Tiling

In geometry, a tiling (also called tesselation, 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.

Mathematically, a tiling of the topological space S consists of a collection B of open subsets of S, such that

• 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.

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.

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.

Table of contents

1 Alternating tilings 1.1 Alternating tilings of type (n,m) 2 Coloured tilings 3 Faultfree tilings 4 Irreptiles 5 N-tesselations 6 Neat tilings 7 Nowhere-neat tilings 8 Penrose tilings 9 Polygons 10 Polysquares 11 Pure tilings 12 Puritiles 13 Rectangles 13.2 Non-congruent rectangles 14 Regular tilings 15 Reptiles 16 Simple tilings 17 Sim-tilings 18 Squares 18.3 Integral squares 19 Symmetries 20 Tetrads 21 Triangles 21.4 Integral triangles 21.5 Pythagorean triangles 21.6 Equilateral triangles 21.7 Other triangles 22 references

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

• 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

• 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.

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

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.

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.

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.

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 (nxm) rectangles⁸.

Irreptiles

An irreptile (derived from 'irregular reptile', definition of reptile see below) is a shape with the property that 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.

The problem to find all irreptiles in the Euclidean plane has been studied in ³, but has not been completely solved yet.

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-tesselations

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.

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

There are many N-tesselations of the plane². We can construct N-tesselations of the plane, the half-plane and the quadrant using only triangles². Also, there are N-tesselations of the plane, the half-plane and the quadrant using only rectangles².

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

• there are nowhere-neat N-tesselations (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-tesselations 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.

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 mininum number of tiles necessary to tile a triangle with triangles in a nowhere-neat way is four².
• The mininum number of tiles necessary to tile a triangle with quadrilaterals in a nowhere-neat way is six².
• The mininum number of tiles necessary to tile a pentagons with quadrilaterals in a nowhere-neat way is twelve².
• The mininum number of tiles necessary to tile a rectangle with squares in a nowhere-neat way is nine².
• The mininum number of tiles necessary to tile a square with rectangles in a nowhere-neat way is five².
• The mininum number of tiles necessary to tile a square with smaller squares in a nowhere-neat way is twenty².
• The mininum 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-tesselations (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.

See Penrose tiling for a detailed description and images.

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.

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.

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.

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.

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 (nalternating 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.

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".

Various conditions can be applied to create mathematical problems. The one most investigated is the "perfect square 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.

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.

Symmetries

See the section titled polygons on this page.
See also: Symmetry

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.

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.

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.

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

External Links:

• Nowhere-neat Squared Rectangles
• Nowhere-neat Squared Squares
• Tiling the Line in Theory and Practice by [[Tom Johnson]
• Self-Replicating Loops by Tom Johnson
• Some Observations on Tiling Problems by Tom Johnson

references

• Tiling and Patterns by B. Grunbaum and G. C. Shephard. 1987