Most people who keep up with advances in science have probably heard of the term self-similarity because it is a key ingredient of fractals — a topic often in scientific news reports during the past two decades. But if you asked for a definition of self-similarity, or some good examples, many of those same people would find it difficult to provide them. This is unfortunate because self-similarity appears to be one of nature’s most favored design principles. Given the ubiquity of this phenomenon in each major realm of nature, it is perhaps excusable to anthropomorphize a bit and say that nature truly adores self-similarity. In fact, were it not for this elegant design strategy, we would be dire straits for at least five reasons. Our bodies would perform poorly and decay due to inadequate circulation if self-similarity did not exist. If our brains and nervous systems could not benefit from self-similar networking, our IQs would be roughly on a par with fence posts. Without the self-similar villi and microvilli in our intestines, our ability to digest food would be compromised. Moreover, there would be little food to eat in the first place because, in the absence of self-similarity, the Earth would be virtually devoid of vegetation, which directly or indirectly provides most of our food sources. Lastly, we would be unable to breathe without the critical self-similar architecture inside our lungs.
Self-similarity is so widespread and important in nature, the arts and society that one can no longer be considered well-informed without at least a conceptual understanding of this remarkable phenomenon. Luckily, the main ideas of self-similarity are not difficult to master and good examples of self-similarity are exceedingly easy to find. When you are done with this article, you will understand the basics of self-similar phenomena and be able to identify them virtually everywhere you look. In preparation we need to spend just a few minutes laying the foundation for that knowledge.
A good place to begin our exploration is with three archetypal examples of self-similarity. Firstly, think of a lone leafless tree standing dark against a gray winter sky. It has branching patterns that are quite familiar to us. There is a single trunk that rises to a region where major branches split off. If we follow any one of these major branches, we see that they also split into smaller branches, which split into even smaller branches, and so on. A “unit pattern” (in this case one length splitting into two or more thinner branches) is repeated on ever-smaller size scales until we reach the treetop. With this picture in mind, we can state a simple generic definition: the fundamental principle of a self-similar structure is the repetition of a unit pattern on different size scales. Note that the delicate veins of leaves also have self-similar branching patterns, as do virtually all root systems. A tree is thus the epitome of self-similarity.
Our second archetype is an intriguing self-similar pattern that was published in 1907 by the natural philosopher E. E. Fournier d’Albe. Think of a pair of dice and focus on a side representing the number 5. The unit pattern involves five dots, four of which are located at the corners of a square and the fifth is located at the center. For simplicity we will call this unit pattern a quintet. Now imagine that each dot is really a miniature quintet, and further, that each dot of the miniature quintet is a microscopic quintet. This constitutes three scales of the self-similar structure, but Fournier d’Albe went much further and imagined the pattern repeating without limits. In this case, the structure we started with would be just one of five quintets of a larger scale “whole”, which would be one of five quintets of an even larger scale quintet, and so on forever. Likewise the quintets-within-quintets hierarchy extends endlessly to ever-smaller scales. If this is hard to visualize, try drawing three scales of this amazing design.
The two examples given above both involve spatial structures, but self-similarity can also occur in temporal processes. For example, imagine the opening theme of Beethoven’s 5th symphony (da da da, DAA) being created on a synthesizer. One could replace the “das” of the first three notes with brief versions of the whole 4-note pattern. One could arrange it so that these compressed 4-note patterns happened fast enough that they could be mistaken for single notes. The individual notes of the compressed 4-note patterns could, in turn, be composed of even faster (instead of smaller) 4-note patterns, and so on. We end up with a sequence of sounds that is self-similar in time, and this could be revealed by playing the sequence at different speeds.
With just a couple more concepts to aid us, we will be able to begin exploring nature’s treasure trove of self-similarity. The curious term self-similarity, which might be improved upon one day (scale-similarity?), was coined in the 1920s by the physicist Lewis Fry Richardson who specialized in fluid turbulence, wherein big eddies have smaller eddies and so on. One way of distinguishing among various types of self-similarity is whether they are based on exact, approximate or merely statistical copies of unit patterns. In Fournier d’Albe’s unbounded quintet creation, for example, exact copies of the unit pattern are repeated on each scale. It is somewhat boggling to realize that, aside from size, all quintets in this design are identical, even in the number of their internal scales, no matter where they occur in the hierarchy of scales. This counter-intuitive property is made possible by the fact that the number of scales in the design is infinite. Tree morphologies, on the other hand, have similar branching patterns on different scales, not identical copies. Other natural examples, such as coastlines, have only vaguely similar patterns (e.g., bays and promontories) on many scales, and this is often referred to as statistical self-similarity.
