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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. [1]

124 relations: Abstract polytope, Aesthetics, Alicia Boole Stott, Ancient Greece, Animation, Apeirotope, Archimedean solid, Arthur Cayley, Augustin-Louis Cauchy, Bartel Leendert van der Waerden, Branko Grünbaum, Chessboard, Compass-and-straightedge construction, Complex number, Complex polytope, Constructible polygon, Convex polygon, Coprime integers, Coxeter group, Cross-polytope, CT scan, Cube, Cubic honeycomb, Cuboctahedron, Dodecahedron, Dual polyhedron, Duncan Sommerville, Edge (geometry), Equilateral triangle, Euclid, Euclid's Elements, Euclidean space, Face (geometry), Facet (geometry), Flag (geometry), Grand stellated 120-cell, Great 120-cell, Great dodecahedron, Great icosahedron, Great stellated dodecahedron, Group (mathematics), Group action, Harold Scott MacDonald Coxeter, Hexagonal tiling honeycomb, Hilbert space, Holography, Honeycomb (geometry), Hyperbolic space, Hypercube, Icosahedral honeycomb, ... Expand index (74 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.

## Aesthetics

Aesthetics (also spelled esthetics) is a branch of philosophy that explores the nature of art, beauty, and taste, with the creation and appreciation of beauty.

## Alicia Boole Stott

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

## Ancient Greece

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).

## Animation

Animation is a dynamic medium in which images or objects are manipulated to appear as moving images.

## Apeirotope

An apeirotope or infinite polytope is a polytope which has infinitely many facets.

## Archimedean solid

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

## Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

## Bartel Leendert van der Waerden

Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician and historian of mathematics.

## 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.

## Chessboard

A chessboard is the type of checkerboard used in the board game chess, consisting of 64 squares (eight rows and eight columns).

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex polytope

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 is accompanied by an imaginary one.

## Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge.

## Convex polygon

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## 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).

## Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions.

## CT scan

A CT scan, also known as computed tomography scan, makes use of computer-processed combinations of many X-ray measurements taken from different angles to produce cross-sectional (tomographic) images (virtual "slices") of specific areas of a scanned object, allowing the user to see inside the object without cutting.

## 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.

## Cuboctahedron

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

## 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.

## Duncan Sommerville

Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer.

## Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

## Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## 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.

## Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

## 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.

## Grand stellated 120-cell

In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol.

## Great 120-cell

In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol.

## Great dodecahedron

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

## Great icosahedron

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of.

## Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## 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.

## 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 tiling honeycomb

In the field of hyperbolic geometry, the hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Holography

Holography is the science and practice of making holograms.

## 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.

## 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.

## Hypercube

In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.

## Icosahedral honeycomb

The icosahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.

## Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

## 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.

## Kepler–Poinsot polyhedron

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

## 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.

## List of regular polytopes and compounds

This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.

## Louis Poinsot

Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist.

## 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.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## 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.

## Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Octahedron

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

## Order-4 square tiling honeycomb

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb, is one of 11 paracompact regular honeycombs.

## Order-5 120-cell honeycomb

In the geometry of hyperbolic 4-space, the order-5 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs).

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## Order-6 hexagonal tiling honeycomb

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.

## Origami

) is the art of paper folding, which is often associated with Japanese culture.

## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

## Orthographic projection

Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions.

## Palindrome

A palindrome is a word, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar.

## 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.

## Pentagram

A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a five-pointed star drawn with five straight strokes.

## Perspective (graphical)

Perspective (from perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye.

## Peter McMullen

Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London.

## Petrie polygon

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 of the facets.

## Physical cosmology

Physical cosmology is the study of the largest-scale structures and dynamics of the Universe and is concerned with fundamental questions about its origin, structure, evolution, and ultimate fate.

## Platonic solid

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

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## 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.

## 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.

## Polytope

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

## Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

## Regular 4-polytope

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

## 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 polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.

## Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter.

## Reinhold Hoppe

Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the University of Berlin.

## Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.

## 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.

## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

## Skew apeirogon

In geometry, an infinite skew polygon (or skew apeirogon) has vertices that are not all colinear.

## Small stellated dodecahedron

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

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Spherical polyhedron

In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons.

## 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.

## Star polygon

In geometry, a star polygon is a type of non-convex polygon.

## String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

## 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.

## 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.

## 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.

## 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.

## Theaetetus (mathematician)

Theaetetus of Athens (Θεαίτητος; c. 417 – 369 BC), possibly the son of Euphronius of the Athenian deme Sunium, was a Greek mathematician.

Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury.

## Université libre de Bruxelles

The Université libre de Bruxelles (in English: Free University of Brussels), abbreviated ULB, is a French-speaking private research university in Brussels, Belgium.

## University of Illinois at Urbana–Champaign

The University of Illinois Urbana–Champaign (also known as U of I, Illinois, or colloquially as the University of Illinois or UIUC) is a public research university in the U.S. state of Illinois and the flagship institution of the University of Illinois System.

## Vertex figure

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

## Virtual reality

Virtual reality (VR) is an interactive computer-generated experience taking place within a simulated environment, that incorporates mainly auditory and visual, but also other types of sensory feedback like haptic.

## 11-cell

In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).

## 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.

## 16-cell honeycomb honeycomb

In the geometry of hyperbolic 5-space, the 16-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs).

## 24-cell

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

## 4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.

## 5-cell

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

## 5-cube

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

## 5-orthoplex

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 cells, 32 5-cell 4-faces.

## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

## 57-cell

In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).

## 600-cell

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

## 8-cube

In geometry, an 8-cube is an eight-dimensional hypercube (8-cube).

## References

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