by Eileen Clegg and Bonnie DeVarco
“The beautiful is a manifestation of the secret laws of nature. When nature begins to reveal her open secret to a person, he feels an irresistible longing for her most worthy interpreter, art.” Johann Wolfgang Goethe
Patterns of Genius
While most of us “read” the language of shape subliminally, many of history’s notable artists, philosophers and scientists did so intentionally enabling their genius creations and discoveries. Leading thinkers in many disciplines explored shapes and their geometries, learning Nature’s language so they could translate sound into music, words into poetry, canvasses into art, and observations into scientific theory.
Page from original copy of Luca Pacioli's 1509 Treatise, De Divina Proportione with sketch of polyhedra by
Leonardo da Vinci. On exhibit in Mlan, ItalyCC Paxelsson
A recent revival of interest in the story of shape focused on a well-known genius of history, Leonardo Da Vinci, who worked at the intersection of many disciplines – art, mathematics, science and engineering. Leonardo painted memorable art pieces including the “Mona Lisa,” designed technologically advanced machinery including an early version of the helicopter, carried forward mathematical innovations based on the work of ancient geometers, and discovered the workings of the human body by dissecting animals. Nearly 500 years after his death, a novel captured the imagination of a culture beginning to rediscover visual language. In 2003, the book Da Vinci Code [i] spun a tale of a historical conspiracy with clues encoded in ancient artifacts.
While the “code” concept makes for great fiction, the reality is far less illicit and far more elegant. Nature speaks the same way to everyone; some just listen more carefully than others. As Goethe said, it is an “open secret.” With today’s tools for analyzing ancient manuscripts and artifacts, we’re discovering that our ancestors mapped the workings of Nature, and made discoveries that are helping today’s scientists. Nature’s patterns appear in pan cultural art throughout time. It seems like a code because it underlies so many disciplines and communication forms.
Shapes in nature –sometimes called “nature’s language”– are not just static forms. They predict the process of growth, the cycles of planets and, and the principles of aesthetics. Conspiracy theories aside, there is some truth to the stories about ancient sects with hidden documents explaining the patterns of Nature. Knowledge about the relationship between geometry, shape, art, architecture and science was passed on through various exclusive organizations over centuries. But the knowledge is not hidden; it can be found with just a little digging into the journals of people who have influenced global knowledge. Many were privy to studies of earlier generations. A few examples:
- The Greek philosopher Plato, known to most Philosophy 101 students as the author of the political treatise “The Republic,” was a student of geometry who wrote about “perfect forms,” and postulated a geometric basis for the universe and in his work Timaeus (360BC)
- German writer Johann Wolfgang von Goethe (1749-1832) generally known for his literary contributions was a polymath who studied science, optics, shape and color and whose philosophy influenced Charles Darwin, who is credited with the theory of evolution.
- Lewis Carroll, author of Alice’s Adventures in Wonderland and Through the Looking Glass was mathematician C. L. Dodgson (1832-1898) whose work in geometry carried forward that of ancient Greek mathematician Euclid. Carroll’s writings are thought to contain references to mathematical and logic problems.
- Composer Claude Debussy (1862-1918) is thought to have created his harmony, rhythm and other musical structures based on his studies of shape and proportion. [ii] A quote attributed to him: “music is the arithmetic of sound as optics is the arithmetic of light.”
From these and other influential creators of the past, we can surmise that the study of shape enhances creativity and innovation. Clearly, innovative thinking comes from crossing disciplines to discover patterns at the intersection. But there is more to it: There is something special about certain shapes that allows them to impact our thinking and imagination in unexpected ways.
Begin with the Golden Mean
The golden mean is fairly well known as a mathematic concept, so it provides a good launch into the exploration of shape of thought. Leonardo and Debussy are among those who based some of their work on the Golden Mean. It appears all around us in Nature, in art, in architecture and even our teeth – and yet the concept is elusive to most in these modern times when the power of shape is not well understood.
