The Study of Patterns is Profound
Published in Leonardo, Volume 40 Number 3, 2007, http://www.leonardo.info
Trudy Myrrh Reagan is an artist who founded YLEM: Artists Using Science and Technology in 1981. In 2004, she started an interest group within YLEM on the subject of patterns, both natural and mathematical. She paints under the name "Myrrh".
ABSTRACT
The author has studied natural patterns both by drawing them and finding analogs for them in crafts materials. Several media will be described: batik, shibori, wrinkled paper painting, paper marbling, constructing a moiré, and painting and engraving on Plexiglas. As well, she will discuss the generation of the patterns in nature, and how scientists’ understanding of them expanded during the period of her own explorations. She recommends this study for enhancing one’s connection to the natural world and the cosmos. The author also explains how she found patterns useful as metaphors for philosophic ideas.
The artist’s eye is captivated even from childhood by rainbow stripes on mud puddles or drifting smoke. The movement of smoke, for example, that mesmerized me when I was small came from my mother’s cigarets or embers in the campfire. Gazing at wisps of smoke is no trivial matter! Drifting up gracefully, smoke obeys laws of physics in a most visible way. It loses momentum and curls around in ever-changing patterns. Like smoke injected into wind tunnels for aeronautical research, it traces out air currents, in particular, the hot air rising from the cigaret. Cool air, which is denser, gently moves toward it to fill the partial vacuum it created. Where the smoke loses momentum, the warm and cool air circle around each other, hovering. The particles of smoke are supported by the invisible atmosphere, principally nitrogen and oxygen molecules. This is why I found an appealing logic in its apparent disorder.
For 45 years, I have explored comparable patterns in nature in my art. They turned out to be manifestations of profound truths, and a vehicle for expressing philosophic ideas as well.
My father, Philip B. King, illustrated his scientific papers and books with pen-and-ink drawings of crags and erosion patterns. He was well-known for his innovations in the means of visualizing sub-surface geology. I carry a memory of his office lined with colorful geologic maps. As well, I saw rainbow slide shows by his friend, projections of mineral thin sections under polarized light. Entering college in 1954 as a skilled representational artist, I was suddenly expected to generate abstract designs. I had scant facility for it. In my senior year, I discovered the just-published book, The New Landscape, by Gyorgy Kepes.[1] In it, things I had seen all my life were honored as worthy art subjects, as much as the nonrepresentational art they resembled.
Nathan Cabot Hale, an art educator, wrote about the dilemma of the representational artist in our time:
"The biggest challenge to the artist today is learning the abstract language of art. Long ago it was enough to copy the surface forms of nature, but now it is our task to get at the root of nature’s meanings. There is no other way to do this than to learn the kind of reasoning that enables us to look beneath the surface of things." [2]
Leonardo did this with his famous sketches of turbulent water. Beginning in the 1880s, Odion Redon and others were inspired by the “landscapes” of cells under the microscope. The surface of the paintings of the cathedral doors at Rouen by Monet, 1904, have a fractal quality, though this was not even a concept or a word before Benoit Mandelbrot began mathematical work on fractals in the 1960s.
Mandelbrot dubbed fractals “the mathematics of wiggles.”[3] They generated novel geometric designs, and when random numbers were added, computer artists found a tool to model nature. Peaks composed of random polygon shapes became “mountains.”[4] However, geologists noted the lack of erosion patterns, and “behaviors” had to be integrated into to fractal algorithms.
One reason I love drawing is that it is an inexpensive way to “own” what I admire. My basic esthetic started with graceful lines in contour maps and patterns, both regular and chaotic, in geologic maps. From science magazine photographs, I mined an amazing array of designs possessing unusual line qualities. Imagine my delight in 1974 to see the patterns I enjoyed analyzed in Peter Stevens’ book, Patterns in Nature.[5]
A 1970 experiment I did morphed one category of line quality into another around the circumference of the E Pluribus Unum skirt, from straight to contour-like to cellular and by stages back to straight lines.
