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Tessellations

One definition of a tessellation is: "the careful juxtaposition of elements into a coherent pattern"* I've named my practice accordingly because that juxtaposition is what I strive for in my work. In addition, tessellations tie into many things that intrigue me, including systems, patterns, abstract designs, puzzles, geometry and other mathematic disciplines, and architecture.

Part 1: Origins and Types of Tessellations

Origins of the Term

The word tessellation originates from the ancient Greek word tessares, meaning four or four-cornered. The Latin word tessera means cube or die and tessella refers to small squares laid in a mosaic. From these ancient words come the use of similar terms in various practical applications.

Types of Tessellations

In mathematics, tessellation refers to the study of "tiling" or how regular shapes can be placed to fill an infinite space with no gaps and no overlapping shapes. This is a mathematical discipline which has been evolving since the early 17th century and formally recognized in the 19th century.

Tessellations can be divided into several categories:

Regular Tessellations

Display of Regular Tessellations and their angles.

Regular tessellations are those which consist of a single shape. Only three types of regular tessellations exist: triangles, squares, and hexagons. These shapes by themselves can fill a surface because their interior angles are exact divisors of 360°. Of these shapes, only the squares line up with one another.

Semi-Regular Tessellations

Display of a Semi-regular Tessellation, showing common angles.

Semi-regular tessellations combine two types of polygons that share a common vertice. For example, a regular hexagon with a 1" side can line up with a 1" square. 9 types of semi-regular tessellations exist.

Replicating Shapes (Rep-Tiles)

Rep-tiles consist of congruent shapes that are rotated to create ever-larger versions of the shape in an infinite series. Often called polyforms, rep-tiles are implicit in such phenomena as the classic illustration of the Golden Mean and the Penrose Tile.

3D Tessellations

Tessellations can take 3 dimensional forms as in truncated octahedrons and in geodesic domes. Such forms can combine combinations of shapes; only five are regular polyhedra (i.e. platonic) shapes.

Non-Periodic Tessellations

Non-periodic tilings have no regular, repetitious patterns but rather evolve as they expand over a plan.

*Webster's New International Dictionary, 3rd Edition.

Next: Practical Applications with Tessellations

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