The golden mean is a geometric ratio found throughout Nature and human design. It is pleasing to the human eye and is one of Nature’s most ubiquitous building tools. In the next set of images, you can a “golden rectangle” at work. A golden rectangle is one in which the long side is slightly more than one and a half times the other. In the image, you see a set of nested rectangles with a spiral inside, demonstrating a unique property of the golden ratio: If you cut a square from the Golden Rectangle, the remaining rectangle would have exactly the same height-to-width ratio as the original rectangle. Cut a square from that, and again you are left with a remaining rectangle that has the same height-to-width ratio, and on and on. Only a rectangle scaled to the Golden Mean has this property. Inside the rectangle, the “Golden Spiral” is also defined by this special proportion. It is the same spiral found in the galaxies and, as shown, in a shell, cauliflower, sunflower and ancient artistic motifs and is known by its symbol, PHI - Φ.[iii]
In the 1980s, British dentist Dr. Edwin Levin applied the golden proportion to make false teeth more natural, an endeavor that led him to create the “golden mean gauge” instrument to measure the ratio on everything from teeth to insects to famous architectures.[iv]
Dr. Edwin Levin's "Golden Mean Gauge" shows the golden ratio in a smile, a butterfly wing, the Great Wall of China.
For more than two millennia, the golden mean has been used by artists, architects and scientists in wildly different ways: To design type fonts and page layouts, to design buildings, and to understand the proportion of the body. Today the golden mean also informs some of our newest technologies. It is being used in biotechnology and nanotechnology to design our smallest human machines; it is also used to understand the structure of the cytoskeleton and micro-crystals and to understand better the workings of our brains.
As a “code” the golden mean has turned out to be, as scientist Mario Livio called it, “astonishing.”[v] As a key for bringing beauty and power to their art, the golden mean was known by Leonardo da Vinci and other artists as “divine.”[vi]
Beyond the Golden Mean
As it turns out, the golden mean is only the beginning of the story. Exploring the geometry of Nature – spheres, spirals and the practical beauty of five-fold symmetry –opens a world of knowledge into the art of human communication.
Startlingly, we’ve now discovered that ancient humans used symbols based on features of Nature they could not see with their eyes or instruments. Is it possible, as some have theorized, that humans intuitively speak the language of Nature – not just through observation? That theory gains credibility with discoveries that made through the electron microscope and the scanning tunneling microscope.
Some of the shapes depicted in ancient artifacts that were invisible to the human eye and came into view only with the advent of 20th century tools included microstructures that are fundamental to the workings of our planet, among them: The Carbon-60 molecule (the buckminsterfullerene), the micro-organism, Emiliania huxleyi,[vii] and the brain protein clathrin[viii].
Each of these organisms resembles the “buckyball” shape, most familiar to people because of the geodesic dome created by architect Buckminster Fuller. Each is a truncated icosahedron (20-faced sphere with 12 pentagons and 20 hexagons). Each serves as a transport mechanism: Buckminsterfullerene in the Universe, Emiliania huxleyi in Earth’s oceans, and clathrin in the brain. And they do so because they have the ability to weave themselves together when necessary and unbind when their task is done.
Unbound, they show another layer of shapes that are also strangely familiar. In fact, these shapes can be seen in the art of the ancient Southeast Asian three-way weave, a part of ancient basketry practices and an important design feature of the woven balls for the game, sepak-takraw, played with a ball based on the same structure of buckyballs – and the same flexible geometry that also inspired Fuller’s geodesic domes.[ix]
Are the similarities in structure just a coincidence? Or are Nature’s structures somehow understood intuitively beyond our logical thinking? The story behind Buckminster Fuller’s geodesic dome offers insight into these questions.
The Architect, the Chemist and the Brain
Architect and philosopher Buckminster Fuller’s life and work demonstrate the power of shape. Born in 1895 and famous as “the world’s friendly genius” by the time he died in 1983, Fuller worked across disciplines and deep into the interrelated aspects of Nature, geometry, and design.[x] He was among the visionaries whose work demonstrates the power of shape—what he called “Nature’s technology”—in cross-disciplinary wisdom. He was known for his philosophy and architecture, most famously for the geodesic dome that would later have a profound influence on chemistry and nanotechnology.
