Tessellation

Tessellations may be loosely defined as patterns made from polygons, which cover a plane two-dimensional surface without leaving any gaps. Some examples of tessellations, shown right, include "regular" tessellations in which the "tiles" are congruent regular polygons, and "semi-regular" tessellations i.e. still using regular polygons but allowing more than one kind. One thing is common for all tessellation patterns - the total of the interior angles of all shapes around a point must add up to 360°. This is especially important when attempting to tessellate regular polygons.

All Triangles, equilateral, isosceles or any other variety, may be tessellated; and all Quadrilaterals may be tessellated as well. These include rectangles, parallelograms, trapezia, and any other irregular quadrilaterals e.g. Convex and Concave quadrilaterals as shown to the left.

But when attempting to tessellate e.g. a regular pentagon, you will find that it is impossible to do so. This is because the value of an interior angle for a regular pentagon, 108°, does not divide exactly into 360°. When this situation arises it is necessary to fill the gaps left by tessellating the regular polygon in question with other polygons. An example of this is filling the gaps left by regular Octagons with Squares.

The centres of rotational symmetry of regular polygons can usually be found very easily. If we join by straight lines the centres of symmetry of adjacent polygons in a tessellation, another tessellation results. This new tessellation is known as the dual of the original tessellation. Tessellations of equilateral triangles and regular hexagons are the dual of each other, as shown above, while a tessellation of squares is self-dual (also shown).