Figure 1 E Pluribus Unum Skirt,
rayon skirt, 1969 (lost), recreated in 2005, 40 in. long
Several drawings I did in this period showed that patterns formed a family of motifs, ones that repeated at many scales. For instance, I did one of a gigantic leaf with "veins" of capillaries, street maps and so on:diverse, and yet so similar! Nature’s tendency toward conservation of energy generates similar forms, whether extremely large or extremely small.[6]
Beginning in 1973, I learned batik and adopted hexagonal patterns as a theme in order to work in modules to create large wall pieces. Hexagons, with their 120¾ angles, tile a plane. Hexagons in nature are plentiful. I found many examples in Ernst Haeckle’s Art Forms in Nature[7] and soon noticed them all around me. One morning I awoke on a camping trip and gazed into the branches of a Red Fir, which has perfect 30¾– 60¾ branching. The batik process added another natural-looking element. Batik is a process of drawing the design on thin fabric in wax, then dyeing it. The waxed areas resist the dye and remain white. Afterwards, the wax is removed from the cloth.
Figure 2 Red Fir, 12” X 15” detail of the batik, Animal, Vegetable, Mineral
During the dyeing process, the wax develops cracks, which the dye enters. This adds a pattern resembling a network of veins. In Red Fir, it resembles the needles of the conifer. This is part of a larger piece, Animal, Vegetable, Mineral.
Figure 3 Animal, Vegetable, Mineral, 1977, 18 batiks,each triangle 9 feet on a side.
As well, I learned the joy of pattern-generating processes of tie-dye. Folding and binding fabric in a systematic way prevents dye from entering the folds. The result always surprises. Complex results can occur from quite simple manipulations.
I like to draw. I was attracted to Japanese shibori (a tie dye variant), where one draws, say, a bamboo leaf, and stitches along the lines of the image with strong thread. It is gathered and secured, using the threads as drawstrings. The tightly-drawn folds are not very deep. Success demands the use of dyes like indigo that do not penetrate well, but remain on the surface of the bound-up cloth. Cutting the threads and ungathering the folds reveals the pattern. An exciting moment! The works had an appearance of not being handmade, but created by some natural process.
When the cloth is tightly drawn up, ruffles in the cloth surrounding the design prevent it from dyeing evenly, creating a halo effect. Kirlian photographs capture a halo effect of natural specimens by placing them on an electrically-charged photographic plate in a dark room. One of my favorite Kirlian photographs was of a large leaf photographed by this process. The Kirlian Effect was my interpretation of it in shibori.
Figure 4 The Kirlian Effect, 1979,
shibori on cotton/polyester: 14 X 27 in
Kirlian uses traditional shibori branching patterns writ large. I then demonstrated shibori could also be used for erosion patterns. My shibori technique demonstrated that the similarity between branching (a growth process) and erosion (a subtractive process) is pronounced, because both involve bifurcation. That is, at certain points in their development, the stem or the ridge becomes divided.
How does the “erosion” pattern develop when sewn? Sometimes stitching follows the lines of a drawing, but another method is stitching perpendicular to the lines. Horizontal rows of stitches create vertical wrinkles that become the design. By offsetting the stitches, branching patterns begin to emerge (mokume shibori, or “wood grain”). Sewing a spiral path in the cloth gathers the wrinkles into something that looked to me like ridges of a deeply-eroded volcano. This is clearly seen in Seismic Fuji, (detail).
Figure 5 Seismic Fuji, detail, 1982,
shibori on cotton: 30 X 30 X 5 in.
