Fractals and Aesthetics: Pattern and Self-Reference

Benoit Baald

Perhaps it's my imagination, but it seems to me that there was a time when philosophers were more concerned with answers than with questions. Now, this may not have been true of Socrates, but definitely of Plato; the "Republic" reads like a "Shell Answer Book" on how to run a state. I believe this is also true of the early scientist-philosophers such as Aristotle, Copernicus, Descartes, Newton; certainly, science is the domain of "answers". What is more important, I believe, is the sense of inclusion exhibited by these early thinkers--the lack of boundaries between science and philosophy before the industrial age.

Even though vestiges of philosophy's alliance with science remain, in the form of schematizations and mechanisms-e.g. Locke's theory of sensation, Sartre's of conscioussness, Foucault's of power structures-philosophy has been moving away from the bent that gives a high priority to the answering of questions. Indeed, the art of asking questions-especially the "right ones"--has gained priority, and the domain of questions asked has been defined by specific agendas and biases, moving from empiricism, positivism, and objectivism --somewhat inclusive focuses-to existentialism, deconstructionism, subjectivism--which are more specialized and exclusionary, amongst themselves as well as non-philosophical disciplines. Perhaps as a result of accumulating vast quantities of knowledge (or more precisely, mental output), focuses or interests have turned into disciplines, then regimes; boundaries have been established--the positivists' bias prevents their effective communication with the phenomenologists, and vice-versa.

The effect of this trend towards exclusionism seems akin to each regime's view of a television screen being limited to a single pixel (dot of resolution), and their thinking that they can figure out the whole picture. The important part missing from such perspectives is the individual perspective's relation to the "whole", where whole can be taken to mean the set of all such limited perspectives, especially, I would believe, in relation to human experience (rather than a tv image). And, it seems that the whole of human experience should be greater than the sum of its perspectives, especially since such specialized perspectives are usually over-abstracted, overcalculated diatribes of arbitrary significance. Here, arbitrary refers to how picking one point of an image to focus on in order to grasp the whole image is arbitrary. In terms of human experience, the relations between perspectives is of utmost importance--knowing each dot on the screen intimately gains little if one can't put them together to form the image. Thus, the whole is not just the sum of its parts, but the sum of its parts plus the relational framework which integrates them.

A closer examination of the notion of "relational framework" might prove insightful. Continuing with the tv-image analogy, if one knows each pixel individually, one hasn't knowledge of the image without their relationships. Yet if one knows the relational framework which entails the individual pixels, one knows the image. I believe this is true regardless of whether one knows the individual pixels or not; the requirements of the pixels are implied by their relational framework--"a black pixel at such and such position is to the left of a pink pixel which is right above a blue one, etc"; the image is formed without explicit knowledge of each element.

Of course, the above analogy is very simplistic. In practice (or "real life"), it is dificult to adhere to a single, narrow perspective without being a solopsist or a nihilist or other such extremist. Boundaries between perspectives and disciplines are crossed, and bridged, frequently. But of great importance is whether this happens intentionally or not, i.e. whether thinkers are moving toward an "inclusive" methodology which would aim at exploring the relations between those perspectives and disciplines from which adherents tend to exclude others. For example, instead of an existential arguing with a positivist about the nature of reality, with each participant vehemently defending and propagating his position, as happens all to often (though not necessarilly with positivists and existentialists) and illustrates what I mean by "exclusion", the participants could--acknowledging that they just might have differences in perspective--try to look for relations between their philosophies, as well as other philosophies. This would not necessarilly preclude their focus on that perspective formerly so guarded, but would include it, among others, in a view of relations that better acknowledged the diversity of human experience.

Now, the philosophic methodology which I am proposing may be regarded as just another of those perspectives that I am arguing against; I don't pretend to have a definitive solution to the problems which plague philosophers. What I wish to explore, specifically, are the relations between a mathematical model known as a fractal, with certain aspects of aesthetics.

(See Appendix for a deeper look at fractal theory)

There are two senses in which discussing aesthetics and fractals side by side relates to the above inclusive methodology. First, it is a "bridge" between the perspectives of science and philosophy, which are disparate if one considers their tendencies toward "objectivism" and "subjectivism", respectively. Fractals are a continuation of the scientist's obsession with measurement and other evidence which are used to support (or discover) theories about "reality" which aspire to objective validity. The philosophic areas I'll examine, however, concern aesthetics--appreciation, interpretation, taste, the nature of consciousness--which, contemporarily, anyway, tend to be of more interest to, or are written about more frequently by, thinkers with a subjectivist bent.

