Tessellation
In mathematics, called tiling (including tiling, paving, or surface -circuit) the gapless, overlap-free coverage of the ( Euclidean ) plane by uniform patches. The concept can also be extended to higher dimensions.
In practical applications, the coverage with the help of primitives ( "primitive" land - forms, if possible with a simple polygon ) is preferred, for which the corresponding limiting term tessellation is used ( English for " mosaic "). If in a technical application, a large sheet in non-primitive sub-areas ( workpieces) is split, it will try to make this so that a tiling exists by dissimilar faces and no waste.
The " cyclical allocation of land " with non-uniform sub-areas ( no polygons ) in the art is very pronounced in front of MC Escher.
Analogous to tiling or tessellation to the plane ( 2D) can also be the three - or higher-dimensional space can be divided.
- 3.1 Platonic tilings
- 3.2 Archimedean tilings
- 3.3 Homogeneous tilings
- 3.4 Inhomogeneous tilings
Definitions
A tile (floor stone, paving stone) is a closed topological disk in the plane. (This will, inter alia, stones with holes and non-contiguous parts are excluded. Well as those more general and stones are occasionally admitted. )
A tiling ( paving, tiling, sometimes called mosaic ) is a ( countable ) set of tiles which both a package ( ie, " no point of the plane lies inside of two or more tiles ", or, in other words, " different tiles have at most boundary points in common " ) and a cover (that is, " every point of the plane belongs to at least one tile " ) is.
Often one restricts the term still further by, for example, requires that all tiles are homeomorphic to the closed disk ( thus, in particular compact and simply connected ), or that each tile is congruent to an element of a finite selection of tiles ( the so-called " proto- tiles " ) is, so that only finitely many tiles occur.
Also tilings in higher dimensions and more general spaces are considered analogously.
Tilings of the plane
Symmetries of a tiling
A congruence ( rigid motion ) of the plane, which maps each tile of a tiling again on a tile is called the " symmetry " of the tiling. The set of all symmetries is called the symmetry group and is a group. Contains the symmetry group of a tiling of two linearly independent shifts, so called the parquetry "periodic" and the resulting symmetry group plane crystallographic group of which there are exactly 17, the so-called wallpaper pattern groups are - otherwise called the parquetry " non-periodic ".
Aperiodic tilings
Sets of proto- tiles (see above) that allow only non-periodic overlaps of the plane, called " aperiodic ". Tilings can be quasi-periodic, that is, the repetition of any size cut-outs without the floor is a whole periodically. An interesting and beautiful example of a quasiperiodic tiling is the Penrose tiling, named after its discoverer Roger Penrose.
Periodic tilings
If one makes certain demands on the basic shapes used in a tiling and its layout to special cases, for which one can then specify all possible tilings arise.
Platonic tilings
If only a regular n-gon as a tile approved and is further restricted in that the tiles edge to edge must be positioned exactly three possible tilings of the plane, the Platonic or regular tilings arise:
Johannes Kepler was the first who examined these tilings and realized that they represent an analogue to the regular polyhedra.
Square lattice
Hex mesh - also honeycomb pattern is called
Archimedean tilings
May as the basic form any regular n-gons with the same edge length are used, resulting in retention of the edge - to-edge rule and the restriction that at each point where the corners meet, always the same combination of polygons (number must collide and order), exactly eight other possible tilings - the Archimedean or semiregular tilings of the plane:
- 2 tilings of triangles and squares
- 2 tilings of triangles and hexagons
- 1 tiling of triangles, squares, and hexagons
- 1 tiling of triangles and twelve corners
- 1 tiling of octagons and squares
- 1 tiling of squares, hexagons and twelve corners
3-3-4-3-4
3-6-3-6
3-3-3-3-6 ( two mirrored variants)
3-4-6-4
3-12-12
4-8-8
4-6-12
Homogeneous tilings
May as the basic form any regular n-gons with the same edge length are used, so there is at maintaining the edge - to-edge rule and the restriction that at the meet, the corners at each point, always the same number of the same polygons (regardless must collide of the order ), other tilings, for example:
- Tiling of triangles and squares ( different pattern than in the corresponding Archimedean parquets )
- Tiling of triangles and hexagons ( different pattern than in the corresponding Archimedean parquets )
- Tiling of triangles, squares and hexagons ( different pattern than in the corresponding Archimedean parquets )
General called tilings be used for any as a basic form of regular n-gons with the same edge length, complying with the edge - to-edge rule and satisfy the restriction that at the meet, the corners at each point, always the same collide ( regardless of order ) the same number of polygons, homogeneous tilings.
Inhomogeneous tilings
Tilings are used for indeed arbitrary as the basic form regular n-gons with the same edge length and complying with the edge - to-edge control, but where where the corners meet at the points, different collide many polygons are called inhomogeneous tilings, for example:
- Tiling of triangles and hexagons ( different pattern than in the corresponding Archimedean parquets )
- Tiling of triangles, squares and hexagons ( different pattern than in the corresponding Archimedean parquets )
- Tiling of triangles, squares, and Twelve Corners
- Tiling of triangles, squares, hexagons and twelve corners
There are infinitely many inhomogeneous tilings. particular examples are:
- Parketierung of a hexagon and a pentagon (which also parketiert alone, Grazebrook tiling )
Aperiodic ( quasicrystalline ) tiling
This was discovered by Roger Penrose. Although is comprehensive, but not periodically, therefore, it is called quasi -crystalline
- Penrose tiling of two diamonds