In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. One usually adds some requirements on the covering shapes, for instance that they all be congruent, or that they all be squares of mutually different size, etc. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

Mathematically, a tiling of the topological space S consists of a collection B of open subsets of S, such that Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

• the shapes in B do not overlap (i.e., are mutually disjoint, have no point in common)
• they 'cover' S (the closure of their union is equal to S)

Most topics in the area of tilings, patterns and packing problems are best known from examples in the two-dimensional Euclidean space, the Euclidean plane. However, many of these problems can be and have been applied to other topological spaces, especially in the area of packing problems. In mathematics, two sets are said to be disjoint if they have no element in common. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

It has been known for some time that all simple regular tilings in the plane all belong to one of the 17 plane symmetry groups. All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. Plane crystallographic groups or wallpaper groups There are seventeen different types of wallpaper patterns. ... (This article is about the Alhambra in Granada, Spain. ... The City of Granada Alhambra, Courtyard of the Lions Granada is a city and the capital of the province of Granada, in Andalusia, Spain (Andalucía, España). ...

This does not exhaust the apparently simple problem of tiling the plane: adding additional constraints or removing the requirement for regularity reveal a large number of interesting problems, some of which are listed here.

The topics are ordered alphabetically.

Alternating tilings

A tiling {T} of a shape S is called alternating if {T} is the union of two disjoint sets {T1} and {T2} of tiles such that In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, two sets are said to be disjoint if they have no element in common. ...

• any tile T adjacent to a tile T1 in {T1} is in {T2} and, vice versa,
• any tile T adjacent to a tile T2 in {T2} is in {T1}.

Example : If we want to tile the plane with squares and dominoes in an alternating way, then we must find a way that Domino redirects here—for other meanings of the word, see Domino (disambiguation). ...

• the plane is fully covered without gaps or overlaps (otherwise it is not a tiling at all) and such that
• no two squares have a side or a part of a side in common (but having a point in common is allowed) and such that
• no two dominoes have a side or a part of a side in common (but having a point in common is allowed)^1,2.

Alternating tilings of type (n,m)

Let {T} be an alternating tiling (see above) of the Euclidean plane made from sets {T1} and {T2}, and let n and m be two natural numbers, n < m. Then T is called alternating of type (n,m), if {T1} are n-gons (polygons with n sides) and {T2} are m-gons. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Several very interesting question arise for tilings of the plane:

• For which n and m do alternating tilings of type (n,m) exist?
• For which n and m do alternating tilings of type (n,m) exist with the additional property that all tiles in {T1} are congruent and all tiles in {T2} are congruent?
• In general, given n and m, how many prototiles do {T1} and {T2} need in order that such an alternating tiling of type (n,m) exists?

The results are not only mathematically interesting; many of the resulting patterns are quite stunning.¹ In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... In a given set S={A} of shapes (e. ...

Coloured tilings

A tiling is called coloured if each tile has a property colour associated with it such that no two adjacent tiles have the same colour. Coloured tilings are also called coloured maps. If one can find such a colouring scheme, we say that we have coloured the tiling. A graph with 6 vertices (nodes) and 7 edges. ...

Examples :

• The most famous problem relating to coloured tilings was the four color problem, which has been solved; see Four color theorem. The problem asks whether one can colour any map in the plane using four colours only.
• Another rich source for interesting problems related to coloured tilings is the area of alternating tilings, see definition on this page.

Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ...

Faultfree tilings

A tiling T={A} of a shape S is called faultfree if there is no fault line in this tiling.
A fault line or breaking line of a tiling is a straight line from one point of the boundary of S to another point of the boundary of S such that the line has no point in common with the interior of any tile of the tiling. The word Boundary has a variety of meanings. ...

Examples :

• The (2n+1)x(3n) rectangle is the smallest rectangle which has a faultfree tiling with (1xn) rectangles⁸.
• The (2nm+m)x(3nm) rectangle is the smallest known rectangle which has a faultfree tiling with (n × m) rectangles⁸.

Irreptiles

An irreptile (derived from 'irregular reptile', definition of reptile see below) is a shape with the property that it "recursively" tiles a larger version of itself, using differently sized or identical copies of itself³. A simple example is a square, because four copies of it tile a larger square. Each triangle also is an irreptile, because four copies of it tile a larger version of this triangle. In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of...

