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# Tessellation

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. [1]

191 relations: Academic Press, Alhambra, Alternated octagonal tiling, Ancient Rome, Andradite, Anisohedral tiling, Annals of Mathematics, Aperiodic tiling, Architecture of Mesopotamia, Basalt, Bell System Technical Journal, Bitruncated cubic honeycomb, Boundary (topology), Bravais lattice, Brickwork, Car door, Cement, Ceramic, Ceramic glaze, Circle Limit III, Classical antiquity, Clay, Concave polygon, Congruence (geometry), Connected space, Convex uniform honeycomb, Conway criterion, Countable set, Cover (topology), Coxeter group, Coxeter–Dynkin diagram, Crelle's Journal, Crystal, Cube, Delaunay triangulation, Dimension, Discrete Mathematics (journal), Disk (mathematics), Dissection problem, Dominoes, Dual graph, Eaglehawk Neck, Edgar Gilbert, Edge (geometry), Edge tessellation, Empty set, Equilateral triangle, Euclidean geometry, Euclidean tilings by convex regular polygons, Evgraf Fedorov, ... Expand index (141 more) »

## Alhambra

The Alhambra (الْحَمْرَاء, Al-Ḥamrā, lit. "The Red One",The "Al-" in "Alhambra" means "the" in Arabic, but this is ignored in general usage in both English and Spanish, where the name is normally given the definite articleالْحَمْرَاء, trans.; literally "the red one", feminine; in colloquial Arabic: the complete Arabic form of which was Qalat Al-Hamra)الْقَلْعَةُ ٱلْحَمْرَاءُ, trans.

## Alternated octagonal tiling

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

## Ancient Rome

In historiography, ancient Rome is Roman civilization from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD, encompassing the Roman Kingdom, Roman Republic and Roman Empire until the fall of the western empire.

Andradite is a species of the garnet group.

## Anisohedral tiling

In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling.

## Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

## Aperiodic tiling

An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches.

## Architecture of Mesopotamia

The architecture of Mesopotamia is ancient architecture of the region of the Tigris–Euphrates river system (also known as Mesopotamia), encompassing several distinct cultures and spanning a period from the 10th millennium BC, when the first permanent structures were built, to the 6th century BC.

## Basalt

Basalt is a common extrusive igneous (volcanic) rock formed from the rapid cooling of basaltic lava exposed at or very near the surface of a planet or moon.

## Bell System Technical Journal

The Bell System Technical Journal was a periodical publication by the American Telephone and Telegraph Company (AT&T) in New York devoted to the scientific and engineering aspects of electrical communication.

## Bitruncated cubic honeycomb

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes).

## Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

## Bravais lattice

In geometry and crystallography, a Bravais lattice, named after, is an infinite array of discrete points in three dimensional space generated by a set of discrete translation operations described by: where ni are any integers and ai are known as the primitive vectors which lie in different directions and span the lattice.

## Brickwork

Brickwork is masonry produced by a bricklayer, using bricks and mortar.

## Car door

A car door is a type of door, typically hinged, but sometimes attached by other mechanisms such as tracks, in front of an opening which is used for entering and exiting a vehicle.

## Cement

A cement is a binder, a substance used for construction that sets, hardens and adheres to other materials, binding them together.

## Ceramic

A ceramic is a non-metallic solid material comprising an inorganic compound of metal, non-metal or metalloid atoms primarily held in ionic and covalent bonds.

## Ceramic glaze

Ceramic glaze is an impervious layer or coating of a vitreous substance which has been fused to a ceramic body through firing.

## Circle Limit III

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came".

## Classical antiquity

Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th or 6th century AD centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world.

## Clay

Clay is a finely-grained natural rock or soil material that combines one or more clay minerals with possible traces of quartz (SiO2), metal oxides (Al2O3, MgO etc.) and organic matter.

## Concave polygon

A simple polygon that is not convex is called concave, non-convex or reentrant.

## 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.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## 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.

## Conway criterion

In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, describes rules for when a prototile will tile the plane; it consists of the following requirements: The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset.

## 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–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).

## Crelle's Journal

Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).

## Crystal

A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.

## 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.

## Delaunay triangulation

In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P).

## 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.

## Discrete Mathematics (journal)

Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications.

## Disk (mathematics)

In geometry, a disk (also spelled disc).

## Dissection problem

In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content.

## Dominoes

Dominoes is a family of tile-based games played with rectangular "domino" tiles.

## Dual graph

In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of.

## Eaglehawk Neck

The Eaglehawk Neck is a narrow isthmus that connects the Tasman Peninsula with the Forestier Peninsula, and hence to mainland Tasmania, Australia.

## Edgar Gilbert

Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, and the Erdős–Rényi model for random graphs.

