Tessellation

A tessellation, or tiling, is the process of covering a surface using geometric shapes called tiles. To be a true tessellation, the tiles must fit together perfectly. This means there can be no gaps between the shapes and no overlaps where one tile sits on top of another.

In mathematics, this concept is studied as planar tiling when it occurs on a flat surface, known as a Euclidean plane. Mathematicians can also extend these ideas to higher dimensions. When tiles fill a three-dimensional space, the pattern is often called a honeycomb.

To understand how these patterns work, we must look at the relationship between tiles. An edge is the line where two bordering tiles meet. A vertex is the specific point where three or more tiles intersect.

Some patterns are edge-to-edge, meaning the sides of the polygons match up perfectly. In these cases, a tile cannot share a partial side with its neighbor. A common example of a non-edge-to-edge tiling is a brick wall, where the long side of one brick meets the corners of two others.

Tessellations are categorized by how much they repeat. A periodic tiling follows a pattern that repeats regularly. These patterns can be organized into seventeen distinct mathematical categories called wallpaper groups.

In contrast, a non-periodic tiling lacks a repeating pattern. There is a special subset called aperiodic tilings. These use a specific set of shapes, known as prototiles, that are mathematically incapable of forming a repeating pattern.

The Penrose tiling is a famous example that uses two different quadrilateral shapes to create this effect.

We can further classify tilings by the types of polygons they use. A regular tessellation is highly symmetric and uses only one type of regular polygon. Only three shapes can form a regular tessellation: the equilateral triangle, the square, and the regular hexagon. A semi-regular, or Archimedean, tessellation uses more than one type of regular polygon. However, it must be isogonal, meaning every vertex looks exactly the same. There are eight known types of semi-regular tessellations. For example, a tiling might have a vertex configuration of 4.8.8, meaning one square and two octagons meet at every corner.

History shows that humans have used these geometric principles for millennia. Around 4000 BC, the Sumerians created wall decorations using clay tiles.

During classical antiquity, Romans used small square blocks called tesserae to create intricate mosaics.

Islamic art also features sophisticated geometric tilings, such as the colorful zellige patterns found in the Alhambra palace in Spain.

In 1619, Johannes Kepler conducted an early documented study of these structures. He explored the hexagonal patterns found in snowflakes and honeycombs. Later, in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling belongs to one of the seventeen isometry groups. This work is considered the unofficial start of the mathematical study of tessellations.

The significance of tessellations extends from art to the natural world. In nature, bees create hexagonal cells in honeycombs to organize their hives.

In human construction, tilings provide durable and water-resistant surfaces for floors and walls. Artists like M. C. Escher used tessellations for incredible visual effects. He often used irregular, interlocking shapes like animals to create complex patterns in both Euclidean and hyperbolic geometry. Even in domestic life, tessellations appear in the decorative patterns of quilts.

Mathematics continues to find new complexities in these patterns. The Swiss geometer Ludwig Schläfli pioneered the study of higher-dimensional versions of these shapes, called polytopes. He created the Schläfli symbol notation to describe them compactly. For instance, a tiling of regular hexagons can be written as {6,3} to show it has six-sided polygons meeting in groups of three. While many rules exist, such as the Conway criterion for periodic tiling, no general rule has been found to determine if any given shape can tile a plane. This leaves many interesting problems for mathematicians to solve.