The following essay was written for the exhibit in Olbrich Botanical Gardens in February 2019

How do bees build the incredibly perfect hexagonal honeycomb cells? What will a spherical soap bubble floating in air become when surrounded by other bubbles in a foam? What do cells look like in a tiny root of a barley plant? These and other mysteries can be explored by studying natural compositions and patterns called tessellations.

The term “tessellation” represents a tiling of shapes on a two-dimensional plane or in a three-dimensional space without any gaps or overlaps between them. These geometries naturally emerge as a consequence of simple and fundamental laws governing our universe.

The first time I noticed the beauty of these tessellated patterns was when I was exploring how thin films of liquid polymers crystallize on a super flat surface. While the substrate was spinning, I dropped a small volume of a dissolved liquid polymer onto it. The solution formed a uniform layer that was thinner than a micrometer. For a comparison, an average diameter of a typical human hair is about 100 micrometers.

The solvent rapidly evaporated and the polymer left on the surface immediately started crystallizing. I observed this process under a polarizing microscope, which enhances the contrast between areas of different thickness with vivid colors and makes it obvious which regions of the polymer film crystallized and which were still amorphous. What I observed was fascinating.

The polymer film did not crystallize uniformly across the surface, but the crystals started developing from a number of random points, which are also called seeds by materials engineers. The polymer molecules arranged themselves into tightly and periodically packed structures. These structures, or crystals, grew from the seeds outwards in all directions in perfect circles. All of the circles in the microscope’s field of view grew at the same rate, only stopping when they reached the edge of their neighboring circles, where they formed crystal boundaries. As a result of each crystalline circle negotiating the available space to occupy, an interesting pattern emerged.

When the growing circles filled up the empty surface of the substrate, the final network of their boundaries consisted of perfectly straight lines, connected at various angles, three or more lines meeting at each junction point. There were no gaps between the crystals and none of them overlapped. They formed a tessellated pattern.

What can be a possible reason we find these tessellated patterns so visually pleasing? Is it the fact that their complexity is a consequence of very simple growth of circles on a surface? The crystals grow at an equal rate in all directions and when they meet their neighbor growing from a different seed, their meeting points form a perfect straight line.

Imagine you live in a large city with many wells randomly distributed through the city. Which well will you use to obtain water? Probably the one located closest to your residence, and everyone in your city does the same. Each residential group using a specific well builds a fence around their neighborhood to keep the well protected and safe. The fence lines, connected and butted up to each other, form a network within the landscape. And when you fly over the city in a hot air balloon, it will be obvious that the network of the fences looks exactly the same as the boundaries between the polymer crystals that I described earlier. The wells represent the crystal seeds. The fences perfectly define the area in which the residents consider the well in the center to be the closest to them. The people who live exactly on the fence line could in theory use both of the neighboring wells. The distance is the same.

In mathematics, this fragmentation of a surface into regions based on distance to a specific point, like the locations of each well to the corresponding fence lines, was described by Georgy Voronoi, a Russian mathematician, in 1908. These patterns are now referred to as Voronoi patterns. However, an earlier mention of this phenomena can be tracked back to 1644 by Rene Descartes, a French philosopher, mathematician, and scientist.

With both examples, the polymer crystals and the fenced wells, the seeds were not uniformly distributed on the surface, so the shape of the tessellated pattern was semi-regular. What if the seeds, or starting points, were equally spaced? The tessellated pattern would be called a perfect periodic pattern. What is a good example of a perfect periodic pattern in nature? Bees’ honeycombs. How is it possible that these little organisms, with tiny brains that are twenty-thousand times smaller than a human brain, can fabricate something so perfect and architecturally advanced?

Despite an extensive literature review, I was having trouble visualizing bees’ movements and their decision processes that would lead to building perfect hexagon cells within a honeycomb. Then one day I built and used a new tool for carving concave spheres in wood. This tool helped me to answer this scientific question through an artistic process of making wood sculptures.

The tool is built from a router used for carving wood and an engine jack used for lifting engines from a car’s engine bay. The router is suspended from the jack’s arm at the end where usually the hook is located. The router hangs on a metal rod connected to the arm with a freely moving joint. This setup allows for the router to swing like a pendulum in all horizontal directions and it can travel up and down by pumping or releasing the hydraulic jack.

Using a thick, flat piece of wood, I first carved a concave partial sphere into the surface. It looked like a circular bowl carved into a flat piece of wood. After cutting the first depression, I moved the wood a little to the side and repeated the cut. That’s when the secret of the mysterious geometry revealed itself. The intersection where these two concave spherical cuts overlapped, was a perfectly straight line. After repeating the cut six times around the first cut, a beautiful hexagonal shape emerged. Without controlling this final form, only repeating cut after cut, I started understanding how bees make their honeycombs hexagon cells.

If we consider only one bee, it seems to be a complicated task to perfectly fit six circles around the center circle, but the power of a bee hive is in their numbers. The bee hive is a super-organism made up of hundreds to thousands of bees and each worker bee is repeating the same task over and over again. In the same proximity, at the same time, they have to negotiate their personal space with other bees and they can measure their distance with their legs or antennas.

