To Boldly Determine a Fractal Dimension

It happens very quickly, and is very easy to miss, unless one is either an inveterate fractalogist or Vulcan, or both.

In the eminently entertaining Star Trek movie just released there is a scene of young Spock's school which appears to be a cavernous room with a floor made up of indented hemispherical shells (as if you were on the inside wall of a very large pimple ball). of each "pimple" while a Vulcan student in each "pimple" listening to a lecture, or reciting a lesson while Mathematical expressions are illuminated on the walls

The film takes us for a brief visit to a few of these math-pimples. In one, a pointy-eared student begins his recitation:

The dimensionality equals the log of N...

The statement is not completed, but clearly this is the beginning of the expression for the Hausdorrf-Besicovitch dimension:

d = Log(N)/log(s)

This expression has many different variants (I am guessing that this is the one used in Vulcan grade schools), and can be used to easily calculate the dimensions of deterministic fractals. So, e.g., the Cantor set weighs in at a dimension of log(2)/log(3) = 0.6309..., while the Sierpinski Carpet is a more robust log(8)/log(3) = 1.8928...

Combined with clever statistical counting techniques, the dimension of random, and even naturally-occurring fractals can also be determined. The boundary of regular Brownian motion has a dimension of 4/3 = 1.33, while the surface of the human brain has a dimension of approximately 2.79.

In case you're wondering what naturally occurring object your brain most closely resembles, note that the dimension of a typical piece of broccoli is 2.66

See this list of fractals sorted by Hausdorff-Besicovitch Dimension for more!

In the lists that regularly appear comparing the mathematical performance of US school kids vs. children in other countries, thank goodness Vulcan is not listed. Not only do they all know about the H-B dimension, they instinctively know the definition of a fractal: an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension.

Live long and prosper, measuring your fractals wherever you find them.

Cabbage Leaves and Temporal Fractals

Fractal tumor on Wild Cabbage LeafI have always considered fractals in time to be related to self-similar music (such as a nested fugue), or just a plain-old self-similar time-series, such as stock market fluctuations, or the corn price fluctuations at the Chicago Mercantile Exchange, whose fractal nature was first noted by Mandelbrot.

Now there's a different way to consider time-fractals - proposed by Carlos Escudero and colleagues of the Institute for Mathematics and Fundamental Physics in Madrid, in their Dynamic Scaling of Non-Euclidean Interfaces

Escudero "performs calculations of the dynamic scaling (how a surface changes in space and over time at several different scales) of growing structures, such as the kind of semiconductor films used in the microchip industry where, even under the most carefully controlled of conditions, rough (non-Euclidean) geometries can exist. He found that the moment-by-moment behavior of the surfaces are strongly effected by the fractal geometry."

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Cephalapod Fractals

Complex SutureSteve LaMonte, a student in my Fall 2007 version of Chaos and Fractals, has noted the fractal-like shapes that are formed by suture lines in ancient cephalopods. He points out the correlation between fractal structure and the ability of the cephalopod to withstand extremes of water pressure. He writes:

One often pictures fractals as consisting of pretty pictures generated by computer programs, but they are quite prevalent in nature. A notable example can be found in the fossils of ancient cephalopods, specifically nautiloids and ammonoids. Nautoloids and ammonoids are the ancient ancestors of modern squids, octopi, and the nautilus. The ancient organisms looked like modern squids and octopi with shells, some elongated and some coiled like a snail. These shells had internal chambers that the organism filled with gas for buoyancy. Each chamber is separated by a wall, or septa. The contact line between the septa and the inner shell wall is called a suture line. The structure of the suture line determines how well the organism can resist water pressure and adjust its buoyancy. The evolution of suture lines follows an increasingly fractal-like pattern from straight sutures to highly undulated sutures. In complex sutures, the dips and folds in the undulations are called lobes and saddles, respectively.

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The Face of Spam

Uugh.. who would want to look at anything remotely representative of two of the most hideously ugly realities of online life?

Alex Dragalescu, that's who. Dragalescu, a Romanian visual artist, often uses analyses of spam and other annoyances to drive visualization schemes, producing highly-organic-looking computer-generated images. (And, in this case, a nice example of meta-imagery: CGI's of nasty things that see their birth, and are spread, via computers.)

For example, Alex uses "the ASCII values found in the text of spam messages determine the attributes and qualities of the Spam Plants."

The graphic in this post is from Spam plants.

His latest series is entitled Malwarez, which is

a series of visualization of worms, viruses, trojans and spyware code. For each piece of disassembled code, API calls, memory addresses and subroutines are tracked and analyzed. Their frequency, density and grouping are mapped to the inputs of an algorithm that grows a virtual 3D entity. Therefore the patterns and rhythms found in the data drive the configuration of the artificial organism.

This is all fascinating, fractal stuff, and is in the spirit of other visualization projects posted on fractalog.

Eine Kleine Nachtfractal

Listen to Mozart by Renata SpiazziThe melodic recursiveness of most music, especially as manifested in the crystalline, almost-mathematical purity of Mozart compositions, suggests the presence of fractal-like structures that exist both in time and the frequency domain - structures that are both solid and ephemeral, logical and otherworldly.

Some may hear (even if they don't articulate) a richness that is reminiscent of fractal construction. Marin Alsop, Music Director of the Baltimore Symphony Orchestra, describes Mozart's popularity as the result of "the depth of the music and ... the fact that Mozart makes contact with our inner selves. Maybe it's because of his organic approach to composition - taking a small, cellular idea and developing it into something beautiful, he takes you by surprise but also comforts you."