The importance of the sphere came to him in a vision in 1927, at a time when he was in the depths of despair over the death of his first daughter and his inability to provide for his new child and her Mother.[xi] He was questioning his will to live when he envisioned himself surrounded by a transparent sphere, and these words came into his mind, “You do not belong to you, you belong to the Universe.” He then dedicated his work to the power of efficient design, which, he discovered could be found in the hidden geometry of the sphere.
The dome captured the zeitgeist of the environmental movement. During the socially turbulent 1960’s, Fuller galvanized a generation of artists and musicians who resonated with his writings about Nature, and his bubble-like Expo dome became an icon of optimism and holistic thinking. A student-created geodesic dome was a feature of the first Earth Day celebration at University of Minnesota, where Fuller was a speaker in 1970 – and the dome would come to be associated with Earth Day and sustainability, which was central to Fuller’s vision for a better world. Fuller’s geodesic dome was so appealing to people that it has been used for world expos, theaters, trade fairs and auditoriums. But the dome had properties that extended well beyond its aesthetic appeal. After his death, in the 21st century, the geodesic dome, affectionately called the “buckyball,” would be considered a key to understanding nanotechnology.[xii]
Fuller lived to see his intuition develop into a design science approach that has inspired the world. But he did not live to see another powerful effect of his vision. It was after his death that the structure of his geodesic dome inspired a significant scientific breakthrough: Scientists Richard Smalley, Harold Kroto and Robert Curl had been puzzling over the 3-dimensional structure of a newly discovered carbon60 molecule, invisible to the eye even with a microscope: They knew it was made up of hexagon shapes, but how could the hexagons curve into a sphere?
Dr. Kroto recalled his experience inside Fuller’s Expo ’67 dome in Montreal that was a key feature of the U.S. Pavilion. He realized that Fuller’s geodesics held the key to understanding the molecule’s 3-dimensional curved shape.[xiii] Fuller introduced pentagons between the hexagons to create his domes, enabling curvature. Richard Smalley called the buckminsterfullerene the “Rosetta stone of nanoscale architecture because of its dynamic properties and unique symmetry. [xiv] It enabled chemists to create carbon nanotubes by introducing pentagons to create curvatures in a lattice structure to close the ends or to braid the nanotubes into “ropes”.
Left to Right: Buckminsterfullerene (C60) molecule. Fuller & Sadao’s Expo ’67 Dome, Montreal, 2004
PSCHologram montage of Buckminster Fuller portrait (circa 1980) and a model of buckyball (by Ellen Sandor,1996).
Fuller had an intuitive vision of a shape that turned out to be a building block of the universe—before scientists were able to perceive it in Nature. Was this an example of understanding “the language of intuition” that is also the language of nature? To Fuller, beauty of design and the workings of Nature were not separate. He often talked about “Nature’s technology” and its resonance with beauty. “When I am working on a problem, I never think about beauty, but when I have finished, if the solution is not beautiful, it must be wrong.”[xv]
One of Buckminster Fuller’s phrases was “invisible architecture,” referring to what he believed to be unseen organizing principles of the universe. He studied geometry in relation to “synergy” (the term he popularized meaning “behavior of whole systems unpredicted by the behavior of its parts considered separately). His own experience demonstrates how synergy may arise from an innate comprehension of the power of shape.
The Invisible Roots of Symbols
Perhaps Fuller would not have been surprised to learn that his geodesic dome reflected an invisible and essential carbon molecule. Or that other shapes that people discovered intuitively would also turn out to have morphological siblings on the micro-scale. It’s a leap to try to explain why. Is it possible that we “feel” or “sense” the language of Nature, and automatically express it in our movements and art?
We have examples of symbols that seem to have universal meaning, without their originators knowing why. The symbol of the triskelion is one. It is included in the brain protein clathrin. As described above, clathrin works by folding and unfolding. When it unfolds, the parts are in a shape of three curved “legs”—the triskelion.