Another distinction among self-similar phenomena is whether they involve discrete, multiple or continuous scaling. Fournier d’Albe’s original quintet structure was rigidly discrete: quintets of neighboring scales always differed in size by a scale factor of 7. In the case of tree branching, on the other hand, there is a multiple, and more random, distribution of unit pattern sizes. There are also cases such as fluid turbulence in boiling water where self-similar motions occur on virtually all size scales, i.e., with a nearly continuous distribution of scale factors,
A further way in which self-similar designs can be distinguished is whether or not there are cutoffs to the hierarchical structure. Fournier d’Albe’s fully infinite design goes on forever; there are no largest or smallest quintets. Tree branching, on the other hand, has upper and lower cutoff limits, so only a finite number of size scales are involved. Lastly we can classify most self-similar phenomena into a few broad categories: branching (e.g., trees and arteries), surfaces (e.g., coastlines and geographic topography), clustering/nesting (e.g., the quintet design and Russian dolls) and temporal (e.g., music and stock market fluctuations).
Perhaps it is also appropriate here to mention a subtle distinction between fractals and self-similarity. A fractal object has structure on so many different size scales that the dimensionality of the object is greater than the Euclidean dimension of its approximate geometric shape. For example, the approximate Euclidean dimensionality for the Earth’s surface is 2, but its fractal dimension is between 2 and 3. On the other hand, an “object” could be called self-similar even with as few as two scales of similar structure. A Russian doll, which is a nested set of dolls within dolls, has virtually exact self-similarity, but is not formally a fractal. Fractals almost always involve self-similarity, but self-similar structures need not be fractal, in the full mathematical sense. Given the ubiquity of fractals in nature, and the many additional instances of non-fractal self-similarity, one has a hard time identifying anything in nature that is devoid of these fundamental phenomena.
Now that we have become familiar with the basic ideas of self-similarity, we are ready to take a look at some actual examples. Let’s say that we are sitting at an outdoor café, sipping double lattes, and amusing ourselves by trying to identify self-similar phenomena in nature. We could hardly do better than to start with our own bodies. As mentioned in the introduction, our circulatory systems are based on a self-similar design. It is crucial that oxygen-carrying blood reaches all parts, and all relevant scales, of our bodies. The branching self-similarity of our circulatory systems, from major arteries to tiny capillaries, accomplishes this critical task in the most efficient way. Like most examples of branching phenomena in nature, the design of the circulatory system involves approximate self-similarity and multiple scaling.
Feeling a bit self-referential, we might then consider the self-similar architecture of our brains and nervous systems. In this case it is electrochemical signals that must be transported throughout our bodies. This occurs via a self-similar branching system of neurons, in good analogy to the case of the circulatory system. Moreover, there is a remarkable degree of nested self-similar networking present in the brain and nervous system. Individual neurons are linked into small clusters, which in turn are components of larger clusters of clusters, and so on. Also, the release of neurotransmitters by individual neurons occurs with similar fluctuations on different time scales, thereby manifesting temporal self-similarity.
If the strong coffee makes our stomachs “growl,” then we might think of an example of self-similarity that occurs in the intestines. To increase the efficiency of nutrient absorption into the blood stream there are small finger-like structures (villi) that line the inside of the intestines. These fingers have even smaller fingers on their surfaces, and by now it should not surprise anyone that there are microscopic fingers on the fingers of the fingers. In our lungs, where the goal is to maximize the transfer of oxygen into the blood stream, it is advantageous to have a very large surface area over which this can take place. This is accomplished by a remarkable self-similar hierarchy of tubular membranes and air passages involving approximately 15 levels of self-similar branching between the largest bronchial tubes and the smallest scale alveoli.
