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Four-dimensional geometric object with flat sides
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure
4-polytope
Four-dimensional analogues of the regular polyhedra in three dimensions
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular
Regular_4-polytope
Class of 4-dimensional polytopes
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra
Uniform_4-polytope
Geometric object with flat sides
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any
Polytope
Four-dimensional analog of the octahedron
convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described
16-cell
Polytope in 8-dimensional geometry
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset
4_21_polytope
Four-dimensional analogue of the tetrahedron
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells
5-cell
Four-dimensional analogue of the cube
regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken
Tesseract
5-dimensional geometric object
geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which
5-polytope
regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. There is only one polytope of
List_of_regular_polytopes
Regular polytope dual to the hypercube in any number of dimensions
2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets
Cross-polytope
Polytope with highest degree of symmetry
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In
Regular_polytope
Regular star 4-polytope with 600 faces
polytetrahedron is a regular star 4-polytope with Schläfli symbol {3, 3, 5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600
Grand_600-cell
the 5-polytope Edge the (n−4)-face of the 5-polytope Face the peak or (n−3)-face of the 5-polytope Cell the ridge or (n−2)-face of the 5-polytope Hypercell
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
needed] 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated
List_of_mathematical_shapes
Isogonal polytope with uniform facets
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means
Uniform_polytope
Isogonal polytope with regular facets
definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled
Semiregular_polytope
24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra
Rectified_24-cell
In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two
Truncated_5-cell
Operation in Euclidean geometry
bitruncated 4-polytope is the same as the bitruncated dual, and will have double the symmetry if the original 4-polytope is self-dual. A regular polytope (or
Bitruncation
Geometric space with four dimensions
regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension, including six found in 4-dimensional
Four-dimensional_space
Regular object in four dimensional geometry
In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra
24-cell
3D shape made of polyhedra sharing a common center
paper New Regular Compounds of 4-Polytopes. Self-duals: Dual pairs: Uniform compounds and duals with convex 4-polytopes: The superscript (var) in the tables
Polytope_compound
Operation in Euclidean geometry
of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded
Rectification_(geometry)
Four-dimensional analog of the dodecahedron
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called
120-cell
Five-dimensional geometric shape
5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets
Uniform_5-polytope
600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell. There are 4 degrees of runcinations of the
Runcinated_120-cells
Complicated polygon
polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes. It has the same edge arrangement
Small_stellated_120-cell
Uniform polychoron
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge
Rectified_5-cell
In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and
3-4_duoprism
geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It
Rectified_tesseract
Type of uniform 4-polytope in four-dimensional geography
In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group.[citation needed]
Prismatic_uniform_4-polytope
Planar surface that forms part of the boundary of a solid object
is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two
Face_(geometry)
Poset representing certain properties of a polytope
mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying
Abstract_polytope
Cartesian product of two polytopes
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions
Duoprism
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two
Truncated_24-cells
or icosaplex is a regular star 4-polytope with Schläfli symbol {3,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is constructed by 5 icosahedra
Icosahedral_120-cell
Uniform 4-polytope bounded by 320 cells
antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300
Grand_antiprism
Edge-joined polygons which fold into a polyhedron
shortest path between two points on a cuboid. A net of a 4-polytope, a four-dimensional polytope, is composed of polyhedral cells that are connected by
Net_(polyhedron)
Regular star 4-polytope
regular star 4-polytope with Schläfli symbol {5/2,3,5}. It is one of 10 regular Schläfli-Hess polytopes. It is one of four regular star 4-polytopes discovered
Great_stellated_120-cell
In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations
Projectively_unique_polytope
Uniform 4-polytope
tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces:
Tetrahedral_prism
Polyhedron formed by joining mirroring pyramids base-to-base
The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following: A is the apex
Bipyramid
Removal of alternate vertices
is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed
Alternation_(geometry)
Shape made by slicing off a corner of a polytope
broadly speaking, is the figure exposed when a corner of a general n-polytope is sliced off. Take some corner or vertex of a polyhedron. Mark a point
Vertex_figure
Skew polygon derived from a polytope
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (but no n) belongs to one
Petrie_polygon
Generalisation of dice with identical faces
geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive
Isohedral_figure
Notation for polytopes and tessellations
squares around each vertex and is represented by {4,3}. A regular 4-dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented
Schläfli_symbol
Uniform 4-polytope
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three truncations, including a bitruncation
Truncated_120-cells
regular star 4-polytope with Schläfli symbol {5,3,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is one of four regular star 4-polytopes discovered
Grand_120-cell
Isogonal honeycomb of uniform polytope facets
terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex
Uniform_honeycomb
Four-dimensional analog of the icosahedron
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known
600-cell
Type of regular star 4-polytope
polydodecahedron is a regular star 4-polytope with Schläfli symbol {5,5/2,3}. It is one of 10 regular Schläfli-Hess polytopes. It has the same edge arrangement
Great_grand_120-cell
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell.