The triskelion is a shape that has become a symbol in many cultures around the globe, dating back to ancient artifacts developed at a time when humans had no way of knowing its role in the human body.
Ancient Tibetan, Celtic and British symbols depicting the triskelion
The folding and unfolding of the brain protein clathrin. [Screenshots from an animation by Tomas Kirchhausen and Allison Bruce, Harvard University
Can Nature Help Shape Our Thoughts?
Amazingly, ancient symbols were based on images of Nature that were invisible to the people who created them. This shifts our thinking about communication. We’ve long known that Nature’s principles enhance art and design—key ingredients in deep communication (engaging the “heart and mind,” the conscious and unconscious). Now we are beginning to discover the reverse is also true: That art and design can give us a glimpse into Nature, perhaps even aspects of Nature as yet undetected.
Nature does have a “code” – it’s comprised of math and geometry, “Nature’s technology” that inspired Fuller and so many other geniuses. Whether uncovered though observation or intuitive leaps of thought, Nature’s technology determines the patterns in the Universe. Should we be surprised to discover that it also applies to our human communication?
Looking back at the scientific observations and artistic of past eras enables discovery of the shapes and symbols that have endured over time. If we “unpack” these shapes and symbols, most have a story to tell. They carry the weight of centuries of human interpretation.
Understanding the interplay between Nature and shape is a step toward understanding how we can use Nature’s technology to re-engineer our communication, data visualization, and information organization at a time of cognitive excess.
It also is a step toward exploring what we can learn about Nature and our own human challenges—which were shared by our ancestors—from shapes, symbols and artifacts that are part of our collective global culture.
[i] Brown, Dan The Da Vinci Code. New York: Random House. 2003
[ii] Howatt, Roy, Debussy in Proportion: A Musical Analysis. Cambridge: Cambridge University Press. 1983.
[iii] The famous “Fibonacci sequence” is intimately linked to the golden mean. In the 12th century, mathematician Leonardo Fibonacci in his studies on rabbit breeding discovered a number series that both reflects PHI and can be used to derive PHI. It is now called the Fibonacci Sequence. This number series is endless, but begins as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... These numbers may look random, but they are not. Starting with 0 and 1, each successive number is the sum of the previous two. The PHI ratio starts to appear after the first eight numbers in this sequence. The magical ratio between the numbers then occurs over and over again.
[v] Mario Livio, Head of the Science Division at the Hubble Space Telescope Science Institute, published the most important recent historical work on the Golden Mean in 2002. The Golden Ratio: The Story of PHI, the Worlds Most Astonishing Number is one of the most extensive additions to the timeline of historical and mathematical treatments of the golden mean.
[vi] The golden mean has been a recurring subject within many classic works. It was first mathematically defined in Euclid’s Elements in the 4th century BCE and discussed in Vitruvius’s 10 books on architecture. The term “divine proportion” was later applied to the golden mean by Fra Luca Pacioli in his book De Divina Proportione. Written in 1499 and published in 1509 and this was the first book completely devoted to the Divine Proportion. With its 60 color plates of polyhedra, Pacioli’s was also the only book illustrated by Leonardo da Vinci. The 2000-year timeline of books on the Divine Proportion is discussed later in the "Shape of Thought" series.
[vii] Although it is hard to see the specific geometry of the coccolithopore, Emiliania huxleyi, the most prevalent form is made of 10 round coccolith platelets self assembled with radial symmetry. The geometry of these platelets creates an internal structural pattern of 20 hexagons and 12 pentagons, the “truncated icosahedron,” one of the most spherical of the family of polyhedra.
[viii] Clathrin coated vesicles are found throughout the human body, with clathrin’s role to bring nutrients into the body and facilitate communication between the cells. To do this, clathrin assembles into geodesic “basket shapes” of varying sizes. But the clathrin protein that is found in the brain – synaptic clathrin - is exactly the same truncated icosahedron shape as buckminsterfullerene and is sometimes referred to as the “biofullerene.” Koruga, Djuro, Stuart Hameroff, et.al. Fullerene C60 – History, Physics, Nanobiology, Nanotechnology. North Holland Publishers. 1993. p142.