The shibori process proved too labor-intensive, but gave me a feeling for what wrinkles would naturally do. This intuition was utilized in my next series of landscapes that looked like satellite photos, dubbed my N.A.S.A. series (Not Actually Science Achievements). Combining what I knew about geology and shibori, I wrinkled thin vegetable paper into “mountainscape” reliefs. These were sprayed from several angles with different colors of spray paints. If water areas were called for, I protected the lowest areas of the relief with a resist of ordinary sand. Unlike the sewn shiboris, these were swiftly executed. The three-dimensionality and degree of detail seemed uncanny to viewers. I was able to mimic certain geologic formations. For instance, in Appalachian II one can see the typical pattern that sedimentary rocks make when uplifted by folding, then truncated by subsequent erosion.
An 1989 article about my work in Kagaku Asahi,[8] a Japanese popular science magazine, raised an interesting question: Could the wrinkling process really be an analog to Earth’s features, which are largely caused by erosion? My hunch is that the answer is yes, because the forces that crumple and lift up the earth’s crust create weaknesses in the strata, guiding erosion by water or ice. David Huffman, a mathematician formerly at U.C. Santa Cruz, did an analysis of crumpled brown paper bags, measuring the angles of folds where they joined together. He found several relations that always hold true. For instance, when many folds meet at a point, there is always an even number of them. In such a group he would number each angle and found that the sum of the degrees of the odd-numbered angles equals that of the even-numbered ones.[9] The crumpling processes must have similarities ro geologic forces uplifting mountains, or it would not be so very easy to create my illusions! Manipulating actual material, paper, easily yielded results more realistic than the first computer-generated fractal landscapes.
Figure 6 Appalachian II, 1987, crumpled paper painting: 14 X 40 in
Fluid dynamics is the name for a set of patterns that have fascinated me since childhood. I was taught in college design courses, “process makes pattern,” such as the deformed water-carved rocks at the base of a waterfall. I spent hours as a child looking at mud puddles rainbowed with oily films, and taking snapshots of rapids in mountain streams. In the 1970s, black and white graphics of fluid flows generated by computers began to appear. Of the three types of flow, laminar, oscillating and turbulent, I gravitated to the oscillating flow diagrams (like flow patterns often observed around bridge supports in a river). These had a natural gracefulness I admired.
This attracted me to paper marbling. In this craft, used in the end papers of fine old books, a substrate of water thickened with carrageenan supports droplets of paint. This substrate is unlike plain water: Diluted paint floats on it well. It is viscous, and supports a design long enough to be captured on paper. The paint, which has a surfactant added to make the paint spread, becomes a film only a few atoms thick. The surface tension, very strong around the edge of each droplet, is maintained even when the droplet is radically deformed. For this reason, neighboring colors do not mix, and complex stripes result when it is combed or blown on. (The result is not unlike computer diagrams of chaos functions).
I was most amazed to see how combing the surface of round droplets led to the mystifying patterns in traditional marbled paper. The same week as I took the marbling course, Douglas Hofstadter in the Scientific American discussed magic. He said essentially that we call it magic when we can sense an underlying pattern but we can't fathom exactly how it arises. Yes, marbling is magic![10]
I tried blowing on the suspended paint with a straw at a very low angle to achieve fluid flow patterns. I did not succeed in making oscillating patterns, but made many mushroom clouds!
Figure 7 Mushroom Cloud Head, 1992,
marbling on cloth, 8 X 10 in
After the paint floating in the pan is has been manipulated into a design, paper treated with an alum solution is lowered onto the surface. When the paper is picked up, most of the paint adheres to it, since it is chemically more attractive to the paint than is the liquid bath.
I used much of this beautiful paper to make paper polyhedra. I love crystals. When I was doing macramé in 1970, a large hanging composed entirely of knots in cotton cord, I saw how different knots aggregated into different shapes, for instance, a helix. I realized that it was similar to the way the various crystal shapes develop depending on their differently-shaped atoms.
I wish I had the skills to grow wonderful crystals. I have learned to mimic crystallization by hand by engraving a 2-D design of closely-spaced lines onto Plexiglas using a rotary power tool, lines which catch the light. I also apply paint to the Plexiglas and carve through it. Lit from behind, my works gleam like stained glass.