The other way in which the discussion of fractals is relevant concerns the nature of the fractal model itself; because fractals carry implications regarding boundaries and "metapatterning" (which will be explained), the fractal in many ways describes the relational framework and nature of boundaries of the inclusive image.

Fractals represent a strange marriage between simplicity and complexity. Although they are the product of very complex mathematics and physics-the field of non-linear dynamics-some of their aspects or functions are simple. They can represent infinite complexity using simple operations. (In a way, they are similar to a virus. What a virus does is very basic--changing a genetic code within a cell to produce likenesses of itself--yet the implications of what that can result in, and the biochemical evolution which enables the process, are nearly unfathomable.) Further, the implications that fractals entail are not only derived from the mathematics behind them, but arise from visual considerations as well; fractals are graphic representations of mathematical processes.

Imagining an image which contains many fractal characteristics should prove easy. Consider a black, uniform, zig-zag line on a white background. Now, picture zooming in on one apparently straight segment of that zig-zag, and as it comes into focus, discovering that it itself is a smaller zig-zag line. Again, zoom in on an apparently straight segment of this, the second (in scale) zig-zag (which is nothing more than how an apparently straight segment of the original zigzag appears close up), and lo and behold, it to is a zig-zag line, and not straight (as it appeared before its scale was changed). Repeat ad infinitum, and you should have an idea of a relatively boring fractal image. It demonstrates the important concepts of self-similarity, metapatterning, and ambiguous boundaries; a (simplified) explanation of the mathematics will help to illuminate these.

In classical mathematics, functions are often graphed on the Cartesian plane, which shows the function in two dimensions, the horizontal and the vertical (quantified by the x and y axes). For example, an increasing population growth over time would be graphically represented by a diagonal line; as time progresses (moves to the right on the xaxis), the population level is marked according to the y-axis directly over the place on the x-axis which represents the appropriate instant in time. A fractal can also be thought of as a similar graph, but with a collection of points instead of a line.

A fractal starts with a difference equation--a mathematical machine with an input and an output. A starting set of numbers is put into the machine which operates on it, returning a different set. This new number is fed back into the machine, and the process repeats until some thing happens, namely until the output stabilizes i.e. the successive outputs quit fluctuating, and converge on a single value. If this fails to happen within a specified number of repititions, the output is said to bifurcate--to keep fluctuating chaotically between extreme values. So, to creae a fractal, one repeats this process for every point on the plane, using the coordinates of the point for the initial set to be put in the difference equation. If a point stabilizes, it gets colored white, otherwise it gets colored black. What emerges is an image that represents the boundary between chaos and stability, and that representation is startlingly intricate, with very interesting properties.

If, for instance, one saw in the fractal a smooth curve where white meets black, and enlarged a single point of that boundary (expecting to see a large region of white next to a large region of black), the new image would look very similar in many ways to the original image-similar patterns, shapes and clusters of points. Further enlargements would yield similar results: The fractal is self similar; its patterns remain similar regardless of thescale at which they are viewed. This concept is closely tied to "metapatterning" which refers to the way the large patterns are composed of smaller patterns, which are composed of smaller patterns, etc.

Another property of fractals that is tied to the above phenomena is the infinite complexity of fractal boundaries. The zig-zag example illustrates this clearly: If the original z-z was terminated at the ends, and one measured it, considering the segments to be straight (as they appear without enlargement), a finite measurement would result; if, however, one set out to measure the length of the outline of the zz, taking into consideration the self-similar detailing resulting from enlargement, the line would be infinitely long--every time an apparently straight, measurable segment was enlarged, it would be found to contain another z-z, which would of course be longer than the pre-enlarged, apparently straight segment. As this continues infinitely, so too does the length of the line.