The problem to find all irreptiles in the Euclidean plane has been studied in ³, but has not been completely solved yet. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

A related set of problems is to find for each irreptile the minimum number of smaller copies such that they tile the original shape. In many cases it is quite difficult to actually prove such a minimality.

N-tessellations

Tesselation is another word for tiling. A tiling of a shape is called an N-tesselation if each tile has an integral area and if for each natural number n there is exactly one tile with area n¹. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

Of course, only shapes with an unlimited area can have an N-tessellation.

There are many N-tessellations of the plane². We can construct N-tessellations of the plane, the half-plane and the quadrant using only triangles². Also, there are N-tessellations of the plane, the half-plane and the quadrant using only rectangles². The word quadrant can mean: A region of the Cartesian coordinate plane with a specific sign for the x and y coordinates: (+, +), (-, +), (-, -) or (-, +). It is the 2-dimensional case of an Orthant Another name for sextant Also see the Wiktionary entry on Quadrant. ...

Even with these restrictions, there are many solutions. For example:

• there are nowhere-neat N-tessellations (see definition of a nowhere-neat tiling on this page) of the plane, the half plane and the quadrant using only rectangles².
• there are N-tessellations of the plane, the half plane and the quadrant using only rectangles of type 1×n, i.e., one unit wide ².

Neat tilings

A tiling {T} of a shape S is called neat if

• each tile T is a polygon and
• adjacent tiles only share full sides, i.e. no tile shares a partial side with any other tile.

Example : The 64 squares on a chess board represent a neat tiling^1,2. From left, a white king, black rook and queen, white pawn, black knight, and white bishop in Staunton chess pieces. ...

Nowhere-neat tilings

A tiling {T} of a shape S is called nowhere-neat if

• each tile T is a polygon and
• adjacent tiles never share a full side, i.e. any tile only shares a partial side with any other tile^1,2.

Examples :

• The minium number of tiles necessary to tile a triangle with triangles in a nowhere-neat way is four².
• The minimum number of tiles necessary to tile a triangle with quadrilaterals in a nowhere-neat way is six².
• The minimum number of tiles necessary to tile a pentagons with quadrilaterals in a nowhere-neat way is twelve².
• The minimum number of tiles necessary to tile a rectangle with squares in a nowhere-neat way is nine².
• The minimum number of tiles necessary to tile a square with rectangles in a nowhere-neat way is five².
• The minimum number of tiles necessary to tile a square with smaller squares in a nowhere-neat way is twenty².
• The minimum number of tiles necessary to tile a square with pentagons in a nowhere-neat way is twelve².
• It is easy to tile the plane with dominoes in a nowhere-neat way.
• There are nowhere-neat N-tessellations (see definition of an N-tessellation on this page) of the plane, the half plane and the quadrant using only rectangles².
• There are nowhere-neat tilings of the plane, the half plane and the quadrant using only squares of different, integral size².

Penrose tilings

Roger Penrose is well-known for his 1974 invention of Penrose tilings, which are formed from two tiles that can only tile the plane aperiodically. In 1984, similar patterns were found in the arrangement of atoms in quasicrystals. Sir Roger Penrose OM (born August 8, 1931) is an English mathematical physicist. ... 1974 is a common year starting on Tuesday (click on link for calendar). ... 1984 is a leap year starting on Sunday of the Gregorian calendar. ... Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. ...

See Penrose tiling for a detailed description and images. A Penrose tiling is pattern of tiles, discovered by Roger Penrose, which could completely cover an infinite surface, but only in a pattern which is non-repeating (aperiodic). ...

Polygons

Tilings using polygons have been studied for many centuries. It has been known for some time that all simple regular tilings in the plane all belong to one of the 17 plane symmetry groups. All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. Wiktionary has a definition of: Polygon A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... Plane crystallographic groups or wallpaper groups There are seventeen different types of wallpaper patterns. ... (This article is about the Alhambra in Granada, Spain. ... The City of Granada Alhambra, Courtyard of the Lions Granada is a city and the capital of the province of Granada, in Andalusia, Spain (Andalucía, España). ...

The artist M. C. Escher has used these symmetries extensively in his frieses and woodcuts. He often modified the polygons in his tilings slightly to turn them into shapes of animals etc. Some of his tilings have an interesting morphing property; e.g., a friese may start as a tiling using fish shapes and slowly turn into a tiling using bird shapes as you go from left top right. Self portrait, 1943¹ Maurits Cornelis Escher (Leeuwarden, June 17, 1898 - Laren, March 27, 1972) was a Dutch artist most known for his woodcuts, lithographs and mezzotints, which tend to feature impossible constructions, explorations of infinity, and tessellations. ... Morphing is a special effect used in motion pictures and animations. ...