I gravitated to this new medium after dyes in my batiks and shiboris faded. Pigments are far more colorfast than dyes, and will stick to Plexiglas if it is sanded to give it “tooth” to hold the paint. The paint needs an additive. I use Golden liquid acrylic paints diluted with liquid acrylic medium and GAC 200, an additive for inflexible surfaces. I select pigments that are translucent. Opaque black gives me the drama of complete darkness, and the exposed plastic, created with lines engraved through the paint layer, gives me something brighter than white—a great dynamic range.
Figure 8 An Essential Mystery: Life Creates 1992, acrylic on Plexiglas, 45" diam. (and detail showing carving and paint application on Plexiglas)
“Biocrystals,” a 1977 article in the Scientific American [11] inspired my painting on Plexiglas, An Essential Mystery: Life Creates. “Biocrystals” discussed chemistry at the boundary of the inorganic and the organic, the mineral and the one-celled protist, the dead and the living. Very intriguing! For resource materials, I used D’Arcy Thompson’s drawing of a radiolarian (minus the projecting spines), and a micrograph from “Biocrystals” showing crystals building a single strut. As described earlier, I drew the “crystals,” line by line, with the power tool. It took weeks, but the thrilling effect lured me on. I learned too late that the crystal structure I had chosen to copy was not of the silicon dioxide (silicate) found in radiolaria, but calcium carbonate.
The painting raised questions in my mind, — soap bubbles' familiar shapes are guided by the mathematics of minimal surfaces. The overall lattice of the radiolarian's crystal structure bore a striking resemblance to bubbles. How could biological growth processes mimic them? A rather new conjecture is that the organism exudes bubbles to form the scaffold for crystal formation. Deciphering how is an active area of research in materials science, some of which has been done since the completion of this painting. Moreover, scientists show us that the crystals formed by microorganisms are very fine, forming rounded shapes, not the coarse, jagged struts shown in my painting. They have a strength unmatched by those that grow inorganically.[12] We humans are trying to emulate these structures to produce stronger industrial materials.
Jacob Bronowski observed,
“There are only certain kinds of symmetries which our space can support, not only in man-made patterns, but in the regularities which nature herself imposes on her fundamental, atomic structures." [13]
My next painting, An Essential Mystery: Number Governs Form, treats the idea that number relationships underlie all that we see. The notion that these are fundamental in nature goes back at least to Pythagoras. The discoveries made by the interplay of observation and mathematics continue to this day.
Computer scientist Rudy Rucker, reflecting upon the work of Stephen Wolfram, goes further:
“But in The Lifebox, the Seashell, and the Soul I’m arguing that we do best to think of computation itself as fundamental. Under this view, logic and mathematics are invented after the fact to explain the observed patterns of the world.[14]
The massively complex computations found in nature, constantly in motion, are deterministic yet revealing something no one would be able to guess. Rucker describes them as “never-repeating lace.”
In the late 1980s, I saw a demonstration of cellular automata by Rucker that elaborated on Mathematician John Horton Conway's Game of Life. Like the Game, it appeared to be a miniature universe generated on a computer grid that evolved as we watched. A few simple rules generated families of patterns, some of which had surprising behaviors. Rucker had enabled a high-resolution, multicolored grid that displayed pulsating patterns in real time, something unheard of on a personal computer of that era, by installing an additional computer chip of his own design. When the designs were symmetrical, I had the sensation of seeing a Persian rug being woven. When asymmetrical, they appeared like fast-growing lichens.
Figure 9 An Essential Mystery: Number Governs Form, acrylic on Plexiglas, 45" diam. (and detail showing carving and paint application on Plexiglas)
In An Essential Mystery: Number Governs Form, the circle represents a geode, with amethyst crystals pointing inward. At the center is a much-enlarged Scotch thistle blossom. With both subjects, I used actual specimens to guide my work. The thistle disintegrated into thistledown.