The most amazing implication that an infinite fractal boundary entails is that its dimension is fractional--between one and two. Consider a sheet of paper, one half white, the other black, the boundary between them a straight line. That line which separates the regions is onedimensional; it has length, but no width, and thus no area. If the boundary were two dimensional, it would either be a third region between the black and white (unallowable ina dualistic system), or it would incorporate bits of each region, thus not bounding them from each other. The fractal, however, squeezes an infinite line into a finite area (since e.g. a circle could be drawn around the z-z fractal, defining an area), and mathematically "almost" has area, thus its fractional dimension (and hence its name). So, the fractal boundary represents a logical paradox; to say that it has no area, or to say that it has definite area are both incorrect. (Nor does it have infinite area, as previously shown by enclosing it in a finite space.) Similarly, any area that it could possibly have can't be part of the areas that it bounds, so where would it come from? It "almost" contains some part of the regions it bounds. It seems then that a fractal be regarded as a relation which implies the existence of the relatants without necessitating their presence.

Another concept related to fractals is "(self)-specification". Earlier I described the mathematical process by which fractals are created by describing the regions of stability and chaos as black and white. Fractals can also be graphed in color, however; the color of a point within a fractal is determined by the number of iterations the numerical set for that point passes through the difference equation. For example, if twenty iterations are required before the output stabalizes, the point might be colored red; if fifty iterations are required, then blue, and so on. There is always a limit, however, to how many iterations a computer will compute a given point set before labeling that point chaotic. So, if that limit is defined as (64) iterations, any point that didn't stabilize after (64) iterations would be colored the specific color chosen to designate chaotic points, usually black. If the limit is (64,000) however, a different looking fractal would emerge, one that appeared to have much more stability -points that take longer to stabilize would have time to do so. One could think of this limit, and the limits that determine the other point-colors, as specifications for resonance.

When a tuning fork or piano string is struck, it starts to vibrate at different frequencies simultaneously, but there is a particular frequency at which it is least resistant to vibration, and thus vibrates more vigorously, actually amplifying the energy that was used to start the vibration. Likewise, it will take longer to stop vibrating at that frequency than it will at others. This phenomena is called resonance. Now, if a device is particularly resonant, so that the smallest input of energy excites it into strong resonance, then when its vibration excites other nearby resonant devices, their vibrations will excite the first device further, and a vicious cycle will ensue, the vibrations growing stronger and stronger until stopped by some external force. This phenomena is known as feedback.

In terms of fractal specification, if the difference equation is nonresonant for a certain point-set, then the output will stabilize after a few iterations, and receive a certain color. If it is very resonant, the output might never stabilize, but break into feedback, and be colored black. If, however, one changes the limits for determining levels of resonance and feedback, a different image will emerge. This is another example of fractal scaling--different degrees of detail resolution can be attained by adjusting the limits of specification.

The way pattern emerges in a fractal is also dependent on specification. Patterns are formed when points cluster around other point of similar specification. So, if different colored points were randomly distributed, it would be dificult to recognize any patterns. "Pattern" in fact implies something which is non-random, and which has some recognizable form and predictable repitition.. Its remarkable that such complex mathematical functions form patterns at all--the result of the computation of a point-set seems hardly predictable; after all, up to millions of iterations may be required to specify a given point. That a group of point of similar specification can cluster next to another cluster of far different specification is just as amazing. Aptly, these clusters are called attractors--areas that attract points of a certain specification. Now, an apparently solid cluster will have, upon enlargement, other, smaller attractors within it; thus, pattern formation--arangement of attractors-in a fractal also depends on specification. The specifying limits of a fractal can be thought of as its genetic code or DNA; the code specifies the growth and reproduction, i.e. scaling, of the fractal in self similar patterns.

In keeping with my previously outlined methodology, I would like to examine correlations between fractals and aesthetics in terms of both empirical and interpretive aspects of artworks; both are important to the appreciation of art, and are actually quite intertwined.

Probably the most obvious visual characteristic of fractals is the form of their boundaries--the infinite detail containedwithin their intertwinings. This can also be found in impressionist painting, and is easily recognizable: Large shapes--faces e.g.--have at their edges smaller areas of color representing planes of skin reflecting light from different directions; within, and at the edges these, are the smaller brushstrokes--the reflection of light off of textured skin; but each of these reflections seems to be made of smaller strokes, actually the texture left by the brush, especially at the edges of each brushstroke, which intertwine with others on a minute scale, the over all effect entrancing in its manipulation of light. Or, considering Klimt's "Judith I", I am first absorbed by the feeling of sensuality captured by Klimt, as if Judith is beckoning to me with her gaze and the glow of her skin. Moving closer, sensuality turns into an air of arrogant pride, as she clutches her trophy--the head of Holofermes. Closer still, looking into her eyes, I am struck with a sense of contempt for her act of succumbing lustily to revenge, and I sense, barely, that she can feel that.

