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102 relations: Abstract polytope, Cantic 5-cube, Convex uniform honeycomb, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Demihypercube, Density (polytope), Dual polyhedron, Duoprism, Dynkin diagram, E9 honeycomb, Emanuel Lodewijk Elte, Expansion (geometry), Gosset–Elte figures, Great duoantiprism, Heptellated 8-simplexes, Hexic 7-cubes, Hexicated 7-cubes, Hexicated 7-simplexes, Hexicated 8-simplexes, Honeycomb (geometry), Hypercube, Hypersimplex, Isogonal figure, Isohedral figure, List of mathematical shapes, List of polygons, polyhedra and polytopes, List of regular polytopes and compounds, Octagon, Omnitruncation, Paracompact uniform honeycombs, Pentic 7-cubes, Polytope, Polytope families, Proprism, Rectified 5-cell, Rectified 5-simplexes, Regular 4-polytope, Runcic 5-cubes, Runcic 6-cubes, Runcic 7-cubes, Semiregular polytope, Snub (geometry), Steric 5-cubes, Steric 7-cubes, Stericated 24-cell honeycomb, Stericated 5-cubes, Stericated 5-simplexes, Stericated 6-cubes, ..., Stericated 6-orthoplexes, Stericated 6-simplexes, Stericated 7-cubes, Stericated 7-orthoplexes, Stericated 7-simplexes, Stericated 8-simplexes, Tesseractic honeycomb, Tetrahedral prism, Triangular prism, Truncation (geometry), Uniform 1 k2 polytope, Uniform 10-polytope, Uniform 2 k1 polytope, Uniform 4-polytope, Uniform 5-polytope, Uniform 6-polytope, Uniform 7-polytope, Uniform 8-polytope, Uniform 9-polytope, Uniform honeycomb, Uniform honeycombs in hyperbolic space, Uniform k 21 polytope, Uniform polyhedron, Uniform tiling, Vertex figure, Wythoff construction, 1 22 polytope, 1 32 polytope, 1 42 polytope, 10-cube, 10-demicube, 2 21 polytope, 2 22 honeycomb, 2 31 polytope, 2 41 polytope, 24-cell honeycomb, 3 21 polytope, 4 21 polytope, 4-polytope, 5-cell, 5-demicube, 5-demicubic honeycomb, 5-polytope, 6-cube, 6-demicube, 7-cube, 7-demicube, 8-cube, 8-demicube, 8-demicubic honeycomb, 9-cube, 9-demicube. Expand index (52 more) »
Abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc.
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Cantic 5-cube
In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube.
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Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.
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Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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Demihypercube
In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn.
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Density (polytope)
In geometry, the density of a polytope represents the number of windings of a polytope, particularly a uniform or regular polytope, around its center.
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Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.
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Duoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher.
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Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).
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E9 honeycomb
In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space.
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Emanuel Lodewijk Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) at joodsmonument.nl was a Dutch mathematician.
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Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.
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Gosset–Elte figures
In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.
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Great duoantiprism
The great duoantiprism is the only uniform star-duoantiprism solution p.
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Heptellated 8-simplexes
In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.
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Hexic 7-cubes
In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube.
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Hexicated 7-cubes
In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-cube.
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Hexicated 7-simplexes
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
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Hexicated 8-simplexes
In eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication (6th order truncation) of the regular 8-simplex.
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Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.
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Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.
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Hypersimplex
In polyhedral combinatorics, a hypersimplex, Δd,k, is a convex polytope that generalizes the simplex.
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Isogonal figure
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.
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Isohedral figure
In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same.
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List of mathematical shapes
Following is a list of some mathematically well-defined shapes.
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List of polygons, polyhedra and polytopes
A polytope is a geometric object with flat sides, which exists in any general number of dimensions.
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List of regular polytopes and compounds
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
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Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.
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Omnitruncation
In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets.
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Paracompact uniform honeycombs
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells.
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Pentic 7-cubes
In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube.
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Polytope
In elementary geometry, a polytope is a geometric object with "flat" sides.
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Polytope families
There are several families of symmetric polytopes with irreducible symmetry which have a member in more than one dimensionality.
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Proprism
In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher.
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Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells.
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Rectified 5-simplexes
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
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Regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope.
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Runcic 5-cubes
In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope.
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Runcic 6-cubes
In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope.
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Runcic 7-cubes
In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube.
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Semiregular polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.
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Snub (geometry)
In geometry, a snub is an operation applied to a polyhedron.
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Steric 5-cubes
In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope.
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Steric 7-cubes
In seven-dimensional geometry, a stericated 7-cube (or runcinated 7-demicube) is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube.
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Stericated 24-cell honeycomb
In four-dimensional Euclidean geometry, the stericated 24-cell honeycomb (or stericated 16-cell honeycomb) is a uniform space-filling honeycomb.
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Stericated 5-cubes
In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.
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Stericated 5-simplexes
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
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Stericated 6-cubes
In six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-cube.
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Stericated 6-orthoplexes
In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex.
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Stericated 6-simplexes
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
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Stericated 7-cubes
In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-cube.
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Stericated 7-orthoplexes
In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex.
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Stericated 7-simplexes
In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex.
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Stericated 8-simplexes
In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex.
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Tesseractic honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol, and constructed by a 4-dimensional packing of tesseract facets.
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Tetrahedral prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope.
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Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.
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Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.
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Uniform 1 k2 polytope
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.
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Uniform 10-polytope
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
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Uniform 2 k1 polytope
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.
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Uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
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Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope.
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Uniform 6-polytope
In six-dimensional geometry, a uniform polypeton (or uniform 6-polytope) is a six-dimensional uniform polytope.
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Uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets.
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Uniform 8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.
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Uniform 9-polytope
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets.
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Uniform honeycomb
In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets.
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Uniform honeycombs in hyperbolic space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells.
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Uniform k 21 polytope
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.
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Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).
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Uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
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Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
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Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
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1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
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1 32 polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
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1 42 polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
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10-cube
In geometry, a 10-cube is a ten-dimensional hypercube.
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10-demicube
In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed.
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2 21 polytope
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.
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2 22 honeycomb
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space.
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2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
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2 41 polytope
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
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24-cell honeycomb
In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells.
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3 21 polytope
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.
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4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
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4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.
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5-cell
In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.
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5-demicube
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
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5-demicubic honeycomb
The 5-demicube honeycomb, or demipenteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space.
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5-polytope
In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets.
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6-cube
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
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6-demicube
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.
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7-cube
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.
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7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.
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8-cube
In geometry, an 8-cube is an eight-dimensional hypercube (8-cube).
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8-demicube
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed.
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8-demicubic honeycomb
The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space.
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9-cube
In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.
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9-demicube
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed.
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Redirects here:
Heptellation, Hexication, Scaliform polytope, Sterication, Sterication (geometry).
References
[1] https://en.wikipedia.org/wiki/Uniform_polytope
