The Spam Artist

Spam Plants - Click to enlargeOne of the most innovative and creative artists working today is Alex Dragulescu, a Romanian "visual" artist who heads  the Experimental Game Lab at the Center for Research in Computing and the Arts at University of California, San Diego. 

His images are absolutely fascinating.  Some of them are beautiful and organic-looking, while others appear to be views of some alien architecture.

But it's not the images that are interesting so much as the process.  Dragulescu uses spam to generate some of his images.  He does this by using words in spam mailings to trigger different structures to be digitally rendered.  The image at the top of this post comes from his Spam Plants series.

Dragulescu definitely pushes the art/mathematics interface to an extreme, with the random detritus of life thrown in for good measure.   He describes his projects as "experiments and explorations of algorithms, computational models, simulations and information visualizations that involve data derived from databases, spam emails, blogs and video game assets."

It's impossible to describe Dragulescu's work without seeing the images, so check his website for much more. 

Be sure to investigate some of the other work being created at the Experimental Game Lab , where "somewhere between media art, scientific visualization and computer gaming a new territory of expression is emerging." Sheldon Brown's Scaleable City project is one such project.

It is hard to imagine how amazing it must be to teach or go to school at a university like UCSD, where anything short of the absolute limits of creativity is viewed as abject failure.

The Frontier of Art and Mathematics: 2006 Mandelbrot Fractal Art Contest

The ubiquity of fractal art can be observed with a simple web search, where a recent google query yielded 640,000 hits. Some will claim that this fact does not convey any artistic status to fractal images, only that an awful lot of folks name their works fractal art.

Which is why I'm happy to see that some big-time mathematics groups are finally recognizing the potential for creative artists to produce captivating art - even if they don't know the mathematics.

Earlier this year (August, 2006) the International Congress of Mathematicians (ICM) - the group that decides the recipient of the Fields Medal (the world's top mathematics prize) included a fractal art exhibit. The exhibitors were chosen after competing in the 2006 Benoit Mandelbrot Fractal Art Contest. Please visit the site where you'll be able to see the winners as well as a number of other entrants.

The promo for the exhibit does a great job of arguing for the existence of fractal art that is truly art:

The exhibition is formed by a collection of computer generated images by a group of artists and/or scientists specialized in fractal art. The mathematical expressions and the parameters used confer a unique and distinctive colour and aesthetics to every image.

Much like painters and sculptors transmit their personality and sensibility to their works by means of their technique, the authors of this exhibition express themselves by means of formulae and algorithms, modifying them progressively until the desired goal is obtained; reaching the frontiers between Art and Mathematics. The synthetic computer

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Number Patterns: From Fundamental Constants to A Fractal Number Popularity Contest

The fundamental constants (e.g. the mass of the electron, the universal gravitation constant, Planck's constant) are given that name for good reason: all computations used to understand or model the physical universe rely on their values.

In addition to the fundamental constants, there are other constants that define our universe. These numbers are really the statistics of the world, e.g. the heights of mountains, lengths of rivers, masses of the planets, etc.

There is a very odd, hard-to-believe-at-first- sight mathematical law that describes the distribution of these constants. In 1938, Frank Benford analyzed over 20,000 numbers taken from the fundamental constants of physics and far-removed areas such as sports stats and street addresses. Benford wanted to measure the frequency distribution of the starting digits for these numbers because of another odd fact - in antique tables of logarithms, the first few pages are often more worn, indicating that the log user was thumbing through the first pages much more frequently than latter pages, i.e. the logs with a "1" as a leading digit.

Now here's the amazing part. Instead of determining that starting digits from 1-9 appeared with approximately the same frequency, Benford found that the numbers he was studying began with a "1" a disproportionate 30% of the time. Benford went further, measuring the entire frequency distribution of starting digits, and developed a formula that predicts this distribution - a formula known as Benford's Law. The Law predicts that the frequency of occurrence of a digit drops off logarithmically with increasing digit size, and therefore numbers starting with 9 appear less frequently than all other numbers .

The explanation for the law comes from the fact that the numbers

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Smilla's Sense of Mathematics

The quotes from G. H. Hardy in the previous post are well known - in mathematical circles at least. Their succinctness is a hallmark of Hardy's desired economy of words in mathematical proofs.

Hardy was a number-theorist, and it seems appropriate to follow up the posting of his quotes on the beauty of mathematics with a very interesting fictional passage about the nature of numbers, and the connection with human life.

The following passage is from Smilla's Sense of Snow, a 1992 novel by Danish author Peter Høeg. A strange mystery that involves a child's murder and an eery trip to Greenland, Smilla is a fascinating heroine, at home in the worlds of mathematics, intrigue, and, obviously, snow...

Do you know what the foundation of mathematics is? ... The foundation of mathematics is numbers. If anyone asked me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why?

Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of longing, and do you know what the mathematical expression is for longing?

The negative numbers. The formalization of the feeling that you are missing something.And human consciousness expands and grows even more, and the child discovers the in-between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce rational numbers. And human consciousness doesn't stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers.

It's a form of madness. Because the irrational numbers are infinite. They can't be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers.