The lack of self-similar patterns in modern archetecture (i.e. of the Bauhaus/monolithic variety) may be one reason why its so unsatisfying to many people. Consider the work of Gaudi. From the shapes of his buildings, down to the doors and windows, even to the shape of ceramic pieces in his mosaics, and other decorative details, his works use selfsimilar patterns which tend to be organically derived. Many people find his work very engaging, possibly because the buildings don't overwhelm as one draws closer; instead, one can find pattern smaller and smaller in scale, so that a sense of proportion can be maintained between the building and the viewer regardless of proximity. Also, because the patterns are self similar, they maintain a coherence with the whole, which may prompt a feeling of "rightness".

The most illustrative examples of metapatterning can be found in music. Consider jazz improvization. Improvization can nearly be reduced to one concept: Phrasing. Phrasing basically concerns a musician's sense of metapatterning--how whole solos are composed of repeated choruses, which may be filled with twoor fourbar phrases, which are built of individual notes, etc. Smaller motifs, both rythmic and melodic, will usually be found within the larger pieces, whiich will be patterned amongst themselves. Yet this doesn't explain why, e.g., John Coltrane's solos are so riveting, involving; why they seem to be a direct communication between his soul and the listener's. Certainly, the amount of patterning might lead one to expect such playing to sound contrived, forced, or redundant; yet, on the contrary, it exhibits a naturalness, a sense of completeness which makes it very satisfying and listenable. I.e., it makes one want to hear more.

Earlier, I stated that patterns in fractals arise because of attractors, areas that attract points of certain colors. Doesn't the same sort of phenomenon occur when considering patterns in art? I believe so, except that the attractors attract ones focus or attention, rather than points of chaos and stability (although I will show how consciousness can demonstrate specification resonance).

For example, many M.C. Escher designs are pattern intensive, with very similar attractors (regions that catch ones attention) which are repeated. On the other hand, consider Maxfield Parrish. A common reaction to his paintings concerns his use of light, e.g. he seems to illumine his subjects in an almost mystical glow. Now, in terms of pattern of design-which I would call empirically observable, and which is exhibited abbundantly in Escher's work--I wouldn't consider Parrish's work richly patterned; yet, the illumination in his paintings is an attractor, attracting not only ones gaze or attention, but also a sense of connection with the viewer, which Escher's work does not, at least to the extent that Parrish's does.

The difference between the above two senses of pattern suggests an aspect of consciousness in common with fractals, namely that of specification; that is, one can consider pattern recognition a form of cognitive resonance or feedback much like that described for fractal specification. Consider the dynamics of consciousness when examining a richly patterned object. Small details that hold ones attention for a fraction of a second, or are barely noticed, could be regarded as exciting weak resonances--they stabilize (are acknowledged or recognized or otherwise give way to present distractions). Large details (or attractors), which "grab" the attention, excite stronger resonances; they don't stabilize as quickly, and represent a lower level of specification--ones attention is more sensitive or responsive to that attractor. Then, there are the attractors which leap out and transfix the viewer, grabbing his attention continually, absorbing the viewer. This is the very resonanceinducing attractor exciting a hypersensitive area of consciousness, leading the resonance into feedback, which is disapated very slowly, as the viewer's attention is drawn to another attractor.

So, when comparing Parrish to Escher, it seems that the attractors in Escher's work excite small (cognitive) resonances from the visual patterns, perhaps larger ones if one is struck by the heterological implications of his work; but the attractors in Parrish's works excite much stronger resonances, vibrating consciousness more vigorously, to the point of feedback-which I think represents the "Aesthetic Experience"--dying out slowly, leaving traces of vibration for some time after the experience.

