Below are two examples of athroisis. The first is a building designed by Antonini Darmon called Arches Boulogne. The images are available here at archdaily. I found the following quote very instructive with regards to Fractal Design:
The width of these balconies varies according to their orientation; they are narrower to the north thus maximizing illumination. However to the South, East and West the arches provide shade against natural overheating in summer.
Athroisis is an effect. The causes are multiple, just as in nature. In this case, while the intent was to vary the lighting, the result is still a clustering of pillars. In nature, plants can cluster together for a variety of reasons. A rhizome may branch out in specific, yet unpredictable ways. A plant may send its seeds out in the same unpredictable way that still reveals a pattern.
The second building Is the Estonian Vallner Lasnamäe Track and Field Center, located in the outskirts of Tallinn, where the area is filled with Soviet Era apartment blocks made of prefabricated concrete panels. I think it is interesting that the highly regulated Soviet complex is softened by the warm, rich and brown columns. Ivy will eventually cover the building, adding an even more natural ambience.
Screenshot
The wild ivy is to cover the building gradually during next two decades, which shall provide greenery in this area where has very little. The face of the building will always change its appearance in a slow speed, representing steady and slow process of physical training.
A full description of the building can be found here.
Jackson Pollock’s drip paintings are considered masterpieces of modern art. This method drips and flicks paint onto a surface laid out on the floor. Pollock employed this technique to produce rhythmic, energetic paintings. Initially his works received mixed reviews by critics, such as the quote from art critic Emily Genauer the paintings ‘resemble nothing so much as a mop of tangled hair I have an irresistible urge to comb out.’ But as the years went by, these paintings were accepted as works of genius. The patterns created are not simple splashes of paint thrown onto a surface, but a deliberate display of nature in an abstract manner.
Physicist and art historian Richard Taylor investigated his works and realized there is a fractal nature to them. The fractal dimensions of Pollock’s earlier drip paintings, Taylor concluded, correspond closely to those found in nature. See full article here at Discover Magazine.
Image property of the Albright-Knox Art Gallery, Buffalo, NY.Autumn Rhythm (Number 30), 1950
An enduring quality to these paintings is their fractal nature. Not only did Taylor find that they exhibit self similarity of scale, an important component of fractal geometry, but their fractal dimension resembles that of coastlines. In fact, Number 14, has a fractal dimension of 1.45, similar to that of many coastlines.
Jackson Pollock Number 14
We humans find this fractal pattern very aesthetic. Taylor and perceptual psychologists in Australia and England asked 50 subjects to evaluate about 40 different patterns. Subjects preferred fractal dimensions between 1.3 and 1.5, roughly 80 percent of the time.
Athroisis in Pollock’s Paintings
I believe Pollock intuited the fractal nature of nature and worked very hard to show it in his work. This includes the concept of athroisis. Fractal patterns, using self similarity of scale, gather in groups of different sizes. Pollock’s paintings show this gathering by color. The result is a dynamic pattern. Can you see in the photo below the similarity between the flowers and the arrangement of colors in Pollock’s paintings?
Field of Flowers
It follows from the concept of athroisis that there is no center. In a field of flowers or a grove of trees, there is no place one can point to and say, “this is the point from which everything has come.” There are only gatherings of similar items. If we happened to be in a forest, for example, we would never say to meet in the center. We might say instead to meet in the heart of the forest, the place where the trees are densest. This may be towards the center of the forest, but it is not necessary.
Pollock’s paintings have no center. This is an important reason for their dynamic nature. Like clouds in the sky, his colors gravitate towards each other and simultaneously pull apart. The result are patterns that are diffuse and permeable, in other words, fractal.
Seventy years ago, Pollock saw an important underpinning in nature. His remarkable observational skills and artistic ability expressed this in his paintings. The world is much richer because of it.
A friend visited Seattle and we stopped by the new Starbucks Roastery and Tasting Room. It is a combination of cafe, roastery and restaurant, along with a souvenir shop. It is dedicated to roasting reserve coffees.
While sitting at a table, I glanced up above the bar and noticed a series of wooden beams that were joined together in an odd arrangement. They were clustered in different amounts.
This is an application of athroisis. Just as a group of trees will gather in varying amounts, the designer decided to do the same with this ceiling. Note the three dimensional application. The thickness of the ceiling resembles a forest, with the thicker portions similar to more trees gathered together.
We humans are linear. We love our straight lines and do our best to find them wherever we look. We like straight roads, straight pillars, tall buildings and straight edges on paper. We try to find straight lines in our scientific endeavors. Sometimes it works, and sometimes the match is way out of line—that pun was unintentional, but I’ll keep it since it works.
