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83 relations: Abstract polytope, Betti number, Cartesian product, Congruence (geometry), Convex polytope, Convex set, Convex uniform honeycomb, Cross-polytope, Cube, Cubic honeycomb, Duocylinder, Duoprism, Edge (geometry), Euclidean space, Euler characteristic, Face (geometry), Flag (geometry), Four-dimensional space, Geometry, Grid (spatial index), Group action, Harold Scott MacDonald Coxeter, Homology (mathematics), Honeycomb (geometry), Hyperbolic space, Hypercube, Isogonal figure, Isohedral figure, J. C. P. Miller, John Horton Conway, Johnson solid, Kepler–Poinsot polyhedron, Line segment, Michael Guy, Michael S. Longuet-Higgins, Net (polyhedron), Norman Johnson (mathematician), Polygon, Polyhedron, Polytope, Prism (geometry), Projection (linear algebra), Projective cover, Pyramid (geometry), Rectified 5-cell, Rectified 600-cell, Regular 4-polytope, Regular polygon, Regular polyhedron, Regular polytope, ..., Rhombic dodecahedral honeycomb, Schlegel diagram, Semiregular polytope, Simplex, Snub 24-cell, Space group, Square, Square tiling, Star polygon, Stella (software), Stereographic projection, Symmetry, Symmetry group, Tessellation, Tesseract, Tetragonal disphenoid honeycomb, Thorold Gosset, Uniform 4-polytope, Uniform honeycombs in hyperbolic space, Uniform polyhedron, Uniform polytope, Uniform star polyhedron, Vertex (geometry), Vertex figure, Weaire–Phelan structure, 11-cell, 120-cell, 16-cell, 24-cell, 3-sphere, 5-cell, 57-cell, 600-cell. Expand index (33 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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Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
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Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
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Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
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Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.
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Convex set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
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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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Cross-polytope
In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions.
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Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
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Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.
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Duocylinder
The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii r1 and r2: It is analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.
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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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Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
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Face (geometry)
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
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Flag (geometry)
In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.
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Four-dimensional space
A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Grid (spatial index)
In the context of a spatial index, a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
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Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
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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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Hyperbolic space
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
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Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.
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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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J. C. P. Miller
Jeffrey Charles Percy Miller (31 August 1906 – 24 April 1981) was an English mathematician and computing pioneer.
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John Horton Conway
John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
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Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform (i.e., not a Platonic solid, Archimedean solid, prism, or antiprism), and each face of which is a regular polygon.
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Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
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Line segment
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
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Michael Guy
Michael J. T. Guy (born c.1942) is a British computer scientist and mathematician.
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Michael S. Longuet-Higgins
Michael Selwyn Longuet-Higgins FRS (December 8, 1925 – February 26, 2016) was a mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute for Nonlinear Science, University of California, San Diego, USA.
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Net (polyhedron)
In geometry a net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.
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Norman Johnson (mathematician)
Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.
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Polygon
In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.
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Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.
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Polytope
In elementary geometry, a polytope is a geometric object with "flat" sides.
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Prism (geometry)
In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.
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Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
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Projective cover
In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.
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Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.
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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 600-cell
In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells.
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Regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope.
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Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
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Regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.
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Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.
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Rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space.
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Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^ through a point beyond one of its facets or faces.
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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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Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
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Snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells.
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Space group
In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.
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Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.
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Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane.
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Star polygon
In geometry, a star polygon is a type of non-convex polygon.
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Stella (software)
Stella, a computer program available in three versions (Great Stella, Small Stella and Stella4D), was created by Robert Webb of Australia.
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Stereographic projection
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.
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Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.
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Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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Tessellation
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.
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Tesseract
In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.
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Tetragonal disphenoid honeycomb
The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells.
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Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.
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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 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 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 polytope
A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
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Uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron.
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Vertex (geometry)
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.
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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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Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles.
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11-cell
In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).
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120-cell
In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.
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16-cell
In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.
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24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
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3-sphere
In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.
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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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57-cell
In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).
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600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
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4-dimensional polytope, 4-polytopes, Hexachoron, Polychora, Polychoron.
References
[1] https://en.wikipedia.org/wiki/4-polytope