Similarly, the patterns of attractors in Coltrane's solos must excite very strong resonances in his audience, leading not just to absorbtion or connection, but what many people describe as communication. Often, people relate experiences of aesthetic communication between an artist and his audience; they know it intuitively, yet can't begin to translate or reiterate it. This seems to be a wonderful metaphor for the excitation of resonances by attractors--it appears that some sort of transmission must occur between the attractors and consciousness; and if it is not translatable, and serves no purpose, it seems that the only use it may have is just to excite resonances and continue the transmission. I think this concept illustrates Gaddamer's notion of "play" in his analysis of the aesthetic experience. (from "The Relevance of the Beautiful") Further, what HGG calls a "work" has a "hermeneutic identity". From my current perspective, "hermeneutic identity" seems intimately related to the effect a "work"'s attractors have on ones consciousness. To be considered a "work", the attractors must be capable of exciting resonances which are strong enough to amplify into feedback, presumably for a large portion of that community which considers it a "work", something which depends on collective specification limits.

Some of Wittgenstein's ideas (from "Lectures on Aesthetics") also reinforce the notions of resonance and specification that I am exploring. First, consider his ideas on interjection and expression of discontent; these seem to be resonant phenomena, e.g., when trying to find the "right" word for a sentence. The resonance one experiences when discontent is removed must be recognizable; a word that is not quite right is like an attractor that excites a resonance of a different specification than the "right" word. As various words are tried, their respective resonances are excited, and either fall short of the anticipated resonance, or do excite it. These various excited resonances may result in an interjection, depending on their relation to the sought after resonance, i suspect.

Finally, L.W. describes the way levels of discontentment may change from day to day. E.g., the word that resonated so "rightly" in some sentence last week leaves one with discontent today, even in the same sentence. This corresponds to a change in specification--the scale and boundaries between different resonant relation has been altered. This also explains why a whole community won't agree on what constitutes a "work", and why works come in and out of favor.

Appendix : Definition of Fractals

Literally, fractals are graphic representations of chaotic behavior in complex systems as studied mathematically. The most basic sense of the fractal as a boundary model describes the boundary between chaotic and non-chaotic regions of a complex dynamical system. But, since the coinage of the term "fractal" by Benoit Mandelbrot in 1975, fractals have been used as model and metaphor in describing boundaries and geometries outside the discipline of mathematics.

The word "fractal" refers to the dimension of these boundaries; where euclidian geometry would separate regions with a one-dimensional line (since a line has no width, this is separation by nothingness) and claim that a twodimensional boundary -a planar area -would either include both regions as part of the boundary, or more likely, wouldn't properly be termed a boundary, a fractal boundary would possess a fractional dimension somewhere between one and two.

Consider the Koch curve as a very simple illustration of fractional dimension. This shape consists of an equilateral triangle to which, at the middle third of each exterior side, other equilateral triangles have been attached. These triangles, similarly, have equilateral triangles appended to their middle-thirds, and so on ad infinitum. Now, this repitition can literally recur infinitely without any of the appended triangles ever touching each other -one could magnify the image, comparing successive magnifications, for eternity and see basically similar images, but not enough detail to determine where the interior stops and the exterior begins; and the shape will never extend beyond a circle circumscribed around the original triangle, so that a finite area would be enclosed by an infinitely long line. This "...infinite length, crowding into a finite area, does fill space. It is more than a line, yet less than a plane. It is greater than one-dimensional, yet less than a two dimensional form.", thus its dimension is fractional. (Gleick, Chaos, p 102) It is more than nothing, yet less than a definite something, in terms of area -a transcendence of the dualistic confines of logic.

This is a very simple, but illustrative model of a fractal -"proper" fractals maintain self-similarity, but never repeat -but serves to illustrate my interpretation of the implications of fractals as boundary models. In a dualistic system, a one-dimensional boundary consists of nothingness, so imagining the boundary necessitates imagining the bounded regions; imagining the nothingness without imagining the regions implies nothing about the qualities, or existence, of the regions. If a two-dimensional boundary is considered, it would imply the nature of the regions by including them as part of the boundary. A fractal, on the other hand, "fills space", yet possesses no area, i.e. possesses none of the bounded area within it. Although other interpretations are surely as valid as mine, I assert that a fractal boundary thus implies the relation of the bounded regions without containing any part of them.

If you don't know what fractals look like, you should. Use any web search engine to search on "fractal." There are millions of them out there.