The stunning rhythms in architecture that consist of straight lines and sharp angles are a joy to view. The colonnade of St. Peter’s Basilica, the Parthenon, a structure by Mies van der Rohe and thousands of other buildings around the world are hypnotizing and seem to reflect a higher order of nature.
But these structures become boring quickly if all we use are straight lines and sharp angles. Eventually we put in a curve here and there to add variety. We add curves because it seems more natural to do that. We know that Nature does not drum a monotonous beat. Too much of the same pattern and we have dreary, staid and lifeless.
Intuitively, we know that Nature is not linear, even though we force her into our straight lines and often get confused when she goes her own, unpredictable way. When wind suddenly changes direction or whirls into a tornado we say the effect was unexpected. We want nature to play nice and thus confuse the map (our beliefs and science) with the territory (Nature.)
We love our strait lines, but the truth is, we also know they’re not quite right. We live in the delicate balance between variety and sameness; we want both a stable relationship and a fiery romance. But the stable relationship gets tedious and the fiery romance quickly fades. We want our Aristotelean mean but never reach it because the mean is not a real entity, but rather just an average of the fluctuations between extremes. Average is a mental state. And it is no surprise that Nature has no average.
Athroisis: Things come together
Nature lives on that boundary of sameness and variety. One such example is what I call Athroisis (pronounced a-THROY-sis), which is Greek for « a gathering process. » (Special thanks to Dr. Richard Martin of Stanford University who helped me with the proper word choice.) When you see objects in nature gather together, they don’t align in perfect squares or circles, they cluster together in a loose arrangement where the number in each cluster is usually different. Sometimes, the cluster will be small, even a cluster of one. Other times the clusters are made up of thousands of individual items.
We find athroisis everywhere in nature. A forest grove, when looked at from above or from a distance, reveals trees interspersed in groups. A pile of leaves on the ground form clumps. The petals on a flower are not completely symmetrical; sometimes they shift a bit to the side. Mountain peaks, river deltas and even spots on a leaf cluster together. We know instinctively that planting our garden requires groupings. An array of flowers and shrubs regularly distributed just seems, unnatural.
A chaos scientist would call athroisis, a strange attractor. An attractor is a point of equilibrium in a system (see page: What is Fractal Design?) A strange attractor is a system where things never quite settle down. There is a sort of equilibrium, but it isn’t stable. And that is what we see in nature. As a system that always in flux, there is never a chance for natures creations to settle down.
One might think that I’m looking for something that really isn’t a phenomenon. After all, the reason plants cluster together is because of seeds, wind patterns, whether or not it is a rhizome and other variables. A river delta is based on the level of the ground.
I argue that the cause is immaterial. The effect is everywhere, so much that we should pay attention to it in our art.
Athroisis in Art
And we do use athroisis in our art. We feel it naturally in our music. We place our songs in a musical key but deviate from that with dissonance. A composition has a theme, but it morphs into different rhythms and slightly different notes. Dancers whirl and twirl across the stage, gathering together for a moment only to break apart and rejoin in a new formation.
We can apply athroisis to design principles just as easily. We can accent our tables and chairs with varying clusters. Chair backings can be composed of varying slats. An end table could vary the number of legs. Le Corbusier used athroisis in two of his buildings: Couvent de la Tourette and Carpenter Center. If you look closely at his windows, you’ll notice varying widths.
Couvent de la Tourette
Couvent de la Tourette — closeup
Notice the fenestration along the second and third floors
Carpenter Center, HarvardCarpenter Center, Harvard
Notice the fenestration along the curved studio space
Groupings are not merely cosmetic. They can be the foundation of a structure. For example, we can design colonnades in groups that vary in distance. Pillars inside of a building can also be grouped. The structure will change to accompany the placement of the pillars. The initial placement of the pillars will determine the levels above it. This can lead to exciting variations in a building’s look merely by the arrangement of the pillars at ground level. This dependence on the initial placement of load-bearing columns is what I call iteration, and it’s a critical component of Fractal Design and fractals in general.
Do not think this is a radical departure from conventional design. I stated earlier that sameness and linearity are critical. Athroisis, and Fractal Design in general are meant to work harmoniously with linear design. When we’re grounded in regularity we can soar to greater heights. Just as we can see that the trees in the forest are « regular but different », we can design with the same principle. In another post I will discuss how to vary athroisis with linear design.
One of Nature’s gifts is that it relaxes our senses and promotes healing. There is a peace and calm that overtakes us when we experience a natural setting. In contrast, an environment that continuously demands concentration and focus is not healthy; the mind and body will suffer over time.
