In the field of [[mathematics]], '''tilings, patterns and packing problems''' have fascinated people for centuries.
It has been known for some time that all simple regular [[tiling]]s in the plane all belong to one of the 17 plane [[symmetry group]]s. 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.
All topics below refer to the two-dimensional [[Euclidean space]].
The topics are ordered alphabetically.
=== Alternating Tilings ===
A tiling {T} of a shape S is called ''alternating'' it is consists of two sets of tiles {T1} and {T2} such that member of {T1} never has a side in common with any tile of {T2}.
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 full 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).<sup>1,2</sup>.
=== 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 amd 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 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, what is the minimum number of different tiles (called ''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 the resulting patterns are quite stunning.<sup>1</sup>.
=== Irreptiles ===
An ''irreptile'' (derived from 'irregular reptile', definition of reptile see below) is a shape with the property that is tiles a larger version of itself, using differently sized or identical copies of itself<sup>3</sup>.
A simple example is a square, because four copies of it tile a larger square.
Each triangle also is a irreptile, because four copies of it tile a larger
version of this triangle.
=== 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<sup>1</sup>.
=== 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<sup>1</sup>.
Example : The 64 squares on a chess board represent a neat tiling.
=== 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 [[quasicrystal]]s.
..... (to be filled) ....
=== Polysquares ===
A ''polysquares'' is a shape that consist of the edge-to-edge joining of squares of same size<sup>3,4,5</sup>.
Polysquares are also called 'polyominoes'.
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.
Eight squares joined make an octomino.
Nine squares joined make an enneomino.
Ten squares joined make a decomino.
=== Puritiles ===
A ''puritile'' (derived from 'purely irregular reptile') is a shape with the property that iin order to tile a larger version of itself, differently sized copies have to be used<sup>3</sup>.
An example of a puritile is the L-shaped hexomino that has a 1x3 rectangle
joint to another 1x3 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 12x1+6x4=36=6x6, hence the larger version is six time bigger than the original. Can you find the tiling?
=== 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<sup>3,4,5</sup>.
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.
=== 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<sup>5</sup>.
Very little is known about these creatures.
Literature:
# 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.)
# Solomon Golomb : Polyominoes, 1994
# [[Journal of Recreational Mathematics]], many articles.
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