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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. [1]

145 relations: Alicia Boole Stott, Alternation (geometry), Antiprism, Archimedean solid, Asterisk, Bitruncation, Branko Grünbaum, Cantellated 120-cell, Cantellated 24-cells, Cantellated 5-cell, Cantellated tesseract, Cantellation (geometry), Cartesian product, Chaim Goodman-Strauss, Convex uniform honeycomb, Coxeter element, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Cube, Cubic honeycomb, Cuboctahedral prism, Cuboctahedron, Decagonal prism, Dimension, Disphenoid, Dodecahedral prism, Dodecahedron, Dual polyhedron, Duoprism, Edmund Hess, Emanuel Lodewijk Elte, Expansion (geometry), Fundamental domain, Geometry, Grand antiprism, Great dodecahedron, Greek language, Harold Scott MacDonald Coxeter, Hexagonal prism, Hyperplane, Icosahedral prism, Icosidodecahedral prism, Icosidodecahedron, Isogonal figure, John Horton Conway, Kaleidoscope, Line segment, Ludwig Schläfli, Michael Guy, ... Expand index (95 more) »

## Alicia Boole Stott

Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish-English mathematician.

## Alternation (geometry)

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

## Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

## Archimedean solid

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes.

## Asterisk

An asterisk (*); from Late Latin asteriscus, from Ancient Greek ἀστερίσκος, asteriskos, "little star") is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often vocalize it as star (as, for example, in the A* search algorithm or C*-algebra). In English, an asterisk is usually five-pointed in sans-serif typefaces, six-pointed in serif typefaces, and six- or eight-pointed when handwritten. It is often used to censor offensive words, and on the Internet, to indicate a correction to a previous message. The asterisk is derived from the need of the printers of family trees in feudal times for a symbol to indicate date of birth. The original shape was seven-armed, each arm like a teardrop shooting from the center. In computer science, the asterisk is commonly used as a wildcard character, or to denote pointers, repetition, or multiplication.

## Bitruncation

In geometry, a bitruncation is an operation on regular polytopes.

## Branko Grünbaum

Branko Grünbaum (ברנקו גרונבאום; born 2 October 1929) is a Yugoslavian-born mathematician and a professor emeritus at the University of Washington in Seattle.

## Cantellated 120-cell

In 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 24-cells

In 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 5-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 the regular 5-cell.

## Cantellated tesseract

In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract.

## Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that bevels a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex.

## 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.

## Chaim Goodman-Strauss

Chaim Goodman-Strauss (born June 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling.

## 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.

## Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

## 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).

## 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.

## 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).

## 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.

## 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.

## Cuboctahedral prism

In geometry, a cuboctahedral prism is a convex uniform 4-polytope.

## Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

## Decagonal prism

In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Disphenoid

In geometry, a disphenoid (from Greek sphenoeides, "wedgelike") is a tetrahedron whose four faces are congruent acute-angled triangles.

## Dodecahedral prism

In geometry, a dodecahedral prism is a convex uniform 4-polytope.

## Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

## 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.

## 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.

## Edmund Hess

Edmund Hess (17 February 1843 &ndash; 24 December 1903) was a German mathematician who discovered several regular polytopes.

## Emanuel Lodewijk Elte

Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) at joodsmonument.nl was a Dutch mathematician.

## 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.

## Fundamental domain

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

## 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.

## Grand antiprism

In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra.

## Great dodecahedron

In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of.

## Greek language

Greek (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean and the Black Sea.

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base.

## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

## Icosahedral prism

In geometry, an icosahedral prism is a convex uniform 4-polytope (four-dimensional polytope).

## Icosidodecahedral prism

In geometry, an icosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Icosidodecahedron

In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces.

## 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.

## 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.

## Kaleidoscope

A kaleidoscope is an optical instrument with two or more reflecting surfaces tilted to each other in an angle, so that one or more (parts of) objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection.

## 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.

## Ludwig Schläfli

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

## Michael Guy

Michael J. T. Guy (born c.1942) is a British computer scientist and mathematician.

## Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

## Octagonal prism

In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.

