Fascinating Fractals

Fractal is a term coined in 1975 by Benoit Mandelbrot in to denote configurations that transcend traditional numerical categories.  Like optical illusions, they show us things that don’t coincide with our assumptions”(Van Eenwyk, 1997, p. 56).  Briggs and Peat (1989) shed some light into its origins and tell us that “the name comes from the Latin fractus, which means irregular, but Mandelbrot also liked the word’s connotations of fractional and fragmented” (p. 90), both of which apply to chaos and fractals, since fractals are “the patterns of chaos” and we can see them expressed visually in Briggs's (1992) beautiful book, Fractals: The Patterns of Chaos.  Briggs and Peat (1989) also make mention of Mandelbrot’s being “stubbornly visual” and note his irregular education, which led him to do research in a number of fields.  Mandelbrot describes his own experience:

Every so often I was seized by the sudden urge to drop a field right in the middle of writing a paper, and to go grab a new research interest in a field about which I knew nothing.  I followed my instincts, but could not account for them until much later.” (Briggs & Peat, 1989, p. 90) 

Boy can I relate, this has happened to me all along this journey.  In a way, fractals bear the stamp of their creator in that they are fascinating, visually beautiful, irregular shapes that occur in many places.  Fractal geometry helps us describe natural forms from the structure of galaxies and the shape of clouds and patterns of weather, to the winding of rivers, the shape of coastlines, trees, and mountains. Even brains, lungs, and blood supplies have fractal dimension, and the closer we look at them, the more the parts look like a miniature version of the whole.  This is called self- similarity across scale. 

Self-Similarity Across Scale

Fractals are self-similar across scale, they look the same, and their details are repeated at different scales or descending levels of magnification.  They resemble themselves across categories as well.  The classic example of a fractal is a coastline.  Coastlines contain bays and peninsulas, which upon closer examination, themselves are made up of bays and peninsulas which are made up of smaller bays and peninsulas on down to boulders and pebbles all the way down to the level of molecules.

The repeating nature of self similarity across scale can also be seen in your cupboard if you happen to have a box of Droste’s Cocoa.  The box shows a girl and a boy who is holding a box of cocoa which shows a girl and a boy holding a box of cocoa, which if you look very closely shows the same thing… well you get the picture.  A video version of this is found in the Mel Brooks (1987) film Spaceballs. This is known as mise-en-abyme or mise-en abîme  (D. L. Miller, personal communication, February 13, 2005).  Mise-en-abyme literally means "put in the abyss." The idea is “taken from heraldry via Andre Gide and is used by strategically Derrida and denotes the repetition-in-miniature of a whole within itself, as in the example of a painting within a painting.” (Surfaces, 2005) In the case of heraldry:

a shield/coat-of-arms will have pictured on it an identical shield/coat-of-arms, which has on it an identical . . . etc. . . . to infinity. A more modern example is that of a TV camera taking a picture of a monitor which displays the camera taking a picture of the monitor . . . etc. (Surfaces, 2005)

In such a model, repetition has, as Derrida would say, "always already" taken place: the regression is synchronic, at once originary and teleological. No matter where the regression halts, there will always be the traces of past and future repetitions” (JHU, 2005). This infinite regress can cause a feeling of falling into an abyss.  Derrida notes that in mise-en-abyme there is “some kind of fragile radical otherness” within the same (Surfaces, 2005).  This harkens back to the whole notion of the uncanny, and possibly is an unconscious reason why strange attractors are called strange. 

Fractal Dimension

Getting back to the coastline example, paradoxically, when you use a finer measurement scale, the length of the coastline increases, so not only are fractals infinitely detailed and self-similar across scale, but they have infinite length.  To deal with this paradoxical situation, the idea of fractal dimension comes into play.  Fractal dimension expresses the in-betweeness of fractals and allows different things to be compared according to their fractal dimension.  As previously mentioned, fractals transcend traditional mathematical categories.  For example, a line is one-dimensional, but if you trace out a line that covers an entire sheet of paper, it comes awfully close to being a plane, which is two-dimensional.  Fractal dimension comes to the rescue and allows you to assign a fractional dimension reflecting the amount of paper covered. In the coastline example, the ruggedness of the coastline can be seen in its fractal dimension, a rougher coastline would have a fractal dimension of say 1.7, where a smoother coastline would be closer to a line with the dimension of 1.3. 

