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Regular 7- polytope
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell
7-orthoplex
Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex 132 polytope
List_of_mathematical_shapes
geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex. There are 16 unique
Runcinated_7-orthoplexes
geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex. There are 24 unique
Stericated_7-orthoplexes
7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex, Hexicated 7-orthoplex
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polytope in 8-dimensional geometry
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 321 polytope
4_21_polytope
geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex. There are 32 unique
Pentellated_7-orthoplexes
geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex. There are unique 7 degrees of rectifications
Rectified_7-orthoplexes
7-polytope
truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex. There are 6 truncations of the 7-orthoplex. Vertices
Truncated_7-orthoplexes
Convex uniform 7-polytope
hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex. There
Hexicated_7-orthoplexes
simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex. Cellitruncated heptapeton
Stericated_6-simplexes
Seven-dimensional geometric object
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 regular 7-polytopes. The
Uniform_7-polytope
7-dimensional hypercube
x4, x5, x6) with −1 < xi < 1. The 7-cube is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family
7-cube
cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex. There are ten degrees of cantellation for the 7-orthoplex
Cantellated_7-orthoplexes
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices
B7_polytope
Uniform 6-polytope
on facets of the pentellated 7-orthoplex. A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations
Pentellated_6-simplexes
including truncations. 4 are most simply constructible from the dual 7-orthoplex. Small rhombated hepteract (acronym: sersa) (Jonathan Bowers) Small birhombated
Cantellated_7-cubes
Convex regular 5-polytope in geometry
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron
5-orthoplex
Four-dimensional analogue of the cube
Polytopes (3rd ed.). Dover Publications. pp. 122–123. See the illustration Fig 7.2C. Hall, T. Proctor (1893). "The projection of fourfold figures on a three-flat"
Tesseract
Type of 7-polytope
on facets of the hexicated 8-orthoplex, . A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations
Hexicated_7-simplexes
Shape with four equal sides and angles
(PDF). Environment and Planning B: Planning and Design. 7 (2): 209–226. Bibcode:1980EnPlB...7..209S. doi:10.1068/b070209. Nakamura, Yuuka; Okazaki, Shigeyuki
Square
Geometric space with seven dimensions
a 7-polytope. The most studied are the regular polytopes, of which there are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A
Seven-dimensional_space
Convex regular 8-polytope
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792
8-orthoplex
cantellations. 8 are more simply constructed from the 7-orthoplex. These polytopes are among 127 uniform 7-polytopes with B7 symmetry. Small prismated hepteract
Runcinated_7-cubes
of the stericated 8-orthoplex. Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers) The vertices of the bistericated 7-simplex can be most
Stericated_7-simplexes
Uniform 7-Honeycomb
7-orthoplex {3,3,3,3,3,4} facets. The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice. The 84 vertices of the rectified 7-orthoplex
7-demicubic_honeycomb
truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex. There are 7 truncation for the 8-orthoplex. Vertices
Truncated_8-orthoplexes
Regular polytope dual to the hypercube in any number of dimensions
In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean
Cross-polytope
simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex. Bitruncated heptapeton
Truncated_6-simplexes
runcinations, and sterications. 16 are more simply constructed relative to the 7-orthoplex. Small terated hepteract (acronym: stesa) (Jonathan Bowers) Teritruncated
Pentellated_7-cubes
simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex. Small biprismated tetradecapeton
Runcinated_6-simplexes
Uniform 6-polytope
more simply positioned in 7-space as permutations of: (0,0,0,0,0,0,1) This construction is based on facets of the 7-orthoplex. The regular 6-simplex is
6-simplex
Shape with six sides
window 24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space Hexagonal crystal
Hexagon
Uniform 7- polytope
located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex. Truncated hepteract (Jonathan Bowers)
Truncated_7-cubes