## Edge (geometry)

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

## Edge tessellation

A tessellation, also known as a tiling, is a set of shapes that must cover the entire plane without the shapes overlapping.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## 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 tilings by convex regular polygons

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

## Evgraf Fedorov

Evgraf Stepanovich Fedorov (Евгра́ф Степа́нович Фёдоров, – 21 May 1919) was a Russian mathematician, crystallographer and mineralogist.

## 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.

## Fibonacci word

A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet).

## Fluorite

Not to be confused with Fluoride. Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2.

## Foam

Foam is a substance formed by trapping pockets of gas in a liquid or solid.

## Four color theorem

In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

## Fracture

A fracture is the separation of an object or material into two or more pieces under the action of stress.

## Frieze group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.

## Fritillaria

Fritillaria (fritillaries) is a genus of spring flowering herbaceous bulbous perennial plants in the lily family (Liliaceae).

## 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.

## Garnet

Garnets are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives.

## Geometric shape

A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object.

## 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.

## Giant's Causeway

The Giant's Causeway is an area of about 40,000 interlocking basalt columns, the result of an ancient volcanic fissure eruption.

## Gilbert tessellation

In applied mathematics, a Gilbert tessellation.

## Girih tiles

Girih tiles are a set of five tiles that were used in the creation of Islamic geometric patterns using strapwork (girih) for decoration of buildings in Islamic architecture.

## Glass

Glass is a non-crystalline amorphous solid that is often transparent and has widespread practical, technological, and decorative usage in, for example, window panes, tableware, and optoelectronics.

Granada is the capital city of the province of Granada, in the autonomous community of Andalusia, Spain.

## 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.

## Halting problem

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running (i.e., halt) or continue to run forever.

## Harmonices Mundi

Harmonices MundiThe full title is Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler's The Harmony of the World).

## Heinrich Heesch

Heinrich Heesch (June 25, 1906 &ndash; July 26, 1995) was a German mathematician.

## Henry Dudeney

Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games.

## 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.

## Hinged dissection

A hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Honeycomb

A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen.

## 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 geometry

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

## Internal and external angles

In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint.

## Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

## Islamic architecture

Islamic architecture encompasses a wide range of both secular and religious styles from the early history of Islam to the present day.

## Islamic art

Islamic art encompasses the visual arts produced from the 7th century onward by people who lived within the territory that was inhabited by or ruled by culturally Islamic populations.

## Islamic geometric patterns

Islamic decoration, which tends to avoid using figurative images, makes frequent use of geometric patterns which have developed over the centuries.

## 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.

## Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

## Jigsaw puzzle

A jigsaw puzzle is a tiling puzzle that requires the assembly of often oddly shaped interlocking and tessellating pieces.

## Johannes Kepler

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

## Lava

Lava is molten rock generated by geothermal energy and expelled through fractures in planetary crust or in an eruption, usually at temperatures from.

## Leonardo (journal)

Leonardo® is a peer-reviewed academic journal published by the MIT Press covering the application of contemporary science and technology to the arts and music.

## 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.

## M. C. Escher

Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically-inspired woodcuts, lithographs, and mezzotints.

## Manufacturing

Manufacturing is the production of merchandise for use or sale using labour and machines, tools, chemical and biological processing, or formulation.

## Marjorie Rice

Marjorie Rice (née Jeuck, February 16, 1923 – July 2, 2017) was an American amateur mathematician most famous for her discoveries in geometry.

## Martin Gardner

Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer, with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literature—especially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.

## Mathematics

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

## Microtechnology

Microtechnology is technology with features near one micrometre (one millionth of a metre, or 10−6 metre, or 1μm).

## Minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area.

## Moors

The term "Moors" refers primarily to the Muslim inhabitants of the Maghreb, the Iberian Peninsula, Sicily, Sardinia, Corsica, and Malta during the Middle Ages.

## Mosaic

A mosaic is a piece of art or image made from the assemblage of small pieces of colored glass, stone, or other materials.

## Mosque–Cathedral of Córdoba

The Mosque–Cathedral of Córdoba (Mezquita-Catedral de Córdoba), also known as the Great Mosque of Córdoba (Mezquita de Córdoba) and the Mezquita, whose ecclesiastical name is the Cathedral of Our Lady of the Assumption (Catedral de Nuestra Señora de la Asunción), is the Catholic cathedral of the Diocese of Córdoba dedicated to the Assumption of the Virgin Mary and located in the Spanish region of Andalusia.

## Motif (textile arts)

In the textile arts, a motif (also called a block or square) is a smaller element in a much larger work.

## Mudcrack

Mudcracks (also known as desiccation cracks, mud cracks or cracked mud) are sedimentary structures formed as muddy sediment dries and contracts.

## Nanotechnology

Nanotechnology ("nanotech") is manipulation of matter on an atomic, molecular, and supramolecular scale.

