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Category: Self-Similarity

  • A New Perspective on Ornamentation

    Over the millennia, ornamentation has been used in architecture and design for countless reasons. For some, the design isn’t complete until it has a few smaller pieces or a border attached to a main structure as a reinforcement of the design’s theme. This is very apparent in Louis Sullivan’s work, where the terra cotta decorations on his skyscrapers reinforced the organic theme of the building.

    Louis Sullivan’s ornamentation on the Van Allen building

    In classical design, one purpose of ornamentation is to reveal the elements of a building that were originally wooden, such as dentils.

    Dentilation used on a classical building

    Another purpose is to show the purpose of the structure. In churches and cathedrals, for example, the ornamentation is a display of Christian themes. 

    The central tympanum of Chartres Cathedral

    Despite the different reasons behind ornaments, one still finds that the borders are fractal. That is, they practice self-similarity. If one looks at the various elements of a structure, such as a wall and a window or a porch and a yard as systems, the merging of those systems requires a border. And the border is most beautiful when that border is fractal. Just as in nature, where two separate systems mix, for example, water vapor (clouds) and air, you will find fractals.

    Modernity Detaches from Ornament

    Many modern architects consider ornamentation as a superfluous addition to a structure or design that adds nothing worthwhile and can thus be discarded. They will point to Baroque and Rococo with their overly dramatic and emotionally intense additions. The purity of true, geometrical form doesn’t need any enhancements, they might argue. The origins of this philosophy began in earnest with Bauhaus, where it emphasized machine technology and function. O. B. Hardison, Jr. writes in “Disappearing through the Skylight”, on the idea that influenced Bauhaus’ founder, Walter Gropius:

    In 1916, Gropius identified the new aesthetic with industrial products: “Accurately cast forms, bare of ornamentation, having clear contrasts and consistency of form and color, become commensurate with the energy and economy of modern life.”

    Bauhaus in Dessau (1925-1932)

    As a result, the borders do not practice self-similarity. One suspects that architects today use these strong, utilitarian lines and mathematical functions to prove that we can move beyond the constraints of Natural Law. For example, Amazon is a trillion dollar company and its headquarters is a group of glass and steel towers rising in defiance of the fractal basis of Nature.

    Amazon headquarters

    Amazon, Virginia. Faustian Man defies Nature.

    This architecture is reminiscent of Spengler’s, “Faustian Man”, where humanity believes it will eventually divorce itself from the constraints of Nature and either reach out to the infinite unknown of space or to embed consciousness into the artificiality of a computer chip.

    Fractals as Ornaments

    In an earlier post I wrote that Nature’s borders are fractal. I believe earlier generations had a strong intuition regarding borders and knew that a building or any design needed self-similarity to soften its edges. They recognized how Nature worked. That is why today’s built environment is a stark contrast to natural forms. The strong, abstract lines, whether Euclidean or based on mathematical functions, clash with the lines of Nature when imposed without self-similarity. However, when a built structure gives way to Nature, such as a derelict or decayed building, you will see the strong lines softened by erosion and vegetation.

    This is one reason why we have intuitively added ornamentation to a structure or design. It makes the object look natural. Rather than having stark, utilitarian lines artificially project, we gently enhance the lines by using self-similarity of scale, or fractals.

    In my walk around Keene, New Hampshire, where older buildings still line the streets, one can easily see the use of ornamentation as a means to soften borders. Windows, porches, eaves, even chimneys all exhibit a self-similarity on the edges.

    Use of ornamentation to soften borders on older structures:

    Compared to modern buildings:

    On the other hand, the modern buildings leave only bare lines that clash with natural form.

    Not just buildings:

    The fractals can be found throughout all parts of exterior furnishings, such as signs and even fences.

    When creating a design, whatever pattern you desire to use, it will have much more resonance and power if you soften its borders with fractals. Use the principles in the design you’ve created and repeat it on the edges. It will be much more beautiful. And you will find that ornamentation, rather than being superfluous or irrelevant, adds harmony and beauty to your creation.

  • Principles of Fractal Design: Self-Similarity of Scale

    When we hear the definition of a fractal, it usually includes the words, self-similarity of scale.

    What does this mean? It sounds abstract, something an engineer or a physicist would use. It’s certainly not an everyday term. However, the truth is completely different. Intuitively, we all know what this means. Our everyday experience tells us the answer. Self-similarity of scale is the notion that the same shape appears in the same pattern no matter the distance you’re looking at it.

    Try an experiment. Take out a sheet of paper and draw a mountain. Start with the main shape, the big view. Now add secondary details, the major components of the mountain. You might want to add two or three, or maybe even five little mountains. Follow up with some details that add relief to the picture.

    Did you find yourself starting with a rough triangle? Did you add a few more triangles within the larger one? And in the details did you add a sharp point here and there? Instinctively you know that a mountain is made up of similar shapes. This is self-similarity of scale. We can also note that while a young mountain will have sharper angles, and older mountains have softer angles, the shapes will be similar throughout each particular drawing.

    You would do the same thing with clouds in the sky or with a tree. In fact, the trick, a friend once told me, to drawing a tree is to begin by drawing a fork from the main trunk. Every so often make another fork in the branches and extend this out until you reach the finer details. If you add leaves, you’ll do the same thing with the veins. (This process has a special name, called bifurcation)

    Self-similarity was first discussed in 1967 by Benoit Mandelbrot in his famous paper, How long is the Coast of Britain? However, Mandelbrot did not come up with the term, fractal until 1975.

    When it comes to design, you’re probably thinking, “Hey, we already do this.” It’s true, we do. Look at many buildings and you’ll see the same shapes echo throughout the building. The Kandariya Mahadeva Temple in India has remarkable detail in each of its domes, as if one is descending spiritually from a higher place.

    Khajuraho: Kandariya Mahadeo tempel (westelijke tempelgroep)

    Palladian architecture, such as Queen’s House, designed by Inigo Jones shows remarkable attention to rectangles. I can’t vouch for the proportions of the windows, colonnades (added later) and the main structure, but it is so pleasing to look at, I imagine the proportions are similar.

    Queens House, by Inigo Jones. Courtesy: Wikipedia

    If you look at a cityscape, you’ll find the fractal nature apparent. Since most buildings are rectangular or square, the varying sizes and widths will give an organic feel to the skyline. Perhaps this is why many people enjoy a cityscape?

    In design, self-similarity often appears because of symmetry (or even asymmetry.) I think it would be advantageous to be more conscious with its use. Take a simple shape and apply it to an object. Use it throughout the piece. Vary the size, but not the shape. If you have smaller ornamentation, make sure it’s there.

  • Look Here, Don’t Look Here

    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.

    Colonnade
    Lobby

    On a larger scale, building complexes can provide tranquility in a stressful environment.