Runcinated_24-cells
Type of tesseract
In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a
Truncated_tesseract
Regular Schläfli-Hess 4-polytope with 600 vertices
polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having
Great grand stellated 120-cell
Great_grand_stellated_120-cell
4-D shape
In 4-dimensional geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space
Uniform_antiprismatic_prism
Abstract regular 4-polytope
mathematics, the 11-cell is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices
11-cell
the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge
Rectified_600-cell
regular star 4-polytope with Schläfli symbol {5/2,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is also one of two such polytopes that is self-dual
Grand_stellated_120-cell
In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of
Cantellated_5-cell
faceted 600-cell is a regular star 4-polytope with Schläfli symbol {3,5/2,5}. It is one of 10 regular Schläfli-Hess polytopes. It has the same edge arrangement
Great_icosahedral_120-cell
Geometric prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms
Octahedral_prism
regular star 4-polytope with Schläfli symbol {5,5/2,5}. It is one of 10 regular Schläfli-Hess polytopes. It is one of the two such polytopes that is self-dual
Great_120-cell
Four-dimensional geometric objects
In 4-dimensional geometry, there are 15 uniform polytopes with H4 symmetry. Two of these, the 120-cell and 600-cell, are regular. Each can be visualized
H4_polytope
4D geometry item
four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell
Cantellated_120-cell
Polytope whose facets are all simplices
simplicial polytopes In geometry, a simplicial polytope is a polytope whose facets are all simplices. It is topologically dual to simple polytopes. Polytopes that
Simplicial_polytope
6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively
E6_polytope
N-dimensional polytope with vertices adjacent to N facets
polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope
Simple_polytope
Four-dimensional geometrical object
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the
Runcinated_5-cell
Compound polyhedron
compound of five tetrahedra is related to the regular 5-cell, the 4-simplex regular 4-polytope, which is also composed of 5 regular tetrahedra. In the 5-cell
Compound_of_five_tetrahedra
Representation of 3D and 4D polytopes
polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope
Schlegel_diagram
the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. The duoprism is a 4-polytope that can be constructed
3-3_duoprism
Convex hull of a finite set of points in a Euclidean space
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional
Convex_polytope
vertex convex 4-polytope composed of 96 irregular cells. Each cell has faces of two kinds: three kites and six isosceles triangles. The polytope has a total
Dual_snub_24-cell
Convex uniform 4-polytope
four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract
Cantellated_tesseract
Geometric operation on a regular polytope
cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each
Cantellation_(geometry)
(pentacontaheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices
57-cell
Polyhedron with four faces
tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme
Tetrahedron
6-dimensional geometric object
six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. A 6-polytope is a closed six-dimensional figure
6-polytope
Number of windings of a polytope around its center of symmetry
densities. There are 10 regular star 4-polytopes (called the Schläfli–Hess 4-polytopes), which have densities between 4, 6, 20, 66, 76, and 191. They come
Density_(polytope)
4-D polyhedral pyramid
regular cells, it is a Blind polytope. Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope. The regular 16-cell has
Octahedral_pyramid
Flat-sided three-dimensional shape
two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several
Polyhedron
geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra
Snub_24-cell
In the study of abstract polytopes, a chiral polytope is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms
Chiral_polytope
16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of
Runcinated_tesseracts
5-dimensional hypercube
deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the
5-cube
four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell
Cantellated_24-cells
Property of a mathematical space
polygon Volume 4 dimensions Spacetime Fourth spatial dimension Convex regular 4-polytope Quaternion 4-manifold Polychoron Rotations in 4-dimensional Euclidean
Dimension
Generalization of a polytope in real space
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension
Complex_polytope
figure for a regular 4-polytope {p,q,r} is an q-gonal antiprism (called an antipodium if p and r are different). For regular 4-polytopes/honeycombs, this
Runcination
Specific set of Hamiltonian quaternions with the same symmetry as the 600-cell
the E8 lattice. This construction shows that the Coxeter group H 4 {\displaystyle H_{4}} embeds as a subgroup of E 8 {\displaystyle E_{8}} . Indeed, a
Icosian
4-D convex polytope
pyramids. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb. The dual four-dimensional polytope of a cubic pyramid is an octahedral
Cubic_pyramid
In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices
B4_polytope
Convex polytope, the n-dimensional analogue of a square and a cube
measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The
Hypercube
Abstract regular 4-polytope with 4 cubic cells
In abstract geometry, a hemitesseract is an abstract, regular 4-polytope, existing in real projective space RP3. It contains one k-face for each pair of
Hemitesseract
4-D uniform polytope
In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms
Dodecahedral_prism
In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices. A4 symmetry
A4_polytope
In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond
Graph_of_a_polytope
6-dimensional hypercube
(the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed
6-cube
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