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The Japanese art of Ikebana, or Kadou, has been celebrated for hundreds of years. Ikebana is the word most Westerners know, and it means ‘flower arranging’ or ‘living flowers.’ Kadou means ‘the way of flowers’ and is another tradition along the lines of Shodou (calligraphy) or Karatedou (Karate). Dou means, The Way.

Ikebana focuses on space and emptiness, line and form and the natural character of the plant. It deliberates on the relationship between heaven, earth, and humanity.

What I found intriguing in Ikebana is the use of athroisis. Flowers are arranged in loose groupings in various locations. Each grouping, often with flowers but also with stems and leaves, maintains a balance with the whole.

The borders of each object in the arrangement: petals, leaves and stems are all fractal. This blurs the boundaries. It is possible to imagine that while the fractal borders end at a certain point, in our minds they may extend further, as we intuitively understand the infinite regression of fractals.

The following pictures were taken from the 1963 book, “Japanese Flower Arrangement Styles” by Lida Webb. The black and white photos add a distinctive clarity to the arrangement of the the flowers.

In the photo below, note the white flowers create a harmony on their own, as one pushes away from the other two, its stem and leaves making it separate, yet within the group of the flowers. Meanwhile, the branches create their own group.

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In the next photo, the entire arrangement is offset to the left of the vase, but enough extends into the empty portion of the vase to give it balance. It is not symmetrical, but instead it gives us a dynamic equilibrium. This asymmetry replicates into the arrangement itself, a self-similarity of scale, where the white flowers are offset on the left of the arrangement. The needles, three separate groupings, balance the flowers by pushing rightwards and upwards.

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The next arrangement I would argue, has three separate groupings: flowers, leaves and needles. The loose clustering of these three adds to the dynamic balance.

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The following image, the branches have flowers. This increases the density of the stems while also grouping the stems closer together, into an almost complete whole. With the heaviness of the flowers, the branches are extended further to the left to provide balance.

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The next photo is similar, except the branches now have leaves. With the flowers in the bottom center, there is more symmetry with this pattern.

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The last photo groups the flowers while a second grouping of leaves fills the empty space left by the flowers. The stark branches pull in opposite directions, focusing on the athroisis pattern of the flowers.

Ikebana is a microcosm of Nature displaying her love for athroisis and self-similarity. By examining the patterns from the various schools, one can glean patterns for designers. There is still symmetry, but it is not precise.

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.

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.

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.

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

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

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)).

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.

From Don Lauritzen.

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.

Information from ArchDaily.

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.