117 relations: Alternation (geometry), Arithmetic group, Artin group, Canada, Cartan matrix, Checkerboard, Clockwise, Commensurability (mathematics), Complete graph, Complex plane, Complex polytope, Complex reflection group, Congruence (geometry), Convex uniform honeycomb, Cornell University Library, Coxeter element, Coxeter group, Coxeter notation, Crystallographic restriction theorem, Danny Calegari, Dihedral angle, Dihedron, Directed graph, Dual polygon, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 (mathematics), E8 lattice, Edge (geometry), Eigenvalues and eigenvectors, En (Lie algebra), Ernest Vinberg, Ernst Witt, Euclidean tilings by convex regular polygons, F4 (mathematics), Face (geometry), Facet (geometry), Fundamental domain, G2 (mathematics), General relativity, Geometry, Golden ratio, Goursat tetrahedron, Gramian matrix, Graph (discrete mathematics), Greatest common divisor, Harold Scott MacDonald Coxeter, Hendrik Lorentz, Heptagonal tiling honeycomb, ..., Hosohedron, Hyperbolic geometry, Hypercubic honeycomb, Hyperplane, Injective function, Jørgen Pedersen Gram, Jean-Louis Koszul, Kaleidoscope, Line segment, List of convex uniform tilings, List of regular polytopes and compounds, List of uniform polyhedra, Ludwig Schläfli, Mirror, Mostow rigidity theorem, Norman Johnson (mathematician), Octahedral symmetry, Orbifold notation, Order (group theory), Order-5 5-cell honeycomb, Order-7 tetrahedral honeycomb, PDF, Perpendicular, Poincaré disk model, Poincaré half-plane model, Point reflection, Polygon, Polyhedron, Polytope, Prism (geometry), Projection (linear algebra), Pyramid (geometry), Rational number, Rectangle, Regular polygon, Regular polytope, Regular Polytopes (book), Rhombus, Root system, Rotations in 4-dimensional Euclidean space, Ruth Kellerhals, Schläfli orthoscheme, Schläfli symbol, Schwarz triangle, Semisimple Lie algebra, Simple Lie group, Simplex, Snub (geometry), Special relativity, Square, Square tiling, Tetrahedron, Toronto, Triangle group, Triangular prism, Ultraparallel theorem, Uniform 4-polytope, Uniform honeycombs in hyperbolic space, Uniform polyhedron, Uniform polytope, Uniform tiling, Uniform tilings in hyperbolic plane, Unitary operator, Vector space, Vinberg's algorithm, Wythoff construction, Wythoff symbol. Expand index (67 more) »

## 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.

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## Arithmetic group

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z).

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## Artin group

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form \begin \Big\langle x_1,x_2,\ldots,x_n \Big| \langle x_1, x_2 \rangle^ &.

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## Canada

Canada is a country located in the northern part of North America.

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## Cartan matrix

In mathematics, the term Cartan matrix has three meanings.

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## Checkerboard

A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of chequered pattern on which English draughts (checkers) is played.

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## Clockwise

Two-dimensional rotation can occur in two possible directions.

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## Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio is a rational number; otherwise a and b are called incommensurable.

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## Complete graph

No description.

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## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

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## 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.

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## Complex reflection group

In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.

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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 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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## Cornell University Library

The Cornell University Library is the library system of Cornell University.

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## Coxeter element

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

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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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## Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold.

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## Danny Calegari

Danny M. C. Calegari (born 24 May 1972) is an Australian-American mathematician who is currently a Professor at the University of Chicago.

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## Dihedral angle

A dihedral angle is the angle between two intersecting planes.

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## Dihedron

A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges.

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## Directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them.

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## Dual polygon

In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.

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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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## E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.

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## E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

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## E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

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## E8 lattice

In mathematics, the E8 lattice is a special lattice in R8.

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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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## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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## En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.

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## Ernest Vinberg

Ernest Borisovich Vinberg (Эрнест Борисович Винберг; born 26 July 1937) is a Russian mathematician, who works on discrete subgroups of Lie groups and representation theory.

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## Ernst Witt

Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.

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## Euclidean tilings by convex regular polygons

Euclidean plane tilings by convex regular polygons have been widely used since antiquity.

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## F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.

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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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## 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.

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## 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.

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## G2 (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups.

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## General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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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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## Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

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## Goursat tetrahedron

In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction.

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## Gramian matrix

In linear algebra, the Gram matrix (Gramian matrix or Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by G_.

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## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

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## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

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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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## Hendrik Lorentz

Hendrik Antoon Lorentz (18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect.

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## Heptagonal tiling honeycomb

In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation (or honeycomb).

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## Hosohedron

In geometry, an ''n''-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

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## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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## Hypercubic honeycomb

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn (or B~n-1) for n>.

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## Hyperplane

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

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## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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## Jørgen Pedersen Gram

Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark.

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## Jean-Louis Koszul

Jean-Louis Koszul (January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex.

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## 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.

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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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## List of convex uniform tilings

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.

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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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## List of uniform polyhedra

In geometry, 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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## 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.

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## Mirror

A mirror is an object that reflects light in such a way that, for incident light in some range of wavelengths, the reflected light preserves many or most of the detailed physical characteristics of the original light, called specular reflection.

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## Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique.

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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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## Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.

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## Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

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## Order (group theory)

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

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## Order-5 5-cell honeycomb

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

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## Order-7 tetrahedral honeycomb

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol.

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The Portable Document Format (PDF) is a file format developed in the 1990s to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.

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## Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

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## Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

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## Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

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## Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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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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## 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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## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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## Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

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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 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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## Regular Polytopes (book)

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

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## Rhombus

In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length.

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## Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

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## Rotations in 4-dimensional Euclidean space

In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4).

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## Ruth Kellerhals

Ruth Kellerhals (born 17 July 1957) is a Swiss mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities.

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## Schläfli orthoscheme

In geometry, Schläfli orthoscheme is a type of simplex.

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## Schläfli symbol

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

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## Schwarz triangle

In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges.

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## Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.

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## Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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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 (geometry)

In geometry, a snub is an operation applied to a polyhedron.

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## Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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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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## 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.

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## Toronto

Toronto is the capital city of the province of Ontario and the largest city in Canada by population, with 2,731,571 residents in 2016.

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## Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle.

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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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## Ultraparallel theorem

In hyperbolic geometry, the ultraparallel theorem states that every pair of ultraparallel lines (lines that are not intersecting and not limiting parallel) has a unique common perpendicular hyperbolic line.

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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 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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## Uniform tilings in hyperbolic plane

In hyperbolic geometry, a uniform (regular, quasiregular or semiregular) hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane 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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## Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

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## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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## Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group.

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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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## Wythoff symbol

In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle.

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## Redirects here:

Arithmetic triangle group, Cocompact Coxeter group, Coxeter diagram, Coxeter-Dynkin diagram, Hyperbolic Coxeter group, Hyperbolic reflection group, Lannér group, Lorentzian Coxeter group, Schläfli Criterion, Schläfli criterion, Schläfli matrix, Schläfli's criterion, Schläflian.

## References

[1] https://en.wikipedia.org/wiki/Coxeter–Dynkin_diagram