Why does Nature have such a calming effect? There are most likely dozens of reasons; fresh air, the sounds of birds and a susurrus wind all come to mind. I think the way Nature arranges herself can also play a role, and that role can be used in design.
Regularity
In an artificial environment we are overwhelmed with regularity. Everything is evenly spaced. Road lines are dashed out in even intervals (for good reason!) Beams overhead are spaced evenly, usually for load displacement, and we are overjoyed with symmetry in all of its forms. Regularity is necessary for information flow. We find it easier to navigate a complex area such as an airport or a subway station when there are regular guideposts. The famous New York City subway map designed by Massimo Vignelli, is considered a masterpiece of for its ability to impart information in a clear and simple visual language.
But regularity, as mentioned above, demands focus. It forces us to look in certain ways at specific points in space. Yet anyone who has looked at the starry sky at night or watched a waterfall knows the joy of letting the mind go and being enraptured by irregularities in nature. A common bit of advice given to students who are suffering from excessive concentration while studying their text books is to occasionally stare into the distance. Focusing on infinity relaxes the eyes.
The Best of Both Worlds
Can we have design that uses the best of both these ideas? I believe we can. I first recognized this possibility when looking at the iron and glass ceilings of European rail stations. The network of iron is regular, but from a greater distance it becomes so intricate that the eye loses its ability to focus. The intricate webwork takes on a sense of randomness, but with an order behind it. It acts like a mandala that is used frequently for meditation in Eastern religions.
Pitt Rivers Museum (Oxford). Architects: Benjamin Woodward and T. N. Deane. 1858. Note the webwork of iron become more complex at a distance. Wikipedia.Iron and Glass Ceiling. The Oxford Museum. Architects: Benjamin Woodward and T. N. Deane. 1858. Photograph 1977 George P. Landow may be used without written permission for any educational or scholarly purpose. Victorianweb.Mandala: an ancient fractal embodying complexity. Wikipedia.
Complexity: Athroisis and SSOS
Complexity leads to a lack of focus. But complexity might be interpreted as randomness, which it is not. It is often \”unpredictable order\” as one would see in a mountain range or a grove of trees. It is also the ever-diminishing detail of a fractal, which leads the eye to infinite smallness. Note that my definition of complexity is not the exact definition that you will find in mathematics and physics.
Fractal Design has within it the power to relax the senses using complexity. What I called, unpredictable order can be simulated with athroisis. Athroisis implies an ordered complexity. A grove of trees is pleasing to look at but one doesn’t usually focus on one tree, but rather the grove of trees. The grove itself is the point of focus. Unlike a row of trees artificially planted, where one’s focus is guided to a vanishing point, the grove of trees stops the eye and gives it a “patch” of rest. We often get lost in these patterns. A starry sky is unpredictable, but the arrangement of the stars within the Milky Way implies an order. Likewise, a mountain range and a waterfall have a sense of unpredictability within order.
A waterfall’s boundaries are also fractal, employing self-similarity of scale. SSOS is different from complexity and athroisis. The patterns are similar enough to guide the eye, but not close enough to demand further attraction. SSOS has the same effect as athroisis. A fractal such as a Koch Snowflake or a Mandelbrot Set throws the mind into ”infinite space” just as easily as a sky of puffy clouds. Straight lines always point to an end. They are saying, ”look over here, look over there.” But when they become so small and intricate the mind loses focus and can give the mind a sense of rest.
Lost in the Design
The joy of fractals and fractal design is to get lost in the design. There is enough regularity and familiarity (order) to guide the eye, but not enough to concentrate the focus.
This can be a pleasure if one’s goal is to create an environment where relaxation is the goal. Dining halls or even restaurants that wish to give a sense of naturalness can benefit. Hospitals, airports or any area where extended waiting is necessary could benefit from this. A danger could result however if traffic flow is critical. A fractal area would be like creating eddies in a current and counter-productive. But, if used with prudence, a fractal environment could ease tension and stress.
Coffee Shop. Arrange tables and dividers in athroitic patterns. Areas with shaded lines are dividers roughly three to four feet high.Clinic. Arrange tables and dividers in athroitic patterns. Areas with shaded lines are dividers roughly three to four feet high.
It should be emphasized, however, that the sense of peace and calm using athroisis and SSOS can also be used on furniture, not just the space itself. Employ it on benches, chairs, furniture, kiosks and anywhere else that would enhance a sense of nature.
ColonnadeLobby
On a larger scale, building complexes can provide tranquility in a stressful environment.
Using the principles of Fractal Design, I’ve sketchd a fireplace. Notice that the sides are not symmetrical. While they are close to even, they are not exact. Also, I’ve used athroisis to vary the width’s and thicknesses of each wooden beam.