In speaking of the chaotic dynamics of symbols, Van Eenwyk (1997) notes that they are like fractal attractors, they appear as “snapshots or ‘slices’ of dynamics: what seems to be a recognizable image is actually a configuration of motions frozen in space and time" (p. 116).  He continues:

By expressing the inexpressible, they can often be hodgepodge affairs as their everyday contents combine with archetypal forms to create unique entities.  Dwarfs, giants, witches, wizards, monsters, and fairies all utilize everyday images, however exaggerated to express the inconceivable.  While exaggeration is certainly a hallmark of the unconscious, it is not the only kind of transcendence of categories that occurs when everyday structures are arranged by archetypal forms.  Symbols also routinely combine elements of everyday life not commonly found together into unique conglomerates, like sphinxes, centaurs, mermaids . . . . Sometimes depictions of the gods rely on combinations of common elements . . . Hathor (cow and human) . . . Aion (lion and human) . . . . All consist of elaborations of the commonplace that become, by virtue of their combinations, anything but common.  Their meaning is synergistic, for it transcends the particulars that make up the image.  Consequently, to discern the meaning of a symbolic image, we must read between the lines.  This comes pretty close to the fractal dimension which either fits in between or transcends the traditional boundaries of logic. (p. 116)

Fractals are highly complex and unique, and yet they can be generated by simple iteration.  Fractals, such as the Mandelbrot and Julia sets are hauntingly beautiful and Briggs and Peat (1989) suggest that they are also parables:

Here float islands of order in a sea of chaos, worlds within worlds.  Are we seeing how simple iteration reveals the way a comprehensible order structures chaos?  Or is it chaos that structures order?  This is the turbulent mirror.  In fact, the generation of Mandelbrot’s mathematical set mirrors how real systems create and destroy the structures of our physical world. (p. 100)

In the Mandelbrot set, we also see that simple iteration “in effect liberates the complexity hidden within it, giving access to creative potential.” When random variations of the iterations are allowed, enabling details to vary from scale to scale, the set more closely mimics nature and “suggests that natural growth is produced through a combination of iteration and chance” (Briggs & Peat, 1989, p. 104).  Perhaps this is why fractals and strange attractors evoke a deep and haunting recognition, and that they resonate with the interwoven and complex designs of ancient cultures and iterative nature of chant patterns and children’s games. Briggs and go on to note: “all great art explores this tension between order and chaos . . . . In confronting the orders of chaos, . . . it appears we are now coming face to face with something that is buried at the foundations of human existence” (p. 110). 

As was previously mentioned, the Shiva Nataraj is considered to be one of the ultimate expressions of art, and perhaps this is why:  It also reflects the tension between order and chaos and the balance between the two.  Jung (1988) notes that dancing is a representation of a creative act, it thus “necessarily symbolizes both destruction and creation.    It is impossible to create without destroying: a certain previous condition must be destroyed in order to produce a new one.”  Shiva is the “great destroyer because he is creative life and as such both creative and destructive” (p. 56). Not only is the dance itself reflective of this in symbolizing the cycles of creation and destruction, but Shiva, was dancing on the back of a dwarf, Mulayakan, who stands for forgetfulness and epilepsy.  Briggs and Peat (1989) tell us that chaos is entirely normal for the brain, and in fact, paradoxically, too much order in the brain, as well as too much chaos can result in chaos.  They give the examples of schizophrenia and epilepsy:

The schizophrenia victim is suffering from too much order—trapped order which paradoxically appears in the epileptic seizure as a massive attack of chaos.  In the case of epilepsy, a small disturbance in the firing pattern of some brain cells creates a bifurcation . . . which oscillate at one frequency, and then are joined by a second frequency; then the first frequency cuts out.  This pattern repeats creating ‘traveling and rotating waves’” (p. 167). 

Shiva’s dance seems to say, my chaotic dance renews the world periodically and balances out too much order or too much chaos.  This mythic theme indeed also reflects something buried at the foundations of human existence, Grof (personal communication, January 30, 2005) feels that many people during LSD sessions were experiencing fractals when they saw the geometric kaleidoscopic shapes, but they did not have the words for their experience then.

End of section, to Archetypes and Strange Attractors