A tessellation (also called tiling) is when shapes are arranged side by side to produce a pattern with no gaps in between. Tessellations have existed since ancient Egyptian times and are still common today in floor and wall tilings. [Photo: pavement tessellation from wikipedia.com]
The artwork of M. C. Esher has popularized the concept of tessellations to the general public, and Islamic tessellations may have influenced modern day origami tessellations. Steve Edwards web site shows an impressive collection of tessellations from Egyptian tombs up to modern day patterns.
The simplist tessellations are the "regular tessellations" where only one shape is used. In order for the shapes to be flush one against another, the shape must be either a triangle, a square, or a hexagon. [Photo: the three possible types of regular tessellations from wikipedia.com].
Tessellations can be made with two or three shapes. These are called semiregular and demiregular tessellations. Tessellations can also be made with irregular shapes.
Most tessellations have a pattern that repeats itself over and over again but some tessellations, like the ones discovered by Robert Penrose do not repeat. These are called "aperiodic tessellations".
[Photo: an aperiodic tessellation from wikipedia.com].
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Origami tessellations can be regular (triangles, hexagons) or irregular (parallelograms, trapezoids). They are all periodic (repeating pattern).