Shape with three equal sides
generates a sequence consisting of the first few numbers: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ... (sequence A000931 in the OEIS) The Padovan sequence
Equilateral_triangle
Polyhedron with 12 faces
{4}{3}}\cdot {\text{Long side}}} Short sides = 7 12 ⋅ Long side {\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}} The eight
Dodecahedron
5-dimensional hypercube
part of an infinite hypercube family. The dual of a penteract is the 5-orthoplex, of the infinite family of orthoplexes. Applying an alternation operation
5-cube
Geometric object
(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 facets)
Uniform_k_21_polytope
Plane figure bounded by line segments
ISBN 0-415-15792-7. Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7. Kenny, Anthony
Polygon
projective polytopes are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example: An apeirotope or infinite
List_of_regular_polytopes
Multi-dimensional generalization of triangle
coordinates as 0 or 1. It can also be seen one facet of a regular (n + 1)-orthoplex. There is a canonical map from the standard n-simplex to an arbitrary
Simplex
Shape with five sides
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 • 7-cube
Pentagon
simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex. E. L. Elte identified
Rectified_6-simplexes
runcinations. 10 are more simply constructed from the 7-orthoplex. This polytope is one of 127 uniform 7-polytopes with B7 symmetry. Small cellated hepteract
Stericated_7-cubes
Polyhedron with four faces
and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48. The Royal Game of Ur, dating from 2600 BC
Tetrahedron
Four-dimensional analog of the octahedron
[citation needed] The 16-cell is the 4-dimensional cross polytope (4-orthoplex), which means its vertices lie in opposite pairs on the 4 axes of a (w
16-cell
10-dimensional hypercube
polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes
10-cube
6-dimensional hypercube
polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed
6-cube
facets of the runcinated 8-orthoplex. Small biprismated octaexon (sibpo) (Jonathan Bowers) The vertices of the biruncinated 7-simplex can be most simply
Runcinated_7-simplexes
Convex regular polytope in 10 dimensional geometry
In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 tetrahedron cells, 8064
10-orthoplex
of the regular 6-orthoplex. There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations. 7 are expressed relative
Runcinated_6-orthoplexes
Convex polytope, the n-dimensional analogue of a square and a cube
ISBN / Date incompatibility (help) Coxeter 1973, pp. 122–123, §7.2 see illustration Fig 7.2C. Miroslav Vořechovský; Jan Mašek; Jan Eliáš (November 2019)
Hypercube
9-orthoplex. There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices
Rectified_9-orthoplexes
Type of 7-polytope
of the 8-orthoplex. This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram: This polytope is one of 71 uniform 7-polytopes
7-simplex
Convex regular 9 dimensional polytope
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032
9-orthoplex
Equiangular and equilateral polygon
regular stars of up to 12 sides are: Pentagram – {5/2} Heptagram – {7/2} and {7/3} Octagram – {8/3} Enneagram – {9/2} and {9/4} Decagram – {10/3} Hendecagram
Regular_polygon
simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex. Small prismated heptapeton
Cantellated_6-simplexes
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 • 7-cube
E6_polytope
10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices
Rectified_10-orthoplexes
eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of
Rectified_8-orthoplexes
Convex uniform 7-polytope in seven-dimensional geometry
on facets of the rectified 8-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2 7. It is also called 04,2 for its
Rectified_7-simplexes
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 • 7-cube
E8_polytope
a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique
Stericated_6-orthoplexes
Uniform 6-dimensional polytope
the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}. Regular polytopes: (convex faces) 1852: Ludwig
Uniform_6-polytope
Solid with twenty equal triangular faces
≈ 5.196 {\displaystyle 3{\sqrt {3}}\approx 5.196} and 2 7 ≈ 5.292 {\displaystyle 2{\sqrt {7}}\approx 5.292} . The regular icosahedron has the thirty-one
Regular_icosahedron
Geometric object with flat sides
regular polytopes Opetope Polytope de Montréal Coxeter 1973, pp. 141–144, §7-x. Historical remarks. Coxeter (1973) Richeson, D. (2008). Euler's Gem: The
Polytope
8-dimensional hypercube
polytopes, called hypercubes. The dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes. Cartesian coordinates
