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# Schläfli symbol

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

## Alternated octagonal tiling

In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane.

## Alternated order-4 hexagonal tiling

In geometry, the alternated order-4 hexagonal tiling or ditetragonal tritetratrigonal tiling is a uniform tiling of the hyperbolic plane.

## 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.

## Angular defect

In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would.

## 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.

## Apeirogonal antiprism

In geometry, an apeirogonal antiprism or infinite antiprism is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane.

## Apeirogonal hosohedron

In geometry, an apeirogonal hosohedron or infinite hosohedron is a tiling of the plane consisting of two vertices at infinity.

## Apeirogonal prism

In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.

## ß

In German orthography, the grapheme ß, called Eszett or scharfes S, in English "sharp S", represents the phoneme in Standard German, specifically when following long vowels and diphthongs, while ss is used after short vowels.

## Beta

Beta (uppercase, lowercase, or cursive; bē̂ta or βήτα) is the second letter of the Greek alphabet.

## Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

## Bitruncation

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

## 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.

## Cantic octagonal tiling

In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane.

## 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.

## Compound of two icosahedra

This uniform polyhedron compound is a composition of 2 icosahedra.

## 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.

## Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

## 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).

## Cross-polytope

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

## 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.

## Dodecagon

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

## 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.

## Duopyramid

In geometry of 4 dimensions or higher, a duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two.

## Elongated triangular tiling

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane.

## Equilateral triangle

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

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Euclidean space

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

## 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.

## 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.

## 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.

## Great duoantiprism

The great duoantiprism is the only uniform star-duoantiprism solution p.

## Harold Scott MacDonald Coxeter

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

## Heptagonal tiling

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane.

## Hexagon

In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six-sided polygon or 6-gon.

## Hexagonal tiling

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex.

## Hexagram

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol, 2, or.

## Hexaoctagonal tiling

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

## 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.

## Hyperbolic geometry

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

## 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 symmetry

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.

## Icosahedron

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

## Infinite-order apeirogonal tiling

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane.

## Infinite-order pentagonal tiling

In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling.

## Infinite-order square tiling

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane.

## Infinite-order triangular tiling

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of.

## Irreducible fraction

An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

## 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.

## 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.

## Norman Johnson (mathematician)

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

## Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane.

## 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.

## 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 of operations

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

## Order-3 apeirogonal tiling

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane.

## Order-4 apeirogonal tiling

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane.

## Order-4 dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs).

## Order-4 heptagonal tiling

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane.

## Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane.

## Order-4 octagonal tiling

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane.

## Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane.

## Order-5 apeirogonal tiling

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane.

## Order-5 hexagonal tiling

In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane.

## Order-5 pentagonal tiling

In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane.

## Order-5 square tiling

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane.

## Order-6 hexagonal tiling

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane.

## Order-6 octagonal tiling

In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane.

## Order-6 pentagonal tiling

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane.

## Order-6 square tiling

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane.

## Order-7 heptagonal tiling

In geometry, the order-7 heptagonal tiling is a regular tiling of the hyperbolic plane.

## Order-7 square tiling

In geometry, the order-7 square tiling is a regular tiling of the hyperbolic plane.

## Order-7 triangular tiling

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of.

## Order-8 hexagonal tiling

In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane.

## Order-8 octagonal tiling

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane.

## Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane.

## Order-8 triangular tiling

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane.

## Palindrome

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

## Pentaapeirogonal tiling

In geometry, the pentaapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r.

## 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.

## Pentahexagonal tiling

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane.

## 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.

## Polygram (geometry)

In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides, so a regular pentagram,, has 5 sides, and the regular hexagram, or 2 has 6 sides divided into two triangles.

## 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.

## 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.

## Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.

## Quarter order-6 square tiling

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane.

## Quasiregular polyhedron

In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex.

## Rectangle

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

## 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 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 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.

## Recursive definition

A recursive definition (or inductive definition) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff).

## Reflection symmetry

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.

## 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 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.

## Regular Polytopes (book)

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

## Rhombicuboctahedron

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

## Rhombihexaoctagonal tiling

In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane.

## Rhombipentahexagonal tiling

In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitetraapeirogonal tiling

In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitetraheptagonal tiling

In geometry, the rhombitetraheptagonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitetrahexagonal tiling

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitetraoctagonal tiling

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitetrapentagonal tiling

In geometry, the rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane.