Amethyst is a type of quartz, whose crystal form is hexagonal in cross section, with six equal rectangular faces. In nature, of course, slight misalignments of atoms in the crystal lattice are commonplace. I took great pleasure in studying the actual specimens and trying to capture their regularity-with-variety. Engraving the lines with a power rotary tool enhanced the crystalline effect. As I mentioned, the plastic had been sanded to accept the paint, making it look frosty. When one looks head-on through a quartz crystal, it is quite transparent, but less so when looking at the oblique faces. My quartz-crystal illusion was greatly enhanced when I polished the plastic on certain areas to restore transparency.
Crystal forms are determined by the atomic properties of their elements. Crystals grow not only in ways determined by the shape of their atoms, like Lego toy pieces, but are constrained by the binding forces between atoms. Some elements like carbon take different forms, diamond and graphite, depending on the binding forces.
When it comes to living things, relatively recent investigations in the physical sciences show that pattern-generating processes, ones that develop through time in an excitable medium, may play a role in generating som biological patterns. Belousov’s 1951 experiments with oscillating chemical reactions, ones that spontaneously produced spirals and target shapes in motion (the BZ reaction), touched off a new line of inquiry, as did a 1952 paper by Alan Turing on morphogenesis that suggested hypothetical chemical reactions giving rise to patterns of stripes and spots. In biology, these relate to “a fibrillating heart’s pulses, the stripes on leopards, and more...all bear a resemblance to oscillating chemical reactions as well as other patterns called Turing patterns that develop through time.”[15] The oscillations are kept going by an interplay between chemicals that activate or inhibit.
The new light this throws on evolutionary theory is worth exploring in some detail. When we say an “eye” pattern forms on a moth’s wing because it serves to scare away predators, what kind of explanation is that? How could variations of shuffled genes through dim recesses of time have turned up that particular card? But experiments with oscillating reactions produce similar patterns! DNA code for a simple pattern-generating recipe using the very nature of the materials at hand is straightforward. Natural selection then favors the particular “eye” configuration. There is no “DNA map of a scary eye” as such. Philip Ball, in The Self-Made Tapestry, elaborates on recent, sometimes controversial, research:
" [T]hese processes suggest that there are certain ‘fundamental’ structures of organisms that are not at all determined by the arbitrary experimentation and weeding out that evolution is thought to involve. Instead, these structures have an inevitability about them, being driven by the basic physics and chemistry of growth. If life were started from scratch a thousand times over, it would every time alight on these fundamental structures eventually. Within the parlance of modern physics, they are attractors—stable forms or patterns to which a system is drawn regardless of where it starts from....[I]f the protagonists of this concept turn out to be validated, that would not by any means bring Darwin tumbling from his pedestal. There is absolutely no question that natural selection operates in the real world and that it has produced the tremendous variety of organisms with which we share the planet....No one argues, meanwhile, that nature’s palette is not constrained by the rules of physics and chemistry. If the formation of patterns by symmetry-breaking proves to pose limitations of evolutionary choices, that will add just one more nuance to Darwin’s towering achievement."[16]
I painted Number Governs Form in 1993. I would later learn of Brian Goodwin’s work (How the Leopard Changed its Spots: The Evolution of Complexity, 1994) and Philip Ball’s book, 1999. However, it was already well known that the thistle is the product of a recursive growth process that produces thorns on the flower at regular intervals that fit the Fibonacci series of numbers. These numbers are also found in measurements relating to the “Golden Rectangle” of geometry.
As well, the Fibonacci numerical patterns observed in a sunflower spiral and elsewhere are associated with a “Golden Angle” of 137.5¾, the angle at which a succession of sunflower seeds form or leaves are offset in a spiral along a growing stem.[17] The sunflower spiral can be replicated by an experiment with magnetic droplets that repel each other.