Polysquares

A polysquare is a shape that consist of the edge-to-edge joining of squares of same size^3,5,6. Polysquares are also called 'polyominoes'.
One square is also called a monomino.
Two squares joined make a domino.
Three squares joined make a tromino.
Four squares joined make a tetromino.
Five squares joined make a pentomino.
Six squares joined make a hexomino.
Seven squares joined make a septomino or heptomino.
Eight squares joined make an octomino.
Nine squares joined make an enneomino.
Ten squares joined make a decomino.
A polyomino is a polyform with the square as its base form. ... Domino redirects here—for other meanings of the word, see Domino (disambiguation). ... A tromino is a geometric shape made from three squares joined along complete edges. ... A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally. ... A pentomino is a geometric shape composed of five (Greek πέντε / pente) identical squares, connected orthogonally. ... A hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. ...

Pure tilings

A tiling T of a shape S is called pure if T contains only one prototile, i.e., if each tile is congruent to any other tile². In a given set S={A} of shapes (e. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

An alternating tiling (see definition on this page) T consisting of two sets of tiles {A} and {B} is called pure alternating if the sets {A} and {B} each contain only one prototile². It is an interesting question to find out for which numbers n,m (n<m) there is a pure alternating tiling of type (n,m) (see definition of alternating tiling of type (n,m) on this page)¹.

Examples :

• The 64 squares on a chess board represent a pure tiling¹.
• Any reptile (see definition on this page) tiles a larger version of itself in a pure way.

Puritiles

A puritile (derived from 'purely irregular reptile') is a shape with the property that in order to tile a larger version of itself, differently sized copies have to be used³.

An example of a puritile is the L-shaped hexomino that has a 1×3 rectangle joined to another 1×3 rectangle. 18 copies of two different sizes are necessary (namely 12 of same size and 6 of twice the size) to tile a larger version of it. Note that 12×1+6×4=36=6×6, hence the larger version is six time bigger than the original. Can you find the tiling?

Rectangles

Non-congruent rectangles

The smallest square that can be cut into (m x n) rectangles, such that all m and n are different integers, is the 11 x 11 square, and the tiling uses five rectangles⁷.
The smallest rectangle that can be cut into (m x n) rectangles, such that all m and n are different integers, is the 9 x 13 rectangle, and the tiling uses five rectangles⁷.

Regular tilings

..... (to be filled) ....

Reptiles

A reptile (or rep-tile, from 'repetitive tiling') is a shape with the property that is tiles a larger version of itself, using identical copies of itself^2,3,5. A simple example is a square, because four copies of it tile a larger square.

Each triangle also is a reptile, because four copies of it tile a larger version of this triangle.

The set of reptiles is a subset of the set of irreptiles.

Simple tilings

..... (to be filled) ....

Sim-tilings

A tiling is called a sim-tiling if all its tiles are similar to each other. Several equivalence relations in mathematics are called similarity. ...

Examples :

• irreptiles (see definition on this page) are those shapes that tile a larger version of themselves with a sim-tiling.
• For a few more examples, see the sub-section other triangles in section triangles on this page.

Squares

Integral squares

A square with integral sidelength is called an integral square. If an integral squares S has been tiled with smaller integral squares, we call this "squaring the square". A square with sides equal to a unit length multiplied by an integer is called an integral square. ...

Various conditions can be applied to create mathematical problems. The one most investigated is the "perfect squared square", see below. Other conditions that yield interesting results are "nowhere-neat" (see link) and "no-touch" squared squares (see definitions below).

If the smaller squares all have different sizes, we call it a "perfect squared square". This is called the squaring the square problem. It is first recorded as being studied by R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, at Cambridge University, and the first perfect squared square was found by Roland Sprague in 1939. A square with sides equal to a unit length multiplied by an integer is called an integral square. ... William Thomas Tutte (May 14, 1917 - May 2, 2002) was a British codebreaker and mathematician. ...

If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.

It is still an unsolved problem, however, whether the plane can be tiled with a set of integral tiles such that each natural number is used exactly once as size of a square tile. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

Symmetries

See the section titled polygons on this page.
See also: Symmetry Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

Tetrads

A tetrad is a (simply connected) shape with the property that four copies of this tetrad can be placed without overlapping in such a way that each copy shares some boundary with each of the other three tetrads⁶. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

Tetrads are rare creatures. Some polysquare reptiles are tetrads³.

