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. [1]
The first part of Hilbert's eighteenth problem asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest [2] 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. [3] However, Heesch then gave an example of an anisohedral tile in the plane in 1935. [4]
Reinhardt had previously considered the question of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while finding the five types of convex pentagon tiling the plane isohedrally. [2] 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. [5]
The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the lowest number orbits (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). [6] 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. [7] Grünbaum and Shephard had previously raised a slight variation on the same question. [8]
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. [9]
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).
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types is aperiodic if copies of these tiles can form only non-periodic tilings.
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.
In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it with no overlaps and no gaps. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 and proposed the more general 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.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry, a parallelogon is a polygon with parallel opposite sides that can tile a plane by translation.
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
Karl August Reinhardt was a German mathematician whose research concerned geometry, including polygons and tessellations. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the Reinhardt polygons.
A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word play on ein Stein, German for "one stone".
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.