For a plant to place the next seed or leaf, it need not “know” the correct angle: The dynamics of growth cause it to keep “discovering” it.[18] Experiments are being done with plant hormones to explain the stop-and-start process of stem development that gives rise to these relationships. The hormones inhibit leaf development as the stem grows. When the stem is longer, hormone concentration diminishes, and leaf development resumes. Now, to discover the peculiar mathematical relationships that develop!
The Fibonacci spiral is a rich template for generating designs. Recently, I used several permutations of it to convey an idea that has held me in its grip since 1975, that “all knowledge is one.” In 1990, E.O. Wilson wrote Consilience, which shows the connections he sees between all branches of knowledge.[19] Therefore, I have named my fourth and most recent version of this piece, completed in 2004, A Vast Consilience
Figure 10 A Vast Consilience, 2004, acrylic on Plexiglas, engraved lines: 48 in. square
The areas of knowledge that I selected to present were:
• the astronomer (who probes the farthest reaches of the universe)
• the biologist (and others who explore the extremely small)
• the artist or composer, who delights in new pattern configurations
• and the mystic blessed with a emotional appreciation of the Whole.
The center of Vast Consilience shows an eclipse, because the whole pattern is unknowable—not by us, not by the culture as a whole, nor by people in a future epoch. It is interesting that both the Greek words logos and cosmos have the idea of an underlying order of the universe imbedded in them.
Of course, how we explain what little we can observe is always under question. Geologists, who can’t “repeat the experiment,” operate with multiple working hypotheses, hoping that evidence will surface that eventually proves one theory more nearly correct than another. Worse, physicists are confronted with the Uncertainty Principle: Measuring light as a wave is incompatible with measuring it as a particle, yet it appears to be both.
Since I think in metaphors, I felt light was a good analogy for The Divine. It manifests itself so differently to people of various temperaments and through the lens of different cultures. Like light, it is more than any of its descriptions, indescribable.
In Divinity, I used a moiré pattern to represent the indescribable Divine. Making the moiré was simple, but tedious; the result magical: When two grid-like patterns are superimposed, a third pattern results. The effect is like the crests and troughs, cancellation and reinforcement patterns, of two sets of waves crossing each other.
For this project, I chose to create a cruciform pattern by superimposing two designs, spots and radiating stripes. I applied the two patterns to two layers of plastic by adhering plastic film to mask portions of the design and spray painting it. I hung the painted layers independently, one in front of the other. One could move them, making the moiré pattern shift dynamically.
Figure 11 Divinity, 1980, acrylic on Plexiglas, flanked by batik panels: 30 X 90 in. triptych.
To the left I created a strong but static cruciform pattern based on the moiré, one that I named "God" and executed it in batik. To the right was a dynamic but less distinct interpretation, “Tao.” I did this piece in 1980, after misreading a sentence. It said, “The Tao is a web.” I read, “The Tao is a verb.” Suddenly, the Judeo-Christian-Muslim God, and even the Greek “cosmos” (the underlying order, as I had understood it) seemed static. The center moiré I think of as “Neither/Both/More.” In this polarized world of loudly competing religious doctrines, Divinity is a plea for tolerance.
Another person studying patterns in nature would pursue different paths, but any inspire a deep connection to the universe larger than ourselves. In the 1970s, for example, Peter S. Stevens and the Philomorph study group at Harvard listed various shapes like spirals and showed why they tend to recur at different scales. They found the wonder of a few pattern types manifesting themselves throughout the universe, in galaxies down to the smallest crystal structures.[20] Even some viruses are in the shape of Platonic solids!
