How Tessellations Work

A jigsaw puzzle offers an easy visual of a tessellation we might commonly encounter.
A jigsaw puzzle offers an easy visual of a tessellation we might commonly encounter.

We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe. Within its figures and formulas, the secular perceive order and the religious catch distant echoes of the language of creation. Mathematics achieves the sublime; sometimes, as with tessellations, it rises to art.

Tessellations -- gapless mosaics of defined shapes -- belong to a breed of ratios, constants and patterns that recur throughout architecture, reveal themselves under microscopes and radiate from every honeycomb and sunflower. Pick apart any number of equations in geometry, physics, probability and statistics, even geomorphology and chaos theory, and you'll find pi (π) situated like a cornerstone. Euler's number (e) rears its head repeatedly in calculus, radioactive decay calculations, compound interest formulas and certain odd cases of probability. The golden ratio (φ) formed the basis of art, design, architecture and music long before people discovered it also defined natural arrangements of leaves and stems, bones, arteries and sunflowers, or matched the clock cycle of brain waves [sources: Padovan, Weiss, Roopun]. It even bears a relationship to another perennial pattern favorite, the Fibonacci sequence, which produces its own unique tiling progression.

Science, nature and art also bubble over with tessellations. Like π, e and φ, examples of these repeating patterns surround us every day, from mundane sidewalks, wallpapers, jigsaw puzzles and tiled floors to the grand art of Dutch graphic artist M.C. Escher, or the breathtaking tile work of the 14th century Moorish fortification, the Alhambra, in Granada, Spain. In fact, the word "tessellation" derives from tessella, the diminutive form of the Latin word tessera, an individual, typically square, tile in a mosaic. Tessera in turn may arise from the Greek word tessares, meaning four.

Mathematics, science and nature depend upon useful patterns like these, whatever their meaning. Beyond the transcendent beauty of a mosaic or engraving, tessellations find applications throughout mathematics, astronomy, biology, botany, ecology, computer graphics, materials science and a variety of simulations, including road systems.

In this article, we'll show you what these mathematical mosaics are, what kinds of symmetry they can possess and which special tessellations mathematicians and scientists keep in their toolbox of problem-solving tricks.

First, let's look at how to build a tessellation.

Shaping Up, or Could You Repeat That Please?

Tessellations run the gamut from basic to boggling. The simplest ones consist of a single shape that covers a two-dimensional plane without leaving any gaps. From there, the sky's the limit, from complex patterns of multiple irregular shapes to three-dimensional solids that fit together to fill space or even higher dimensions.

Three regular geometric shapes tessellate with themselves: equilateral triangles, squares and hexagons. Other four-sided shapes do as well, including rectangles and rhomboids (diamonds). By extension, nonequilateral triangles tile seamlessly if placed back-to-back, creating parallelograms. Strangely enough, hexagons of any shape tessellate if their opposite sides are equal. Therefore, any four-sided shape can form a gapless mosaic if placed back-to-back, making a hexagon.

You can also tessellate a plane by combining regular polygons, or by mingling regular and semiregular polygons in particular arrangements. Polygons are two-dimensional shapes made up of line segments, such as triangles and rectangles. Regular polygons are special cases of polygons in which all sides and all angles are equal. Equilateral triangles and squares are good examples of regular polygons.

All tessellations, even shapely and complex ones like M.C. Escher's, begin with a shape that repeats without gaps. The trick is to alter the shape -- say, a rhomboid -- so that it still fits snugly together. One simple approach entails cutting a shape out of one side and pasting it onto another. This produces a shape that fits together with itself and stacks easily. The more sides you alter, the more interesting the pattern becomes.

If you're feeling more adventurous, try doodling a wavy line on one side, and then copying the same line to the opposite side. This approach may require some tweaking to get the pieces to interlock properly. For example, if your polygon has an odd number of sides, you might want to divide the leftover side in half and then draw mirror-image shapes on either side of the split. This creates a side that interlocks with itself.

Try your luck with two or more shapes that tessellate. You can do this geometrically, or simply fill the page with any shape that you like, and then imagine an image that fits the negative space. A related method entails filling a known tessellating shape with smaller shapes. There are even fractal tessellations -- patterns of shapes that fit together snugly and are self-similar at multiple scales.

Don't worry if your initial results seem a bit nonsensical. It took Escher years to master these mad mosaics, and even he had pairings that didn't always make sense.

Now that we've laid the groundwork, let's take a look at some of the special tessellations that researchers use to solve tricky theoretical and applied problems.

Tiling the Universe: Special Tessellations

This Voronoi tessellation is looking at the photon density of a particular region. Each dot in the cell represents a photon.
This Voronoi tessellation is looking at the photon density of a particular region. Each dot in the cell represents a photon.
Image courtesy NASA

As researchers explored tessellations and defined them mathematically, they identified certain types that excel at solving difficult problems. One popular example is the Voronoi tessellation (VT) also known as the Dirichlet tessellation or the Thiessen polygons.

 A VT is a tessellation based on a set of points, like stars on a chart. Each point is enclosed by a polygonal cell -- a closed shape formed from line segments -- that encompasses the entire area that is closer to its defining point than to any other point. Cell boundaries (or polygon segments) are equidistant to two points; nodes, where three or more cells meet, are equidistant to three or more defining points. VTs can tessellate higher dimensions as well.

The resulting VT pattern resembles the sort of honeycomb a bee might build after an all-night nectar bender. Still, what these cockeyed cells lack in beauty, they more than make up for in value.

Like other tessellations, VTs pop up repeatedly in nature. It's easy to see why: Any phenomenon involving point sources growing together at a constant rate, like lichen spores on a rock, will produce a VT-like structure. Collections of connected bubbles form three-dimensional VTs, a similarity researchers take advantage of when modeling foams.

VTs provide a useful way to visualize and analyze data patterns as well. Closely clustered spatial data will stand out on a VT as areas dense with cells. Astronomers use this quality to aid them in identifying galaxy clusters.

Because a computer processor can build a VT on the fly from point source data and a set of simple instructions, using VTs saves both memory and processing power -- vital qualities for generating cutting-edge computer graphics or for simulating complex systems. By reducing required calculations, VTs open the door to otherwise impossible research, such as protein folding, cellular modeling and tissue simulation.

A close relative to the VT, the Delaunay tessellation also boasts a variety of uses. To make a Delaunay tessellation, begin with a VT, and then draw lines between the cell-defining dots such that each new line intersects a shared line of two Voronoi polygons. The resulting lattice of chubby triangles provides a handy structure for simplifying graphics and terrain.

Mathematicians and statisticians use Delaunay tessellations to answer otherwise incomputable questions, such as solving an equation for every point in space. Instead of attempting this infinite calculation, they compute one solution for each Delaunay cell.

In his Jan. 27, 1921, address to the Prussian Academy of Sciences in Berlin, Einstein said, "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 perfection. Nevertheless, they enable progress by reducing otherwise unwieldy problems to a form manageable by current computation power. More than that, they remind us of the underlying beauty and order of the cosmos.

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