Anisohedral tiling
Encyclopedia
In geometry
, a shape is said to be anisohedral if it admits a tiling
, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
The second part of Hilbert's eighteenth problem
asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest that Hilbert was assuming that no such tile existed in the plane. Reinhardt answered Hilbert's problem in 1928 by finding examples of such polyhedra, and asserted that his proof that no such tiles exist in the plane would appear soon. However, Heesch
then gave an example of an anisohedral tile in the plane in 1935.
Reinhardt had previously considered the question of anisohedral convex polygon
s, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagon
s, while finding the five types of convex pentagon tiling the plane
isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.
(equivalence classes) of tiles in any tiling of that tile under the action of the symmetry group
of that tiling, and that a tile with isohedral number k is k-anisohedral. Berglund asked whether there exist k-anisohedral tiles for all k, giving examples for k ≤ 4 (examples of 2-anisohedral and 3-anisohedral tiles being previously known, while the 4-anisohedral tile given was the first such published tile). Goodman-Strauss considered this in the context of general questions about how complex the behaviour of a given tile or set of tiles can be, noting an 8-anisohedral example of Kari. Grünbaum and Shephard had previously raised a slight variation on the same question.
Socolar showed in 2007 that arbitrarily high isohedral numbers can be achieved in two dimensions if the tile is disconnected, or has coloured edges with constraints on what colours can be adjacent, and in three dimensions with a connected tile without colours, noting that in two dimensions for a connected tile without colours the highest known isohedral number is 10.
Joseph Myers has produced a truly remarkably collection of tiles with high isohedral numbers, particularly a polyhexagon with isohedral number 10 (occurring in 20 orbits under translation) and another with isohedral number 9 (occurring in 36 orbits under translation).http://www.srcf.ucam.org/~jsm28/tiling/
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a shape is said to be anisohedral if it admits a tiling
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
The second part of Hilbert's eighteenth problem
Hilbert's eighteenth problem
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space....
asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest that Hilbert was assuming that no such tile existed in the plane. Reinhardt answered Hilbert's problem in 1928 by finding examples of such polyhedra, and asserted that his proof that no such tiles exist in the plane would appear soon. However, Heesch
Heinrich Heesch
Heinrich Heesch was a German mathematician. He was born in Kiel and died in Hanover.In Göttingen he worked on Group theory. In 1933 Heesch witnessed the National Socialist purges among the university staff...
then gave an example of an anisohedral tile in the plane in 1935.
Reinhardt had previously considered the question of anisohedral convex polygon
Convex polygon
In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...
s, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
s, while finding the five types of convex pentagon tiling the plane
Pentagon tiling
In geometry, a pentagon tiling is a tiling of the plane by pentagons. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108 is not a divisor of 360...
isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.
Isohedral numbers
The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the least number of orbitsGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
(equivalence classes) of tiles in any tiling of that tile under the action of the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of that tiling, and that a tile with isohedral number k is k-anisohedral. Berglund asked whether there exist k-anisohedral tiles for all k, giving examples for k ≤ 4 (examples of 2-anisohedral and 3-anisohedral tiles being previously known, while the 4-anisohedral tile given was the first such published tile). Goodman-Strauss considered this in the context of general questions about how complex the behaviour of a given tile or set of tiles can be, noting an 8-anisohedral example of Kari. Grünbaum and Shephard had previously raised a slight variation on the same question.
Socolar showed in 2007 that arbitrarily high isohedral numbers can be achieved in two dimensions if the tile is disconnected, or has coloured edges with constraints on what colours can be adjacent, and in three dimensions with a connected tile without colours, noting that in two dimensions for a connected tile without colours the highest known isohedral number is 10.
Joseph Myers has produced a truly remarkably collection of tiles with high isohedral numbers, particularly a polyhexagon with isohedral number 10 (occurring in 20 orbits under translation) and another with isohedral number 9 (occurring in 36 orbits under translation).http://www.srcf.ucam.org/~jsm28/tiling/
External links
- John Berglund, Anisohedral Tilings Page
- Joseph Myers, Polyomino, polyhex and polyiamond tiling