Our cellular DNA, which acts as a blueprint for building an organism and as a control center for biological processes, contains examples of self-similarity in the frequencies of base pair occurrences and in long-range correlations. The outer membranes of our cells have sodium, potassium and calcium channels whose open/shut patterns reveal a temporal self-similarity. Viruses that sometimes afflict us, and the immune system’s macrophages that fight back often have fractal surfaces that are statistically self-similar. We might go on with further examples of self-similarity within our bodies, but perhaps we would like to expand our horizons a bit and explore the outer world for new examples of this remarkable design strategy. Let’s say we decide to use our mind’s eye to view nature from the local stellar neighborhood to the largest observable scales of the universe.
Galaxies, the vast spiral and elliptical systems measuring on the order of 1,000,000,000,000,000,000 miles in diameter, are distributed in statistically self-similar patterns. Individual galaxies are usually found in small clusters, which are the components of larger clusters of clusters, which are nested into “superclusters” of clusters of clusters. On a much smaller scale, the stars within each galaxy are distributed in an analogous self-similar pattern: individual stars, small clusters of stars, clusters of clusters, etc. Likewise the countless numbers of atomic ions that make up stars and their immediate environs are in high energy plasma states with rapidly shifting self-similar clustering. At interstellar scales the Hubble Space Telescope has revealed beautiful, and somewhat eerie, interstellar gas/dust clouds with nested self-similar “pillars” and clumps occurring over six orders of magnitude in scale. The Eagle Nebula is an awe-inspiring example of vast colorful pillar-shaped clouds set against the blackness of space. The largest pillars have small pillars sprouting from their sides, and the small pillars have even smaller pillars studding their surfaces. These last four examples are members of the nesting category, with primarily statistical and nearly continuous self-similarity. The solar wind, consisting mostly protons flowing away from the Sun along its magnetic field lines, displays comparable fluctuations on different time scales, i.e., temporal self-similarity. Of further interest is the fact that the distributions of galaxies, of stars within galaxies, and of interstellar atomic ions often have comparable filamentous or “honey-comb” patterns: relatively high density linear filaments separated by low density regions.
Examples with higher degrees of similarity can also be found in space, such as the analogy between planet/moon systems and the much larger Sun/planet system. We might also consider the strong similarities between galactic scale quasars and stellar scale microquasars, or the fact that neutron stars are in many ways like giant atomic nuclei. But perhaps we are ready to come back from the far reaches of the universe and look for further examples of self-similarity in the middle distance.
So we lean back in our chairs and gaze up at the Earth’s atmosphere, which provides another rich source of examples in our hunt for self-similarities. Clouds, especially cirrus (wispy) and cumulus (puffy) clouds, are prime examples of nested self-similarity wherein the biggest wisps or puffs are composed of smaller scale wisps or puffs, and so on. The sky also abounds with the turbulent motions of different air masses. Not surprisingly, since a turbulence expert coined the term self-similarity, turbulent motion provides archetypal examples. Richardson even wrote a nice piece of doggerel poetry (a take-off on Jonathon Swift’s rhyme about self-similarity in biology: “Big fleas have little fleas…”) to convey his ideas.
“Big whorls have little whorls,
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity”
On smaller scales within the atmosphere, snowflakes often display self-similar branching patterns or hexagonal crystals within crystals, and aggregating dust particles have growth patterns that are statistically self-similar. On very small scales, there is chaotic Brownian motion wherein big molecules and microscopic particles are buffeted around by the smallest and fastest moving air molecules in erratic zigzag patterns that are self-similar on micrometer to nanometer scales.
Lowering our gaze, we might look out on an ocean whose surface usually has a self-similar hierarchy of waves upon waves upon waves, with heights ranging from meters to millimeters. Bodies of water also have an abundance of self-similar turbulence: whorls within whorls within whorls, as in the case of atmospheric turbulence. The mixing of fluid masses with differing densities, temperatures, or chemical contents often occurs in a self-similar hierarchy of interpenetrating “fingers”. The tributary and drainage systems of rivers usually exhibit branching self-similarity, and as mentioned before, coastlines are a classic example of statistical self-similarity.
If we see a distant mountain range, we could count its profile as yet another example of statistical self-similarity because of the hierarchy of peak/valley morphologies on size scales ranging from kilometers to centimeters. In fact, nearly all of the surfaces we encounter have self-similar structure over at least some range of size scales. Consider the vast craters-within-craters surface of the Moon displaying self-similar divots that range in diameter from kilometers to millimeters, or the miniature analogue of the pocked bricks under our table. Erosion processes in desert regions yield a wealth of self-similar surfaces, and areas of snow and ice, from the vast continent of Antarctica to the piles of snow next to our driveways, have self-similar topographies. Well chosen aerial or ground photographs of Antarctica with different scales between hundreds of kilometers and hundreds of centimeters can be surprisingly difficult to order in terms of size if reference objects are not included.
