These are the analogues to polygons and polyhedra in spaces with more dimensions. The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. Mathematically, tessellations can be extended to spaces other than the Euclidean plane. No general rule has been found for determining whether a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. These tiles may be polygons or any other shapes. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. The elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain that attracted the attention of M. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Many other types of tessellation are possible under different constraints. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.

There are only three shapes that can form such regular tessellations: the equilateral triangle, square and the regular hexagon. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile. The tessellations created by bonded brickwork do not obey this rule. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another. Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.Ī rhombitrihexagonal tiling: tiled floor in the Archeological Museum of Seville, Spain, using square, triangle, and hexagon prototiles Tessellations are sometimes employed for decorative effect in quilting. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the Moroccan architecture and decorative geometric tiling of the Alhambra palace. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor, or wall coverings. A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.Ī real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. A tiling that lacks a repeating pattern is called 'non-periodic'. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups.

Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.Ī periodic tiling has a repeating pattern. An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles.Ī 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.

Escher tessellation cubes