Tessellations

Patterns covering the plane by fitting together replicas of the same basic shape have been created by Nature and Man either by accident or design. Examples range from the simple hexagonal pattern of the bees' honeycomb or a tiled floor to the intricate decorations used by the Moors in thirteenth century Spain or the elaborate mathematical, but artistic, mosaics created by Maurits Escher this century. These patterns are called tessellations.

What is a tessellation? In geometrical terminology a tessellation is the pattern resulting from the arrangement of regular polygons to cover a plane without any interstices (gaps) or overlapping. The patterns are usually repeating.

Afamous Irish artist Escher, used diverse animal figures.

There are three types of tessellation. Regular Tessellations Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons.

Semi-regular Tessellations Semi-regular tessellations are made up with two or more types of regular polygon which are fitted together in such a way that the same polygons in the same cyclic order surround every vertex. There are eight semi-regular tessellations which comprise different combinations of equilateral triangles, squares, hexagons, octagons and dodecagons.

Non-regular Tessellations Non-regular tessellations are those in which there is no restriction on the order of the polygons around vertices. There is an infinite number of such tessellations. http://www.mathpuzzle.com/Tessel.htm