## Octahedral prism

In geometry, a octahedral prism is a convex uniform 4-polytope.

## Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

## 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.

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

## Overdetermined system

In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns.

## Paracompact uniform honeycombs

In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells.

## Pentagon

In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon or 5-gon.

## Pentagonal antiprism

In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

## Pentagonal prism

In geometry, the pentagonal prism is a prism with a pentagonal base.

## Pi

The number is a mathematical constant.

## Pieter Hendrik Schoute

Pieter Hendrik Schoute (21 January 1846, Wormerveer &ndash; 18 April 1923, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry.

## Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

## Point group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.

## Point groups in four dimensions

In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.

## 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.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## 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.

## Radian

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

## Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

## Rectified 120-cell

In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell.

## Rectified 24-cell

In geometry, the rectified 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 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.

## 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.

## Rectified tesseract

In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra.

## Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

## Regular icosahedron

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices.

## 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).

## 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.

## Rhombicosidodecahedral prism

In geometry, a rhombicosidodecahedral prism or small rhombicosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Rhombicosidodecahedron

In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

## Rhombicuboctahedral prism

In geometry, a rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Rhombicuboctahedron

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces.

## Runcinated 120-cells

In four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.

## Runcinated 24-cells

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 5-cell

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 regular 5-cell.

## Runcinated tesseracts

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract.

## Runcination

In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers.

## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

## 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.

## 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.

## Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol.

## 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.

## Snub cube

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles.

## Snub cubic prism

In geometry, a snub cubic prism or snub cuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Snub dodecahedral prism

In geometry, a snub dodecahedral prism or snub icosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Snub dodecahedron

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

## Square antiprism

In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

## Square pyramid

In geometry, a square pyramid is a pyramid having a square base.

## Star polyhedron

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

## Stella (software)

Stella, a computer program available in three versions (Great Stella, Small Stella and Stella4D), was created by Robert Webb of Australia.

## Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

## 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.

## 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.

## Symmetry number

The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, i.e. the order of its symmetry group.

## 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.

## Tetrahedral prism

In geometry, a tetrahedral prism is a convex uniform 4-polytope.

## Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

## Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 &ndash; December 1962) was an English lawyer and an amateur mathematician.

## Triangle

A triangle is a polygon with three edges and three vertices.

## Triangular cupola

In geometry, the triangular cupola is one of the Johnson solids (J3).

## 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.

## Tridiminished icosahedron

In geometry, the tridiminished icosahedron is one of the Johnson solids (J63).

## Truncated 120-cells

In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell.

## Truncated 24-cells

In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.

## Truncated 5-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.

## Truncated cube

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid.

## Truncated cubic prism

In geometry, a truncated cubic prism is a convex uniform polychoron (four-dimensional polytope).

## Truncated cuboctahedral prism

In geometry, a truncated cuboctahedral prism or great rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Truncated cuboctahedron

In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron.

## Truncated dodecahedral prism

In geometry, a truncated dodecahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid.

## Truncated icosahedral prism

In geometry, a truncated icosahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Truncated icosahedron

In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.

## Truncated icosidodecahedral prism

In geometry, a truncated icosidodecahedral prism or great rhombicosidodecahedral prism is a convex uniform 4-polytope (four-dimensional polytope).

## Truncated icosidodecahedron

In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

## Truncated octahedral prism

In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope.

## Truncated octahedron

In geometry, the truncated octahedron is an Archimedean solid.

## Truncated tesseract

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

## Truncated tetrahedral prism

In geometry, a truncated tetrahedral prism is a convex uniform polychoron (four-dimensional polytope).

## Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid.

## 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.

## Uniform antiprismatic prism

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 hyperplanes, connected by uniform prisms cells between pairs of faces.

## Uniform honeycombs in hyperbolic space

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells.

## 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).

## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

## Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

## Victor Klee

Victor L. Klee, Jr. (September 18, 1925, San Francisco &ndash; August 17, 2007, Lakewood, Ohio) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics.

## Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

## 120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

## 16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

## 24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

## 5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

## 600-cell

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

## References

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