## Non-Euclidean geometry

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

## Nonagon

In geometry, a nonagon or enneagon is a nine-sided polygon or 9-gon.

## Octagonal tiling

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

## Octahedron

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

## 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.

## Parallelogram

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides.

## Pattern

A pattern is a discernible regularity in the world or in a manmade design.

## Patterns in nature

Patterns in nature are visible regularities of form found in the natural world.

## Pavement (architecture)

Pavement, in construction, is an outdoor floor or superficial surface covering.

## Penrose tiling

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles.

## 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 tiling

In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.

## Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

## Physical Review Letters

Physical Review Letters (PRL), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society.

## Pinwheel tiling

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Plateau's laws

Plateau's laws describe the structure of soap films.

## Platonic solid

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

## Plesiohedron

In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set.

## 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.

## Polyiamond

A polyiamond (also polyamond or simply iamond) is a polyform whose base form is an equilateral triangle.

## Polyomino

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge.

## 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.

## Prototile

In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.

## Pythagorean tiling

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides.

In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners.

## Quasicrystal

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.

## Quilt

A quilt is a multi-layered textile, traditionally composed of three layers of fiber: a woven cloth top, a layer of batting or wadding, and a woven back, combined using the technique of quilting, the process of sewing the three layers together.

## Quilting

Quilting is the process of sewing two or more layers of fabric together to make a thicker padded material, usually to create a quilt or quilted garment.

## Recreational mathematics

Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity.

## Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

## 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).

## Rep-tile

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape.

## Rhombic dodecahedron

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

## Rock (geology)

Rock or stone is a natural substance, a solid aggregate of one or more minerals or mineraloids.

## Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

## Sébastien Truchet

Jean Truchet (1657 – 5 February 1729), known as Father Sébastian, was a French Dominican priest born in Lyon, who lived under the reign of Louis XIV.

## Schläfli symbol

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

## 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.

## Scientific American

Scientific American (informally abbreviated SciAm) is an American popular science magazine.

## Self-organization

Self-organization, also called (in the social sciences) spontaneous order, is a process where some form of overall order arises from local interactions between parts of an initially disordered system.

## Self-replication

Self-replication is any behavior of a dynamical system that yields construction of an identical copy of itself.

## Sheet metal

Sheet metal is metal formed by an industrial process into thin, flat pieces.

## Snub trihexagonal tiling

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

## Spain

Spain (España), officially the Kingdom of Spain (Reino de España), is a sovereign state mostly located on the Iberian Peninsula in Europe.

## 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 trigonometry

Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.

## 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.

## Squaring the square

Squaring the square is the problem of tiling an integral square using only other integral squares.

## Substitution tiling

In geometry, a tile substitution is a method for constructing highly ordered tilings.

## Switzerland

Switzerland, officially the Swiss Confederation, is a sovereign state in Europe.

## 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.

## Tangram

The tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes.

## Tasman Peninsula

The Tasman Peninsula is a peninsula located in south-east Tasmania, Australia, approximately by the Arthur Highway, south-east of Hobart.

## Tasmania

Tasmania (abbreviated as Tas and known colloquially as Tassie) is an island state of Australia.

## Tessellated pavement

In geology and geomorphology, a tessellated pavement is a relatively flat rock surface that is subdivided into more or less regular rectangles, blocks approaching rectangles, or irregular or regular polygons by fractures, frequently systematic joints, within the rock.

## Tessera

A tessera (plural: tesserae, diminutive tessella) is an individual tile, usually formed in the shape of a cube, used in creating a mosaic.

## 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.

## Thin film

A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness.

## Tiling puzzle

Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps).

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Translational symmetry

In geometry, a translation "slides" a thing by a: Ta(p).

## 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.

## Truchet tiles

In information visualization and graphic design, Truchet tiles are square tiles decorated with patterns that are not rotationally symmetric.

## Truncated octahedron

In geometry, the truncated octahedron is an Archimedean solid.

## Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

## Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

## Uniform boundedness

In mathematics, a bounded function is a function for which there exists a lower bound and an upper bound, in other words, a constant that is larger than the absolute value of any value of this function.

## 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 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 (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

## Vertex configuration

In geometry, a vertex configuration by Walter Steurer, Sofia Deloudi, (2009) pp.

## Voderberg tiling

The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg.

## Voronoi diagram

In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane.

## Wallpaper group

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern.

## Wang tile

Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems.

## Weaire–Phelan structure

In geometry, the Weaire–Phelan structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles.

## William Thomson, 1st Baron Kelvin

William Thomson, 1st Baron Kelvin, (26 June 1824 – 17 December 1907) was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824.

## Woodcut

Woodcut is a relief printing technique in printmaking.

## Wythoff construction

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

## Zellige

Zellige (الزليج; also zelige or zellij) is mosaic tilework made from individually chiseled geometric tiles set into a plaster base.

## References

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