geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a shape is said to be anisohedral if it admits a

tiling Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...

, but no such tiling is

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

(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.

Existence

The first part of Hilbert's eighteenth problem asked whether there exists an anisohedral polyhedron in

Euclidean 3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...

; Grünbaum and Shephard suggestGrünbaum and Shephard, section 9.6 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.

Convex tiles

Reinhardt had previously considered the question of anisohedral

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

s, showing that there were no anisohedral convex

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

s but being unable to show there were no such convex pentagons, 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.

Isohedral numbers

The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the lowest number

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

(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 a 10-anisohedral example of Myers. 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 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

References

External links

* John Berglund
Anisohedral Tilings Page
* * Joseph Myers
Polyomino, polyhex and polyiamond tiling
{{Tessellation Tessellation