Tiling
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
- 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)
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.
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}.
- 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?
Coloured tilings
- 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) rectangles^{8}.
- The (2nm+m)x(3nm) rectangle is the smallest known rectangle which has a faultfree tiling with (nxm) rectangles^{8}.
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 itself^{3}. 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 n^{1}.
Of course, only shapes with an unlimited area can have an N-tessellation.
There are many N-tesselations of the plane^{2}. We can construct N-tesselations of the plane, the half-plane and the quadrant using only triangles^{2}. Also, there are N-tesselations of the plane, the half-plane and the quadrant using only rectangles^{2}.
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 rectangles^{2}.
- 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.
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}.
- The mininum number of tiles necessary to tile a triangle with triangles in a nowhere-neat way is four^{2}.
- The mininum number of tiles necessary to tile a triangle with quadrilaterals in a nowhere-neat way is six^{2}.
- The mininum number of tiles necessary to tile a pentagons with quadrilaterals in a nowhere-neat way is twelve^{2}.
- The mininum number of tiles necessary to tile a rectangle with squares in a nowhere-neat way is nine^{2}.
- The mininum number of tiles necessary to tile a square with rectangles in a nowhere-neat way is five^{2}.
- The mininum number of tiles necessary to tile a square with smaller squares in a nowhere-neat way is twenty^{2}.
- The mininum number of tiles necessary to tile a square with pentagons in a nowhere-neat way is twelve^{2}.
- 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 rectangles^{2}.
- There are nowhere-neat tilings of the plane, the half plane and the quadrant using only squares of different, integral size^{2}.
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 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 tile^{2}.
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 prototile^{2}.
It is an interesting question to find out for which numbers n,m (n
Examples :
- The 64 squares on a chess board represent a pure tiling^{1}.
- 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 used^{3}.
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
Regular tilings
Reptiles
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
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 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 tetrads^{6}.
Tetrads are rare creatures. Some polysquare reptiles are tetrads^{3}.
Triangles
Integral triangles
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 congruent^{2}.
The plane can be tiled with Pythagorean triangles such that no two of these triangles are congruent^{2}.
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 size^{4}.
Other triangles
A square can be tiled with eight 30-60-90 triangles of mutually different sizes.
- Karl Scherer : New Mosaics, 1997 (see http://karl.kiwi.gen.nz)
- Karl Scherer : Nutts And Other Crackers, 1994 (see http://karl.kiwi.gen.nz)
- 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.)
- 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)
- Solomon Golomb : Polyominoes, 1994
- Journal of Recreational Mathematics, many articles.
- Journal of Recreational Mathematics, 28:1, p.64.
- Journal of Recreational Mathematics, (?:?), 1980, p.4.
- 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
- Tiling the Line in Theory and Practice by Tom Johnson, PDF
- Self-Replicating Loops by Tom Johnson
- Some Observations on Tiling Problems PDF by Tom Johnson
- Tiling problems in music theory by Harald Fripertinger PDF
- Tiling problems in music composition: Theory and Implementation PDF by Moreno Andreatta, Carlos Agon, and Emmanuel Amiot from IRCAM
References
- Tiling and Patterns by B. Grunbaum/Gruenbaum and Geoffrey C. Shephard. 1986. ISBN 071671194X.