The Tiling reference article from the English Wikipedia on 24-Jul-2004
(provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Tiling

Support a children's charity online
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.

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

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 groupss. 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
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
14 Regular tilings
15 Reptiles
16 Simple tilings
17 Sim-tilings
18 Squares
19 Symmetries
20 Tetrads
21 Triangles
22 External links
23 Music
24 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

Example : If we want to tile the plane with squares and dominoes in an alternating way, then we must find a way that

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:

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 :

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 :

Irreptiles

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

Neat tilings

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

Example : The 64 squares on a
chess board represent a neat tiling1,2.

Nowhere-neat tilings


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

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 groupss. 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 size3,5,6. 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 :

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 itself2,3,5. 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 :

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 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.

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

Music

References