## Rhombitriapeirogonal tiling

In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr.

## Rhombitriheptagonal tiling

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane.

## Rhombitrihexagonal tiling

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane.

## Rhombitrioctagonal tiling

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane.

## 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.

## 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.

## Simplex

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

## Snub (geometry)

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

## 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 apeiroapeirogonal tiling

In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane.

## Snub cube

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

## Snub heptaheptagonal tiling

In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub hexahexagonal tiling

In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub hexaoctagonal tiling

In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane.

## Snub octaoctagonal tiling

In geometry, the snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub order-6 square tiling

In geometry, the snub tetratritetragonal tiling or snub order-6 square tiling is a uniform tiling of the hyperbolic plane.

## Snub order-8 triangular tiling

In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane.

## Snub pentahexagonal tiling

In geometry, the snub pentahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub pentapentagonal tiling

In geometry, the snub pentapentagonal tiling is a regular tiling of the hyperbolic plane.

## Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane.

## Snub tetraapeirogonal tiling

In geometry, the snub tetrapeirogonal tiling is a uniform tiling of the hyperbolic plane.

## Snub tetraheptagonal tiling

In geometry, the snub tetraheptagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub tetrahexagonal tiling

In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub tetraoctagonal tiling

In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub tetrapentagonal tiling

In geometry, the snub tetrapentagonal tiling is a uniform tiling of the hyperbolic plane.

## Snub triapeirogonal tiling

In geometry, the snub triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of sr.

## Snub triheptagonal tiling

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane.

## Snub trihexagonal tiling

In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane.

## Snub trioctagonal tiling

In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane.

## 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.

## Square tiling

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane.

## Star polygon

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

## Stellated octahedron

The stellated octahedron is the only stellation of the octahedron.

## 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.

## Tetraapeirogonal tiling

In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r.

## Tetrahedral symmetry

A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

## 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.

## Tetraheptagonal tiling

In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane.

## Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane.

## Tetraoctagonal tiling

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

## Tetrapentagonal tiling

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane.

## Trapezohedron

The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism.

## Triangle

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

## Triangular tiling

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane.

## Triapeirogonal tiling

In geometry, the triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r.

## Triheptagonal tiling

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling.

## Trihexagonal tiling

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons.

## Trioctagonal tiling

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling.

## 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 cube

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

## Truncated cuboctahedron

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

## Truncated heptagonal tiling

In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated hexagonal tiling

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane.

## Truncated hexaoctagonal tiling

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated infinite-order square tiling

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane.

## Truncated infinite-order triangular tiling

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t.

## Truncated octagonal tiling

In geometry, the Truncated octagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated octahedron

In geometry, the truncated octahedron is an Archimedean solid.

## Truncated order-3 apeirogonal tiling

In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t.

## Truncated order-4 apeirogonal tiling

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-4 heptagonal tiling

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-4 hexagonal tiling

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-4 octagonal tiling

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-4 pentagonal tiling

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-5 hexagonal tiling

In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-5 pentagonal tiling

In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane.

## Truncated order-5 square tiling

In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-6 hexagonal tiling

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-6 octagonal tiling

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-6 pentagonal tiling

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-6 square tiling

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-7 heptagonal tiling

In geometry, the truncated order-7 heptagonal tiling is a regular tiling of the hyperbolic plane.

## Truncated order-7 square tiling

In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-7 triangular tiling

In geometry, the Order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane.

## Truncated order-8 hexagonal tiling

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated order-8 octagonal tiling

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated order-8 triangular tiling

In geometry, the Truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane.

## Truncated pentahexagonal tiling

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex.

## Truncated tesseract

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

## Truncated tetraapeirogonal tiling

In geometry, the truncated tetrapeirogonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated tetraheptagonal tiling

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated tetrahexagonal tiling

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated tetraoctagonal tiling

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated tetrapentagonal tiling

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane.

## Truncated triapeirogonal tiling

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr.

## Truncated triheptagonal tiling

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane.

## Truncated trihexagonal tiling

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane.

## Truncated trioctagonal tiling

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane.

## 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 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).

## Vertex figure

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

## 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.

## 3-7 kisrhombille

In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane.

## 4-5 kisrhombille

In geometry, the 4-5 kisrhombille or order-4 bisected pentagonal tiling is a semiregular dual tiling of the hyperbolic plane.

## 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.

## References

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