In geometry

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 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...

 is a family of shapes – called tile

Prototile

In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others...

s – that cover the plane (or any other geometric setting) without gaps or overlaps. Such a tiling might be constructible

Constructibility

In mathematics, there are several notions of constructibility. Each of the following is by definition constructible:* a point in the Euclidean plane that can be constructed with compass and straightedge...

 from a single fundamental unit

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...

 or primitive cell

Primitive cell

Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions...

 and is then called periodic. An example of such a tiling is shown in the diagram to the right (see the image description for more information). Every periodic tiling has a primitive cell that can generate it. A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles can be called aperiodic tiling

Aperiodic tiling

An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. Properly speaking, aperiodicity is a property of particular sets of tiles; any given finite tiling is either periodic or non-periodic...

s.

The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. Please note that this list of tiles is still incomplete.

Explanations

Abbreviation	Meaning	Explanation
E2	Euclidean plane	normal flat plane
H2	hyperbolic plane Hyperbolic space In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...	plane, where the parallel postulate Parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...  does not hold
E3	Euclidean 3 space Three-dimensional space Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...	space defined by three perpendicular coordinate axes
MLD	Mutually locally derivable	two tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or inserting an edge)

List

Image	Name	Number of tiles	Space	Publication Date	refs	Comments
	Trilobite and cross tiles	2	E2	1999		Tilings MLD from the chair tilings
	Penrose P1 tiles	6	E2	1974		Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
	Penrose P2 tiles	2	E2	1977		Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
	Penrose P3 tiles	2	E2	1978		Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
	Binary tiles	2	E2	1988		Although similar in shape to the P3 tiles, the tilings are not MLD from each other, developed in an attempt to model the atomic arrangement in binary alloys
	Robinson tiles	6	E2	1971		Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No image	Ammann A1 tiles	6	E2	1977		Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
	Ammann A2 tiles	2	E2	1986
	Ammann A3 tiles	3	E2	1986
	Ammann A4 tiles	2	E2	1986		Tilings MLD with Ammann A5.
	Ammann A5 tiles	2	E2	1982		Tilings MLD with Ammann A4.
No image	Penrose Hexagon-Triangle tiles	2	E2	1997
No image	Golden Triangle tiles	10	E2	2001		date is for discovery of matching rules. Dual to Ammann A2
	Socolar tiles	3	E2	1989		Tilings MLD from the tilings by the Shield tiles
	Shield tiles	4	E2	1988		Tilings MLD from the tilings by the Socolar tiles
	Square triangle tiles	5	E2	1986
No image	Sphinx tiles	91	E2
	Starfish, ivy leaf and hex tiles	3	E2			Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
	Robinson triangle	4	E2			Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
	Danzer triangles	6	E2	1996
	Pinwheel tile Pinwheel tiling Pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway.They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.... s		E2	1994		Date is for publication of matching rules.
No image	Wang tile Wang tile Wang tiles , first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems... s	104	E2	2008
No image	Wang tiles	56	E2	1971		Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
	Wang tiles	32	E2
	Wang tiles	16	E2
	Wang tiles	14	E2
	Wang tiles	13	E2	1996
No image	Decagonal Sponge tile	1	E2	2002		Porous tile consisting of non-overlapping point sets
No image	Horocyclic tiles	85	H2	2005
	Goodman-Strauss hyperbolic tile	1	H2	2005		Weakly aperiodic
No image	Goodman-Strauss strongly aperiodic tiles	26	H2	2005
No image	Böröczky tile	1	Hn	1974		Not strongly aperiodic
	Schmitt tile	1	E3	1988		Screw-periodic
	Schmitt-Conway-Danzer tile	1	E3			Screw-periodic and convex Convex 'The word convex means curving out or bulging outward, as opposed to concave. Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
	Socolar Taylor tile	1	E3	2010		Periodic in third dimension
No image	Penrose rhombohedra	2	E3	1981
No image	Wang cubes	21	E3
No image	Wang cubes	18	E3
No image	Wang cubes	16	E3
No image	Danzer tetrahedra	4	E3	1989
	I and L tiles	2	En for all n ≥ 3	1999

External links

• Stephens P. W., Goldman A. I. The Structure of Quasicrystals
• Levine D., Steinhardt P. J. Quasicrystals I Definition and structure