[ix] Subsequent installments of the "Shape of Thought" series explores the influence of the Southeast Asian three-way weave on Fuller’s work in detail.
[x] From 1927 to 1983 R. Buckminster Fuller devoted his life to what became a lifetime experiment that he called “Guinea Pig B” – B for Bucky. During his 56-year experiment, Fuller left an incredible legacy of artifacts, patents and publications, as well as what the Smithsonian called the “most extensive personal archive in existence.” See “Life, Facts & Artifacts” by Bonnie DeVarco, an essay on Fuller’s archive from the companion Web Site for the American Masters Biography: R. Buckminster Fuller: Thinking Out Loud.http://www.thirteen.org/bucky/devarco.html
[xi] Fuller’s young daughter, Alexandra, died from complications from polio and spinal meningitis in 1922. His second daughter, Allegra, was born in 1927 when Fuller was 32 years old.
[xii] “For Fuller there were no flat planes. Everything was curved, from space to shape. Everything was in motion and was continually shifting. But the classical shapes remain as guideposts to form found throughout the Universe. His approach the same problems that earlier geometricians and artists tackled by "tiling the plane" or building polyhedra was through the closest packing of spheres. Delineating vector lines within closest packed spheres, Fuller defined the basic polyhedra that could be used as dynamic building blocks on a larger scale.” DeVarco, Bonnie. “Energetic Architecture – Buckminster Fuller’s Geometry of the Sphere.” fromInvisible Architecture. Online Publication. 1996. http://members.cruzio.com/~devarco/invisible.htm
[xiii] “Kroto remembered the day he had walked around with his son inside the Expo Geodesic Pavilion in 1967. He had marveled at the triangular latticework of the huge sphere that surrounded him, a sphere that needed no internal supports. Almost 20 years later as Kroto and Smalley discussed Fuller's geodesic dome and its possible relevance to their finding, Smalley decided to check out a book on Fuller from the library and take a closer look at his domes. He found the secret in a photo of Fuller's 1958 Union Tank Car Dome in Baton Rouge, the largest clear span enclosure of its day. Fuller used pentagons! Once they figured that out, it was easy for them to build a cage structure of sixty atoms whose shape exhibited 12 pentagons and 20 hexagons. Kroto and Smalley felt it most appropriate to name it “buckminsterfullerene” for its striking resemblance to Fuller's geodesic domes” DeVarco, Bonnie. “The Discovery of Buckminsterfullerene” from Invisible Architecture. Online Publication. 1996. http://members.cruzio.com/~devarco/nature.htm See a longer account of the discovery of buckminsterfullerene in Chapter 2 “September 1985” in: Aldersey-Williams, Hugh. The Most Beautiful Molecule – The Discovery of the Buckyball. Canada: John Wiley & Sons, Inc. 1995. pp52-92.
[xiv] In an early lecture on nanotechnology in 1996, the late chemist and Nobel Laureate Richard Smalley described the importance of buckminsterfullerene’s natural structure to the still emergent field of nanotechnology, "Carbon has an incredible ability to spontaneously assemble to form these objects. That's what we really discovered. The more we think about that, and how neat these objects are, the more we are beginning to realize that we can find ways of tricking nature in to self assembling carbon into other fullerene-like shapes as well, and that these new materials may well have major practical as well as theoretical significance. "In fact, it emerges that buckyball was (and is) a sort of Rosetta Stone of what we now realize is an infinity of new structures made of carbon one way or another . . . And the deciphering of the C60 Rosetta Stone has led us to start dreaming of all sorts of new structures that truly are geodesic architecture on a nanometer scale, and to scheme about how to make them."
Smalley , Richard. "From Tubes to Ropes." Presentation to the American Institute of Chemical Engineers, S. Texas January 1996 http://cnst.rice.edu/aiche96.html
[xv] Fuller’s original oft-quoted comment in full is: "When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." This quote is attributed to numerous sources in Fuller’s live talks and transcripts.