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Wada Wada Wada

Wada - minus the reflecting ballsNormally, symmetric fractals doen't possess the type of arresting, intriguing beauty of the asymmetric ones. Wada basin fractals, which are often symmetric, do have a unique visualization that I find more aesthetically appealing.

Perhaps the best thing about Wada basins is that they can be simulated with easy-to set-up lighting and symmetric arrangements of reflecting balls - e.g. Christmas ornaments. Some instructions for doing this are at The Optical Gasket Lab , a fun module from the Yale Fractals course designed by M. Frame and B. Madelbrot himself.

The real instantiation of the basin boundaries allows for interesting comparisions of these real optical images and computer-generated graphics. See the Secret Inner Life of Mirrored Spheres: Wada Basins by Miquel.com for a set of beautiful images. (The image at the top of this post is from this site)

Flaming Symmetric Fractals

Apophysis Fractal Flame (click to enlarge)I've seen the flames - fractal flames - and they are amazing. Originally an outgrowth of the ideas developed in Symmetry in Chaos by Field and Golubitsky, fractal flames are related to the Chaos Game because they are created by tracking the iterates of starting points as they are mapped into other points. As in the Chaos Game, the iterates reveal, over time, the structure of the attractor associated with the map. Flame fractals use a combination of non-linear maps and very inventive approaches to coloring/visualization.

Flame mathematics is interesting enough on its own, but the intrigue of flames is the ability to generate images of surreal organic beauty. In the following excerpt from The Fractal Flame Algorithm by Scott Draves for the Cosmic Recursive Fractal Flames site, aesthetics are essential:

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Tessellatin' Rhythm and Fractal City Maps

Portland - The Fractal (Click to enlarge)One of the craziest art efforts out there is the geospatial art of Nikolas Schiller. Schiller takes satellite photos of cityscapes and melds them into quilts, morphs them onto spherical surfaces, and, basically anything else he can think of. The net result is a set of amazing images of familiar cities looking as if viewed through kaleidoscopes. Many of the images remind me of Escher, only with buildings and landscape features serving as the interlocking escher-figures, receding to infinity at the edges.

Maybe more insane is Schiller's pace: a new map every few days for several years now, all posted on his Daily Render blog, subtitled A Digital Scrapbook for Past, Present, and Future.

Schiller also works with old maps, e.g. combining 16th century maps with current images.

The fractal connection is an obvious one, and Schilling has a special section devoted to images that are more fractal-like. (See the Dupont Circle tessellation, e.g.)

Schiller's motivation is artistic and political. As described in a Washington Post article by D. Montgomery,

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The Right of First Recusal

Some odd serendipity at work (isn't serendipity always just a bit odd?). While cleaning out an incredibly messy office that I can't stand being wink-wink-nudge-nudge described as chaotic, I came across an article written by Lance Morrow for Time magazine in 1990. Titled Let Us Recuse Ourselves Awhile, it is a paean to removing ourselves from all of the random facts, emotional baggage, and general neuronal detritus that clutter our brain, in effect stunting our ability to be curious and creative.

The closing paragraphs may be one of the first metaphorical appearances of fractals to appear in mainstream print:

History proceeds in gossip and fractals. Fractals are the mysterious and apparently irrational forms proposed by the mathematician Benoit Mandelbrot, who says that reality has shapes undreamed of by Euclid and surprises that ridicule the idea of order. The shape of a mountain is not a cone. Clouds, coastlines, tree branches, commodity prices, word frequencies, turbulence in fluids, stars in the sky, reputations, fame, the passage of history itself (think about the past ten months) -- all these are fractal shapes.

The mind is the grandest, most mysterious fractal. It takes its shape from what it holds, and therefore, Zen-like, sometimes grows more graceful because of what it has kept out.

While reducing my own physical clutter I find justification to reduce my mental clutter, which frees my fractal-like brain to post about reducing clutter, allowing me to throw away the Morrow article which sat in a drawer for 17 years, uncluttering my fractal-like brain enough to undertake a more massive drawer cleanup, a drawer which containing endless layers of yellowed articles torn from old magazines, fractal-like. Serendipity, indeed.

And what type of cranial clutter did Morrow wish to recuse himself from?

The answer is frighteningly obvious, because we all suffer from the same flotsam callosum: any and all news of Donald Trump.

The Terrible Tao of Chaotic Career Moves

Myron Cope and his Terrible Towel: Pittsburgh broadcaster or Chaos Theorist?With a field of study as rich in language and imagery as chaos and fractals, it is inevitable that whole bodies of research will develop that find the theory and results of chaos & fractals applicable in totally improbable situations. It used to be that quantum physics was the leader in this phenomena, with the Tao of Physics  by Fritjof Capra the ur-text that promised a much more balanced outlook on life informed by wave/particle duality. (And I will note that I still have my copy.) Given the history of this text, I need to introduce a new category of post, which I openly steal from all Pittsburgh friends and readers - The Terrible Tao. The T-Tao designation is given to applications of chaos and fractals - and I might as well throw in complexity - to the most unlikely social situation.

My goal here is not to criticize these efforts, because they represent attempts to find models for social behavior that are grounded in a well-established field - chaos and fractals - that just happens to yield a range of behaviors that are remarkably similar to human and institutional behavior. Actually, with many of the articles appearing in journals well outside of the natural sciences, the writing often contains a self-contained expository section on nonlinear dynamics because a general knowledge of chaos and fractal theory on the part of the journal's audience cannot be assumed. So I am glad that the ideas of chaos and fractals reach a larger audience.

With that said, I often find that the modeling is more a use of chaos and fractals as metaphor - a way to describe human situations with exotic terms such as bifurcation, or homoclinic tangle. As a result, I rarely see any predictive value in the modeling, which, as a result, leaves me no farther along in understanding the situation being modeled.

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