8-cube
Regular 6 dimensional polytope
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell
6-orthoplex
honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets. A quadritruncated 7-cubic honeycomb, , contains
7-cubic_honeycomb
Stericated penteract / Stericated 5-orthoplex / Stericated pentacross Expanded penteract / Expanded 5-orthoplex / Expanded pentacross Small cellated
Stericated_5-cubes
9-dimensional hypercube
polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes. Cartesian coordinates
9-cube
facets of the tritruncated 9-orthoplex. The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets. Octadecazetton
Truncated_8-simplexes
Polytope contained by 7-polytope facets
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 regular 8-polytopes. The topology of any given
Uniform_8-polytope
geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex. There are 8 runcinations
Runcinated_5-orthoplexes
truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices
Truncated_5-orthoplexes
Uniform 4-polytope bounded by 320 cells
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 • 7-cube
Grand_antiprism
and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified
Rectified_9-cubes
Uniform 7-polytope
facets of the truncated 8-orthoplex. Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers) The vertices of the bitruncated 7-simplex can be most simply
Truncated_7-simplexes
are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 vertices respectively. They can be visualized
D7_polytope
truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex. There are 5 degrees of truncation for the 6-orthoplex. Vertices
Truncated_6-orthoplexes
Solid with eight equal triangular faces
between a triangle and a square is arctan ( 2 ) ≈ 54.7 ∘ {\displaystyle \arctan({\sqrt {2}})\approx 54.7^{\circ }} . Therefore, for the regular octahedron
Regular_octahedron
5-dimensional geometric object
5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes, or 5-pyramids, can
5-polytope
Uniform 7-polytope
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is
7-demicube
In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms. Small cellated
Pentic_7-cubes
half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), . Stericantellated 6-demicube Stericantellated demihexeract Cellirhombated
Pentic_6-cubes
Trirectified 7-orthoplex Trirectified heptacross (acronym: sez) (Jonathan Bowers) Cartesian coordinates for the vertices of a trirectified 7-cube, centered
Rectified_7-cubes
Type of geometrical object
3} - 10-simplex {4,3,3,3,3,3,3,3,3} - 10-cube {3,3,3,3,3,3,3,3,4} - 10-orthoplex There are no nonconvex regular 10-polytopes. The topology of any given
Uniform_10-polytope
Type of tesseract
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 • 7-cube
Truncated_tesseract
Regular 5-polytope
or (0,1,1,1,1,1). These constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively. A lower symmetry form is a 5-cell pyramid
5-simplex
Polytope with highest degree of symmetry
or 4-orthoplex 5. Regular triacontakaiditeron (pentacross) or 5-orthoplex ... An n-orthoplex has 2n vertices. The process of making each orthoplex can
Regular_polytope
polychora based on the icositetrachoron (24-cell) - Model 23, George Olshevsky. 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23, George
Rectified_24-cell
seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope is vertex-transitive
Cantic_7-cube
Type of geometric object
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 nonconvex regular 9-polytopes. The topology of any given
Uniform_9-polytope
Polygon with 14 edges
double covering polygons 2{p/q}, namely: t{7/6}={14/6}=2{7/3}, t{7/4}={14/4}=2{7/2}, and t{7/2}={14/2}=2{7}. An isotoxal polygon can be labeled as {pα}
Tetradecagon
Four-dimensional geometric object with flat sides
doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1. Coxeter 1973, p. 141, §7-x. Historical remarks. Coxeter 1973, pp. 292–293, Table I(ii): The sixteen
4-polytope
Uniform 6-polytope
leaves the 5-orthoplex in its alternated form: (211), . Every simplex facet touches a 5-orthoplex facet, while alternate facets of the orthoplex touch either
2_21_polytope
Uniform Polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin
2_31_polytope
Uniform polytope
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 • 7-cube
Uniform_1_k2_polytope
Class of eight-dimensional polytopes
1,1,1,1,2). This construction is based on facets of the stericated 9-orthoplex. Acronym: sobcane (Jonathan Bowers) The Cartesian coordinates of the vertices
Stericated_8-simplexes
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 • 7-cube
Runcinated_tesseracts
7 ORTHOPLEX
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