Understanding Tessellations

We explore mathematics not only for its beauty and elegance, but also for its ability to encapsulate the patterns that govern the universe. In its numbers and formulas, those with a secular view see order, while those with a religious perspective hear the distant whispers of the language of creation. Mathematics reaches sublime heights, and with tessellations, it even becomes a form of art.

Tessellations – seamless mosaics of geometric shapes – are part of a vast family of ratios, constants, and recurring patterns found across architecture, visible under microscopes, and emanating from the structure of everything from honeycombs to sunflowers. Break down any number of formulas in geometry, physics, probability, statistics, or even chaos theory, and you'll encounter pi (π) as a central element. Euler's number (e) frequently appears in calculus, the mathematics of radioactive decay, compound interest, and in certain aspects of probability. The golden ratio (φ) is the foundation for art, design, architecture, and music, long before it was revealed to also explain natural formations, from leaves and stems to bones, arteries, sunflowers, and the cycle of brain waves [sources: Padovan, Weiss, Roopun]. Additionally, it is closely linked to the Fibonacci sequence, which produces its own distinctive tiling pattern.

Tessellations appear in science, nature, and art, much like π, e, and φ. These repeating patterns are everywhere, from everyday elements like sidewalks, wallpaper, and tiled floors to the renowned works of Dutch artist M.C. Escher, or the stunning tilework of the 14th-century Moorish Alhambra fortress in Granada, Spain. The term "tessellation" comes from tessella, a diminutive form of the Latin word tessera, referring to a single square tile in a mosaic. Tessera may even trace its roots to the Greek word tessares, meaning four.

Mathematics, science, and nature rely on meaningful patterns like these, regardless of their significance. In addition to their captivating elegance in mosaics or engravings, tessellations have practical uses in various fields such as mathematics, astronomy, biology, botany, ecology, computer graphics, materials science, and even in simulations like road systems.

This article will introduce you to the concept of mathematical mosaics, explaining the types of symmetries they exhibit and the special tessellations that mathematicians and scientists use in their problem-solving arsenal.

To start, let’s explore how to create a tessellation.

Shaping Up, or Could You Repeat That Please?

Tessellations range from simple to mind-boggling. The simplest ones involve a single shape that fills a two-dimensional plane completely, without any gaps. From there, the possibilities are endless, including intricate patterns of multiple irregular shapes or three-dimensional solids that seamlessly occupy space, and even higher dimensions.

There are three regular geometric shapes that can tessellate on their own: equilateral triangles, squares, and hexagons. Other four-sided shapes, such as rectangles and rhomboids (diamonds), can do this too. Interestingly, non-equilateral triangles can also tile perfectly when placed edge to edge, forming parallelograms. In fact, hexagons of any type will tessellate as long as their opposite sides are equal. Therefore, any four-sided shape can create a seamless mosaic by aligning back-to-back, much like a hexagon.

A plane can also be tessellated by combining regular polygons or by arranging regular and semiregular polygons in specific patterns. Polygons are two-dimensional shapes composed of line segments, such as triangles and rectangles. Regular polygons are special polygons where all sides and angles are equal, like equilateral triangles and squares.

Every tessellation, even complex and beautiful ones like M.C. Escher's, starts with a basic shape that fits together perfectly without gaps. The key is modifying the shape—like a rhomboid—so it continues to fit snugly. One simple method involves cutting part of the shape from one side and attaching it to the other. This results in a new shape that fits with itself and can stack easily. The more sides you modify, the more fascinating the resulting pattern.

For a more creative approach, try drawing a wavy line on one side, then copy it to the opposite side. You may need to adjust it to ensure the pieces fit together properly. For example, if your polygon has an odd number of sides, you might split the remaining side in half and mirror the shapes on each side of the split. This creates a side that interlocks seamlessly with itself.

Challenge yourself with two or more shapes that tessellate. You can do this geometrically or simply cover a page with any shape you like and then visualize an image that fits into the negative space. Another method is filling a tessellating shape with smaller shapes. There are even fractal tessellations, where patterns of shapes fit together perfectly and are self-similar at different scales.

Don’t be discouraged if your first attempts seem a little strange. It took Escher years to perfect his intricate mosaics, and even he had combinations that didn’t always quite make sense.

Now that we’ve established the basics, let’s explore some of the unique tessellations that researchers use to tackle challenging theoretical and practical problems.

Tiling the Universe: Special Tessellations

As researchers delved into the study of tessellations and began defining them mathematically, they discovered certain types that proved to be particularly effective in solving complex problems. One widely recognized example is the Voronoi tessellation (VT), also referred to as Dirichlet tessellation or Thiessen polygons.

A VT is a tessellation based on a set of points, such as stars on a graph. Each point is surrounded by a polygonal cell, a closed shape formed by line segments, that covers the entire area closest to its defining point compared to any other point. The boundaries between cells (or polygon segments) are equidistant from two points, and nodes where three or more cells meet are equidistant from three or more defining points. VTs can also tessellate in higher dimensions.

The resulting VT pattern looks like the kind of honeycomb a bee might construct after a long night of drinking nectar. Despite their quirky appearance, these oddly-shaped cells more than make up for their lack of beauty with their usefulness.

Like other tessellations, VTs frequently appear in nature. It’s easy to understand why: Any process involving point sources expanding at a steady rate, such as lichen spores growing on a rock, will form a structure similar to a VT. When bubbles connect together, they form three-dimensional VTs, a pattern that researchers use to model foams.

VTs are also a valuable tool for visualizing and analyzing data patterns. When spatial data points are closely clustered, they become clearly visible on a VT as regions filled with densely packed cells. Astronomers take advantage of this feature to help identify galaxy clusters.

Since a computer processor can generate a VT in real-time from point source data and a set of straightforward instructions, using VTs conserves both memory and processing power. These are critical resources when creating advanced computer graphics or simulating intricate systems. By minimizing the necessary calculations, VTs enable previously unfeasible research, including protein folding, cellular modeling, and tissue simulation.

A close cousin to the VT, the Delaunay tessellation offers a range of applications. To create a Delaunay tessellation, start with a VT, then draw lines between the defining points of the cells so that each new line intersects an existing line from two Voronoi polygons. The resulting arrangement of plump triangles forms a useful structure for simplifying graphics and terrain.

Mathematicians and statisticians use Delaunay tessellations to solve otherwise uncomputable problems, such as finding an equation for every point in space. Rather than trying to compute this infinite task, they calculate one solution for each Delaunay cell.

In his speech on Jan. 27, 1921, to the Prussian Academy of Sciences in Berlin, Einstein remarked, "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Clearly, tessellated approximations fall short of perfect accuracy. However, they facilitate progress by transforming otherwise intractable problems into manageable forms, thanks to modern computational power. More importantly, they serve as a reminder of the inherent beauty and order of the universe.