The hive's worker bees deposit the wax in circles, from a central point outwards. Slowly, they make these circles bigger and bigger, eventually covering the entire surface and coming into contact with the wax circles built by the neighboring bees. At this instance, they must compromise between each other, so they don’t overlap their circle with the other circle. The result of this negotiation becomes a straight line cut down the middle. A hexagon emerges when seven circles are packed tightly together. This is consistent with the polymer crystals I observed and the concave spheres I cut into wood.

Sometimes, bees don’t start their circles in the perfect pattern of six around the central circle and the result becomes interesting. They end up with pentagonal or heptagonal honey cells. That would be fine, but these irregular cells do not fit into the hexagonal pattern and leave large gaps between cell walls, which is not an efficient use of space. This type of design would not be a tessellation by definition. What consequence do gaps between the cells have for the bees?

There would be a lot of wasted space between cells, and a worker bee would have to expend a lot of energy filling in those gaps with wax. That is inefficient. Bee wax is an energetically expensive material because a bee has to use approximatelly eight units of honey to make one unit of wax. So to make their labor-intensive effort as productive as possible, they must develop a pattern with no wasteful gaps. The hexagons with shared walls are the best pattern that uses the least amount of wax. That is one of the most stunning tessellated patterns in nature.

Tessellated patterns come in all shapes and forms. Another fascinating pattern that consists of one type of shape was invented by mathematician Heinz Voderberg. Voderberg tessellations are complex patterns of tiles grouped together in a spiral. The individual tile is a complex, elongated, irregular enneagon, or a nine-sided shape.

We all are familiar with tessellated spirals. Think of a sunflower, a pineapple, or a pinecone. There are so many other examples in nature, because all these fruits and seed pods follow the rules described above. For example with sunflowers, individual seeds and cells start growing from microscopic points in the flower and as they increase in size, they negotiate and fill up every available space. As these seeds grow from the fixed boundary, such as the stem, they push the previous seeds outwards. This gets repeated over and over, and the emerged shape ends up being a spiral composition.

The world of plants is an excellent place to find intricate and fascinating tessellations. Again, there is no blueprint of the final structure that each growing cell would follow. They just keep growing in size as they keep dividing. They are filling up the space in all available directions, eventually reaching their neighbors, exerting pressure on them. The final tissue becomes a tessellated group of individual cells whose boundaries are flat planes. This is evident when one looks at a cross section of a tree and inspects its xylem or phloem under a microscope. Plant's roots, such as barley, are tessellated. Under a microscope, the cross-section of a barley root looks like a large circular window found on historic cathedrals.

In nature, we think of tessellations forming from growth and expansion. Tessellations also can be formed through shrinkage. In this reverse movement, nature follows the same tessellation rules. Have you ever seen a dry lake bed or a dry puddle of mud? While drying up, no mud is removed. Its volume gets smaller because water evaporates, but all the mud particles remain there. The particles pack closer together and cracks develop. The individual mud pies fit perfectly together and form a tessellated pattern.

In mathematics, this particular phenomenon was studied by Edgar Gilbert, an American mathematician and coding theorist, in the 1960s. He developed a model for the the formation of the crack network, which starts with a set of randomly distributed points. At each point, a crack starts forming in both directions at random orientations until they meet another crack. In the 1970s, it was proven that Gilbert’s model correctly described a majority of mud cracks that form in nature.

How can we imagine what these tessellated patterns look like in the three-dimensional space? A great way to visualize it is to do a simple experiment in a kitchen sink or a bathtub. Take a straw and blow air into soapy water. You will create foam. Not one bubble, but hundreds or thousands of bubbles in a small space, touching and pushing on each other. Look closely. As they pack together, they reach an equilibrium at which they share their walls with their neighbors. What do the bubbles look like now? Their walls become planar surfaces. The bubbles that were spheres in open air now became shapes called polyhedra. Amazing and beautiful!

Earlier, the bees taught us that the most economical way to tessellate a two-dimensional plane with the cells of the same size is to divide it into hexagons. In the case of three dimensions, what is the most efficient way to divide the space with polyhedral bubbles of the same size? What would these bubbles look like if we want to tessellate the space with the smallest amount of soapy water?

The first person asking this question was Lord Kelvin, a physicist and engineer. His research led to the 1887 discovery of what is called the truncated octahedron, a polyhedron with six square faces and eight hexagonal faces. This discovery was so brilliant that it took over 100 years to come up with a more efficient way to tessellate foam with the smallest amount of soapy water. It is called the Weaire–Phelan structure, which is a complex 3-dimensional structure representing an idealized foam of equal-sized bubbles, and it improved Lord Kelvin’s approach only by 0.3%!

Overall, the geometric organization of tessellated patterns is pleasing to a human eye. Its attractiveness does not only symbolize efficient construction and packing, but more importantly, humankind's dream to live in a society where its members fall into each other's side without any voids or conflicting overlaps - physically, as well as spiritually.

(c) Peter Krsko 2019