George Johnson writes in Fire in the Mind,
"Imagine that instead of 92 elements, a continuum. In place of neatly arranged cells, Mendeleev’s table would become a continuous band. But if the energy levels of electrons are not continuous [quantum theory], if only certain values are allowed, we can explain why the same forms keep turning up in nature."[21]
In a recent work, Catastrophe!, I murdered a previous painting on Plexiglas by stretching a wrinkled cotton sheet over it and pouring rubbing alcohol (an acrylic paint solvent) on it. I pulled the cloth up, and found a portrait of an explosion that took relatively little effort to revise. Post-Katrina, this is how I expressed the “perfect storm.” The prototypical avalanche triggered by just one additional grain of sand, so familiar in chaos theory, is discussed in Ubiquity: Why Catastrophes Happen.[21] Its author, Mark Buchanan, shows that phenomena like earthquakes happen continually. On a Bell curve, both the miniscule and mammoth ones rest at the extreme edges, that is, are very rare. Moderate ones are common. Which kind will be triggered by the “last grain of sand”? It’s utterly unpredictable, he declares. Some argue that such unpredictability is not only more artistically intriguing, but closer to the true nature of reality. I would argue that even different catastrophes have their pattern “signatures,” just as one can distinguish “modern jazz” or “kletzmer” from surprising combinations of improvised notes.
Figure 12 Catastrophe!, 2006, acrylic on Plexiglas, engraved lines: 45 in. circle
It is fashionable to denigrate “scientific truth,” because so much remains unknowable. The “unknowable” is the subject of my work, Divinity. However, I believe that physical and mathematical rules for pattern generation have great explanatory power, taking us beyond, for instance, the “random process of evolution” to explain the beauties we observe. We should not abandon our search for truth and universality!
My curious eye has attracted me to certain patterns in nature; learning about them plunged me ever deeper into fundamental questions about how they arise, and which have to do with the very fabric of space and time themselves. The study of patterns in nature is profound.
References and Notes
1. Gyorgy Kepes, The New Landscape in Art and Science (Chicago, Paul Thobald & Co. 1956)
2. Nathan Cabot Hale, Abstraction in Art and Nature (New York: Watson Guptil, 1980), p.13.
3. Benoit Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, San Francisco, 1983 ed.) p. 5.
4. For example, Loren Carpenter, Vol Libre, a 2-minute computer animation classic (1980).
5. Peter Stevens, Patterns in Nature (Boston: Atlantic Monthly Press Book, Little, Brown, 1974).
6. Ray Pestrong, “Nature’s Angle,” Pacific Discovery, Vol. 44 #2, p.14.
7. Ernst Haeckel, Art Forms in Nature, (NY, Dover __, orig. pub. 1913),
8. Itsuo Sakane, “Wrinkled Mountain Ranges,” Kagaku Asahi, Vol. 49 #1, p. 70. (2/1/89)
9. David Huffman, lecture at Stanford math department, 1982 or 3.
10. Douglas Hofstadter, "The Music of Frédérick Chopin: Starling Aural Patterns that also Startle the Eye,” title of "Metamagical Themas" column in Scientific American, Apr. 1982, p. 16: "...[P]henomena perceived to be magical are always the outcome of complex patterns of nonmagical activities taking place at a level below perception. In other words, the magic behind the magic is pattern."
11. Shinya Inoue and Kayo Okazaki, “Biocrystals,” Scientific American, April 1977, pp. 82-4.
12. Philip Ball, The Self-Made Tapestry: Pattern Formation in Nature (Oxford: Oxford University Press, 1999), p. 42-3
13. J. Bronowski, The Ascent of Man, (Boston: Little Brown & Co. 1973) p. 174
14. Rudy Rucker, The Lifebox, the Seashell, and the Soul (New York: Thunder’s Mouth Press, 2005) p. 10.
15. Tapestry, p. 69
16. Tapestry, pp.102-4.
17. Patterns in Nature, pp. 159-66
18. Tapestry, pp. 104-9.
19. E.O. Wilson, Consilience (NY: Knopf 1998)
20. Patterns in Nature, pp. 3-4.
21. George Johnson, Fire in the Mind: Science, Faith and the Search for Order (New York: Vintage Books, 1995), p. 43.
22. Mark Buchanan, Ubiquity: Why Catastrophes Happen (NY: Three Rivers Press 2000) p. 44-7