Having noted several examples of statistical self-similarity in the inorganic world, we now decide to return to living organisms wherein the degree of self-similarity is sometimes high, if not quite exact. The plant kingdom provides one of the richest sources of such examples. Ferns and cedar boughs have pleasing multi-leveled designs that are strongly self-similar. The overall shape of the fronds or boughs is roughly that of a broad “sword” with sharp tip and a stem running along its midline. But if we look more closely at the overall shape, we see that it is actually composed of many small swords oriented at right angles to the main stem. These second-level swords, in turn, are divided into third-level swords lying roughly perpendicular to the second-level stems. The Western Red Cedar tree has five scales of self-similar sword-shapes, if one includes the overall triangular shape of the tree itself. When the three-scaled self-similarity of ferns was first pointed out to me, I realized that I had been looking at ferns for decades without really seeing and appreciating their true design.
The Dill plant affords a pleasing example of self-similarity with two scales of structure. A main stem rises to a node from which many secondary stems radiate. At the end of each second level stem there is a miniature copy with radiating third level stems capped by flowers. Many types of cactus plants, such as the Saguaro, have growth patterns that produce a hierarchy of self-similar lobes. Another impressive example of self-similarity is found in Broccoli Coral. Amazingly thick trunks repeatedly subdivide into ever-smaller branches in an impressive hierarchy that reminds one of the bronchial tubes of the lungs.
Having had a fairly broad sampling of self-similarity in various realms of nature, we now decide to try something new by seeking out more abstract examples of this design in social systems, mathematics and the arts. If one thinks about how our governments, judicial systems, law enforcement agencies or various economic systems work, one sees that they have fairly discrete hierarchical arrangements with federal, state and local levels often being the major scales. Basically a similar type of activity is occurring at the different scales and so we find a familiar design in a new context. The main principle of self-similarity — “same” thing on different scales — can be found in countless examples from the social sectors. The Internet has grown according to the laws of self-similarity and its complex multi-scaled networking is reminiscent of the self-similar networking in the brain. Likewise the growth patterns and interconnectedness of cities exhibit similar phenomena on different scales, the hallmark of self-similarity.
In the world of art, self-similarity is a common theme. It is found in the floor and wall art of medieval churches and mosques, in the drip paintings of Jackson Pollack, in African art and sculpture, in the drawings of M.C. Escher, in the tradition of Russian dolls, and in a host of other artistic forms. Self-similarity is a standard motif in music, most notably in the works of J.S. Bach. His “well-tempered” tuning, which facilitates the playing of an instrument in many keys, is grounded in temporal self-similarity. In mathematics, where recursive operations are common, self-similarity pops up everywhere from proofs of the Pythagorean theorem to logarithmic spirals. By far the most impressive example is the incomparable Mandelbrot set, with its infinite hierarchy of M-sets within M-sets within M-sets. Interestingly, in Albert Einstein’s last scientific paper, written for a ‘50th anniversary of relativity’ conference in Italy, he noted that the equations of general relativity had an intrinsic self-similarity to them. He struggled to understand the strange implications of this finding, but his time on Earth ran out before he reached an answer. Perhaps he had once again foreseen the right path toward a new understanding of nature.
Well, we could go on with further examples of self-similarity such as lightning bolts, designs on shells, aggregation of bacteria or metal ions, surfaces of cancer cells, crystallization patterns in agate, scores of scaling laws in biology, quantum particle paths, gamma-ray burst fluctuations, species distributions or abundances, drop formation, renormalization in quantum electrodynamics, and so on. But alas, our lattes are drained and our brains are overflowing with nearly 80 examples of self-similarity, so we decide it’s time to adjourn this session. No doubt we will have many opportunities in the future to add to our collection because, beyond any reasonable doubt, nature adores self-similarity.
Robert L. Oldershaw
Amherst, MA 01002
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Barnsley, M., Fractals Everywhere, Academic Press, Boston, 1988.
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