Triangles

Integral triangles

A triangle with three integral sidelengths is called an integral triangle. There are squares that can be tiled with integral triangles such that no two of these triangles are congruent². The plane can be tiled with integral triangles such that no two of these triangles are congruent².

Pythagorean triangles

A right triangle with three integral sidelengths is called a Pythagorean triangle. In mathematics, the Pythagorean theorem or Pythagorass theorem, is a relation in Euclidean geometry between the three sides of a right triangle. ...

There are squares that can be tiled with Pythagorean triangles such that no two of these triangles are congruent².

The plane can be tiled with Pythagorean triangles such that no two of these triangles are congruent².

Equilateral triangles

The mathematician William Tutte showed that one cannot tile an equilateral triangle with a finite number of smaller regular triangles, all of different size. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

On similar lines, it can be shown that one cannot tile the plane with regular triangles, all of different size, if one of them has a smallest size⁴.

Other triangles

However, it is possible to tile the plane with enlargements of one single triangle, all of mutually different size².
The isosceles right triangle (angles 45, 45, 90 degrees) solves this problem².
The half regular triangle (angles 30, 60, 90 degrees) also solves this problem².
The enlargements can be chosen to be all integers². But there are also solutions where these enlargements are not all integers².

A square can be tiled with eight 30-60-90 triangles of mutually different sizes.

Literature

1. Karl Scherer : New Mosaics, 1997 (see http://karl.kiwi.gen.nz)
2. Karl Scherer : Nutts And Other Crackers, 1994 (see http://karl.kiwi.gen.nz)
3. Karl Scherer : A Puzzling Journey to the Reptiles And Related Animals, 1986 (see http://karl.kiwi.gen.nz) (Written as a fiction story, this is the only book which investigates into the realm of puritiles.)
4. Karl Scherer : The impossibility of a tessellation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal.(Article in journal Elemente der Mathematik, 1984)
5. Solomon Golomb : Polyominoes, 1994
6. Journal of Recreational Mathematics, many articles.
7. Journal of Recreational Mathematics, 28:1, p.64.
8. Journal of Recreational Mathematics, (?:?), 1980, p.4.
9. Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. "The Dissection of Rectangles into Squares." Duke Math. J. 7, 312-340, 1940
10. Tiling and Patterns by Branko Grünbaum/Gruenbaum and Geoffrey C. Shephard. 1986. ISBN 071671194X.

The Journal of Recreational Mathematics is a peer reviewed journal dedicated to recreational mathematics. ... The Journal of Recreational Mathematics is a peer reviewed journal dedicated to recreational mathematics. ... The Journal of Recreational Mathematics is a peer reviewed journal dedicated to recreational mathematics. ...

See also

Penrose tiling, aperiodic tiling, quasiperiodic tiling A Penrose tiling is pattern of tiles, discovered by Roger Penrose, which could completely cover an infinite surface, but only in a pattern which is non-repeating (aperiodic). ... An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... See Penrose tiling for a mathematical viewpoint. ...

External links

• Nowhere-neat Squared Rectangles (http://karl.kiwi.gen.nz/prosqtre.html)
• Nowhere-neat Squared Squares (http://karl.kiwi.gen.nz/prosqtsq.html)

Music

• Tiling the Line in Theory and Practice (http://homepage.mac.com/javiruiz/English/articleslecturesenglish.html) by Tom Johnson, PDF (http://homepage.mac.com/javiruiz/Articles/Tiling%20the%20line%20in%20theory.pdf)
• Self-Replicating Loops (http://homepage.mac.com/javiruiz/Articles/selfreplicatingloops.html) by Tom Johnson
• Some Observations on Tiling Problems PDF (http://www.ircam.fr/equipes/repmus/mamux/documents2002-2003/TomTiling.PDF) by Tom Johnson
• Tiling problems in music theory (http://www-ang.kfunigraz.ac.at/~fripert/tilings.html) by Harald Fripertinger PDF (http://www-ang.kfunigraz.ac.at/~fripert/papers/myepos.pdf)
• Tiling problems in music composition: Theory and Implementation PDF (http://www.ircam.fr/equipes/repmus/moreno/AndreattaAgonAmioticmc2002.pdf) by Moreno Andreatta, Carlos Agon, and Emmanuel Amiot from IRCAM