolved
Uniform_5-polytope
5-dimensional geometric object
the semiregular 5-polytope, their elements are: The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube honeycomb
5-polytope
{\displaystyle \Delta _{d,k}} is a convex polytope that generalizes the simplex. It is determined by two integers d {\displaystyle d} and k {\displaystyle
Hypersimplex
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
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
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
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
Shape with five sides
by: H = 5 + 2 5 2 t ≈ 1.539 t , W = D = 1 + 5 2 t ≈ 1.618 t , W = 2 − 2 5 ⋅ H ≈ 1.051 H , R = 5 + 5 10 t ≈ 0.8507 t , D = R 5 + 5 2 = 2 R
Pentagon
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
Type of uniform space-filling tessellation
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
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
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
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
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
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
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
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
Tiling of five-dimensional space
uniform honeycombs in 5-space: 5-demicubic honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb de Bruijn, N
5-cubic_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
Five-pointed star polygon
Honor – Highest award in the United States Armed Forces Pentachoron – the 4-simplex Pentagram map – Discrete dynamical system on polygons in the projective
Pentagram
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
Shape with six sides
have d 1 2 + d 4 2 = d 2 2 + d 5 2 = d 3 2 + d 6 2 = 2 ( R 2 + L 2 ) , {\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\le
Hexagon
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
In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices. Each can
A5_polytope
Four-dimensional geometrical object
semiregular polytope. Runcinated 5-cell (Norman Johnson) Runcinated pentachoron Runcinated 4-simplex Expanded 5-cell/4-simplex/pentachoron Small prismatodecachoron
Runcinated_5-cell
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
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
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
Geometric object
9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell
Uniform_k_21_polytope
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
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
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
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
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
diagram: . The 122 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root
1_22_polytope
Solid with twenty equal triangular faces
expressions are: A = 5 3 a 2 ≈ 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
Regular_icosahedron
Uniform 6-polytope
leaves the 5-simplex, . Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), . Every simplex facet touches
2_21_polytope
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
Uniform_6-polytope
Polyhedron with 12 faces
y|{\Big \}}} and α = 3 − 5 2 , β = 5 − 1 2 = 1 φ . {\displaystyle \alpha ={\frac {3-{\sqrt {5}}}{2}},\qquad \beta ={\frac {{\sqrt {5}}-1}{2}}={\frac {1}{\varphi
Dodecahedron
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
5-dimensional hypercube
In five-dimensional geometry, a 5-cube (or penteract) is a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and
5-cube
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
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
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
Polytope in 8-dimensional geometry
rings in this Coxeter-Dynkin diagram: . The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the
4_21_polytope
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
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
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
Medical condition
Epidermolysis bullosa simplex (EBS) is a disorder resulting from mutations in the genes encoding keratin 5 or keratin 14. It is one of the major forms
Epidermolysis_bullosa_simplex
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
Seven-dimensional geometric object
There are exactly three such convex regular 7-polytopes: {3,3,3,3,3,3} - 7-simplex {4,3,3,3,3,3} - 7-cube {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex
Uniform_7-polytope
24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122
E6_polytope
figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2{34}, . The face figure is the vertex figure of the edge figure, here
2_22_honeycomb
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
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
Species of beetle
Iphiclus simplex is a Neotropical species of pleasing fungus beetle (family Erotylidae. Within its family, it belongs to subfamily Erotylinae. In the
Iphiclus_simplex
Equiangular and equilateral polygon
diagonals is 1 2 n ( n − 3 ) {\displaystyle {\tfrac {1}{2}}n(n-3)} ; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide
Regular_polygon
Uniform 6-polytope
rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole
6-demicube
Uniform polychoron
(Thorold Gosset) Dispentachoron Rectified 5-cell (Norman W. Johnson) Rectified 4-simplex Fully truncated 4-simplex Rectified pentachoron (Acronym: rap) (Jonathan
Rectified_5-cell
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
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
American fire protection company
Tyco bought Simplex Time Recorder Company on January 5, 2001, for US$1.15 billion and merged it with Grinnell Fire Protection, forming SimplexGrinnell. The
SimplexGrinnell
Plane figure bounded by line segments
24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122
Polygon
Geometric object with flat sides
regular n-simplex, in any number of dimensions, with Schläfli symbol {3n}. These include the equilateral triangle {3}, regular tetrahedron {3,3}, and 5-cell
Polytope
Uniform Polytope
Coxeter-Dynkin diagram, . Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations
2_31_polytope
Regular 6 dimensional polytope
or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry
6-orthoplex
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 located
Rectified_6-simplexes
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
Convex regular 8-polytope
3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group. A lowest symmetry
8-orthoplex
Cartesian product of two polytopes
The 3-3 duoprism is the vertex figure for the second, the birectified 5-simplex. The fourth figure is a Euclidean honeycomb, 222, and the final is a paracompact
Duoprism
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
) ≈ 109.5 ∘ . {\displaystyle \arctan({\sqrt {2}})+\arctan({\sqrt {2}})=2\arctan({\sqrt {2}})=\arccos \left(-{\frac {1}{3}}\right)\approx 109.5^{\circ }
Regular_octahedron
hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined
Rectified_6-orthoplexes
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