Once on a trip across the country, my plane was flying across a cloudless sky. Up ahead I could see a flat layer of clouds. From a distance, the clouds looked like someone had unrolled a carpet in the sky. The cloud was one, single blanket that abruptly started and continued for miles.
However, as we passed over the layer, it became clear that there was no fixed border for the clouds. In fact, the edge became porous, and thinned as it moved outward. Each portion of the fringe had a sculpted look that resembled the main border. The edge was expressing self-similarity.
Nature’s boundaries are fractal. Wherever you see two natural objects meet, you will find fractals. Something as simple as the boundary of a tree is fractal. What is the tree connected to? Just as a cloud borders the air, so does a tree. The bark is a fractal as well as the branches and the leaves.
Alder Bark and Oregon Grape flowers and leaves. Every border in this photo has a fractal component.
A river bank is fractal. Water of every variety, when bordering the land will produce fractals. Likewise, the surface of water is fractal.
In the photo above, the water surface is fractal, the grass is fractal and even the goose feathers are fractal.
Intuitively, we know that Nature is fractal. It is what makes something look natural.
A human body’s exterior is fractal. The hair, certainly, but also the skin, with its fractal layering of cells. Any connection between water and earth will be fractal.
Even the atmosphere, bordering space itself, seems to be fractal.
Every border that I have shown has its own reasons to behave in a fractal manner. Tree bark is not the same as the edge of Earth’s atmosphere, yet we can see that self-similarity plays a role.
In my previous post on Order and Disorder, I discussed the idea that Nature is perceived as disorder because, when left on its own, it will destroy our created works. But nature is a fractal system, so it is not really disordered. Perhaps one could say that the fractal, chaotic system of nature is a superset of our Euclidean system. As it has all the advantages of a living system, it will eventually overtake our own artificial environment.
When looked at this way, the natural and artificial realms are simply two different systems. We know intuitively that, when utilized with discretion, one system can enhance the other. Looking out my window, I see a short, concrete wall with a metal fence. The plain concrete and black metal bar by itself is not attractive in the least. However, planted in front of this fence are bushes that look quite wild. The juxtaposition of these two systems works very well.
Transition Zones in Physics
Without going into excessive detail, physical equations can often vary depending on the distance or on the amount of time. For example, gravity inside the earth increases linearly and outside the earth it decreases exponentially. The equation shifts on the surface of the earth. In electromagnetism and optics, scientists discuss the ”near field” and the ”far field”. There is a transition zone as one shifts from near to far. Often, this is where the interesting phenomena occur.
With the juxtaposition of our two systems, we can also see that there is a beauty in the ”transition zone.” Rob Bear wrote an article at Curbed on the home of the architect George Post. One of his photographs focused on an outdoor wall made of stone. Like any ordinary wall, grass will grow to the wall. What they did in this case was to make a dirt barrier around the wall. The grass continued to grow and encroach on the dirt barrier, creating a fractal boundary. The dirt itself is also a fractal boundary; it, I believe, complements the irregularity of the stone wall.
Home of architect George Post
Fractals soften borders
Any homeowner will also find that there is no crack between concrete slabs that doesn’t find grass or weeds poking out. The same with a crack in any building. While annoying, there is a beauty here. The strong, Euclidean shapes are softened by the fractals of the weeds.
Vatican Piazza
Vatican Piazza
Note however, that the same building when viewed from a perspective where smaller buildings surround it produces a softening effect, because the irregularity of buildings is fractal.
St Peter’s Basilica, city view. Courtesy Wikipedia
Below are further examples where Nature has provided her own softening of Euclidean lines. Moss and leaves are common sources. Plants with thin branches or leaves can also be used for a softening effect.
There is no reason why this cannot be done artificially. One can use fractal design as a complement to traditional design. I do not mean that we add vegetation around the building, but add artificial structures that resemble fractals. This could simply be a colonnade or smaller buildings arranged with athroisis. Remember that athroisis, or groupings, is a strange attractor. Strange attractors are fractals
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.
Seattle’s Museum of Pop Culture (formerly known as the Experience Music Project) was designed by Frank Gehry. The inspiration behind the building was a smashed guitar, in honor of Seattle’s Jimi Hendrix. Gehry purchased several electric guitars, sliced them into pieces, and used them as building blocks for an early model design.
Three-thousand panels, made up of 21 thousand individually cut and shaped stainless steel and painted aluminum shingles, encase the outside of the building. Their individual finishes respond to different light conditions and appear to change when viewed from different angles, reminding audiences that music and culture is constantly evolving.
The exterior undulates in parts and the top of the red section is meant to look like guitar frets. The building has brought on a mixed response of admiration and derision. This is true for many of his buildings. Gehry is not satisfied with traditional design and his works defy categorization—which leads to the ambiguity of criticism.
Like Wright, I think Gehry has intuited the fractal nature of the world. The Euclidean approximations that we have used since buildings began aren’t necessarily the only way to construct a building. With the freedom offered to us with new materials and also with new design and construction techniques, it is easier than ever to change the shape of a structure. Gehry has taken advantage of these. For instance, traditional methods of handling the loads of a building are calculated with CATIA, a software used at Boeing that analyzes the stresses and strains of airplanes.
Fractal Dimension of Crumpled Paper
As I’ve discussed in previous posts, fractals take on a dimension between the Euclidean dimensions. Natural, 3-dimensional objects fall within 2.0 and 3.0 rather than the Euclidean 3rd dimension. According to a study by Zhang, Samal and Brandle, the investigation of the fractal dimension of three different tree crowns ranges 2.24 to 2.45. Study: A Method for Estimating Fractal Dimension of Tree Crowns from Digital Images. Link to the study.
What is very interesting is that the Hausdorff (fractal) dimensions of a crumpled sheet of paper is 2.5, which is very close to a tree’s fractal dimension. From Wikipedia:
When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the ISO 216 A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made. Creases will form at all size scales (see Universality (dynamical systems)).
Los Angeles Philharmonic Association – Rehearsals
Photos by Craig T. Mathew/Mathew Imaging
Crumpled paper is the inspiration for some of Frank Gehry’s work, including the set for Don Giovanni performed by the Los Angeles Philharmonic in 2012.
The natural shapes emerging from modern architecture and design, look to me at least, to be a recreation of the natural world. Gehry and Berkus have intuited the fractal nature of the world and are implementing it in their works. While the aesthetics aren’t visible, there is a possibility that this could lead to interesting and beautiful architecture and design.
The 20th Century architect, Marcel Breuer designed and built the St. John’s Abbey Church in Collegeville, Minnesota (near St. Cloud.) Breuer states that ‘this cast-in-place concrete marvel is a stepping-stone in modern design of religious architecture in the United States. One must admire the great concrete trees that support the ceiling and the dominant bell banner that shields the church.’
The church is a technological feat as well. Incorporating a massive use of cast and steel-reinforced concrete, it was fundamentally constructed by local carpenters who made the forms into which the concrete was cast. The large bell banner at the north side, the main entrance to the church, houses a cross made of oak harvested from the woods at Saint John’s Arboretum, and the five bells that call people to worship. It is 112 feet high, stands on thin parabolic arches, and announces that this is indeed a special place.
The wooden slats used to cast the concrete were 1 inch by 3 inches. The wet concrete seeped between the slats before hardening, giving a softening texture to the walls.
When examining the cast concrete I noticed the patterns on the concrete formed athroisis patterns.
Athroisis is everywhere. It is a random pattern, found both in Nature and in human endeavors. It is chaotic and fractal. Use it wherever you find a design too artificial.
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.