It doesn't stop. It never stops. Because now, on the spot, we expand the real numbers with imaginary square roots of negative numbers. These are the numbers we can't picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it's possible to explain satisfactorily the crystal formation of ice. It's like a vast, open landscape. The horizons. You head towards them and they keep receding.

Greenland icecap. Photo by Evert Wesker.Luckily for all of us, the madness hasn't stopped. The complex number system is essential to fractals, and forms the mathematical plasma in which algorithms for determining the Mandelbrot set are calculated. This set is like a vast, open landscape with horizons replaced by fractal basin boundaries. As you head towards them, they don't recede. Rather, they get more complex. Without complex numbers, then, we'd be living in world bereft of color and self-similarity - an uninteresting world of snow.

Notes - Thanks to Sharon Armstrong, of La Salle's Psych department, for reminding me of Smilla's mathematical musings...Julia Ormond played Smilla in the movie version. Click here for a film clip of the mathematics scene...And be sure to try this excellent fractal basin applet.

Mathematics is Art: Make No Apology

"Ikebana" by Tim Fadden

Recently, the following provocatively simple set of questions was added to an earlier post describing Jackson Pollock and fractals - one of a series of posts that debate whether technology & mathematics -produced art can actually be ART:

why does the mathematics have to be producing art? why can't the mathematics be art itself? would that actually solve any of the problems discussed here.

This question is from Jonathan "Fish" Fisher - co-winner of the 2005 Duke poetry award. Fish has posted a number of interesting comments and questions, adding his voice to the dialogue of whether or not art can be produced using mathematics and technology. He recently posted his Poem For Benoit Mandelbrot, A Connoisseur of Chaos, a moving poem that ends that concludes with his own awareness of the beauty and mystery of mathematical forms and structure via Mandelbrot's fractals:

Briefly I taste the salt sting
Of equations I'll never understand,
But a wave of awe
Sweeps me up as if divine
Artistry had finally
Conformed to a function of
Some rigid geometry.

Can any man be more than an artist?

I am always heartened when a non-mathematician senses the beauty of mathematics, which Fish makes evident in his poem, and his suggestive question: why can't the mathematics be art itself?

There are mathematicians who will go into full rapture mode when describing the beauty, in their eyes, of mathematics. This is especially true when they describe mathematical proofs that are considered "elegant" - proofs whose logical path leads inexorably from axiom to conclusion using an economy of steps, with leaps to other areas of mathematics hitherto unrelated, demonstrating a dizzying web of connections among the farthest reaches of mathematics.

This view is typified by G.H. Hardy, an early-mid-20th-century British number theorist, and the author of A Mathematician's Apology, his 1941 memoir and a book that should be read by all mathematicians. Hardy writes...

In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.

But what about beauty? And where is the art? Hardy lived in a world of pure thought and number - far-removed from the real world earthiness of mathematical modeling, and just as far from the computer-produced display of impossibly intricate fractals. Yet for him, it is the essence of pattern - not, as we've seen, a pattern in space such as a fractal pattern, but a pattern of abstraction and logic that is the substrate of mathematics, and it is there that beauty can, and MUST be found:

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

For Hardy, mathematics is surely art. To those who still aren't convinced, and who would claim that only mathematicians can see the "beauty," hence it is not art, isn't this an argument similar to those who can't see the landscape in an abstract painting, or who can't hear a melody in the serial music of Schoenberg?

Applying some logic here, mathematics must be art because it has an art-essence that is independent of the observer. If you don't see it, it must be there.

Stanislaw Lem: The Passing of a Deep Spirit

Stanislaw Lem, the great Polish "science fiction" writer died on March 27, 2006, at the age of 85. Two of Lem's works have played a role in the Chaos and Fractals course, and I describe the connection below. First, though, must be a tribute to this extraordinary writer.

Even though I know many who read, or have read a great deal of science fiction, I know very few who have read any of Lem's works. This is very odd, given that Lems' works have been translated into over 40 languages, with an estimated 27 million sold. (Some do read and prosper: Will Wright, the creator of the wildly popular SimCity simultaion game credits Lem's The Cyberiad as inspiration. )

With sci-fi readers (in the U.S., at least) not paying attention, what hope is there for more readership of this essential 20th-century author who is usually listed as I wrote above - a science -fiction writer, only without the quotes.

It has always been unfortunate that Lem's works are described as science fiction. This is itself a fiction. Lem - a brilliant scientist, writer, and thinker - told wonderful tales with an unnerving mixture of darkness, humor, philosophy, and theology that just happened to be placed deep in space, or inside a computer. While the location and time period of his stories are essential to their plots, Lem's stories are often more relevant to our current time and place because of his ability to paint rich characters in situations that are paradoxically both imaginable and impossibly strange.

Lem's life as a scientist and writer growing up in Poland, through Nazi occupation and Soviet rule, is much of the reason for his chosen genre, as described in the Times of London obituary -

He began to write fiction, his first works being in the tradition of socialist realism acceptable to the authorities. But he graduated to literary "fantasies", which he succeeded in hoodwinking the humourless and dogma-bound authorities into believing were innocuous, though they were in fact highly subversive and satirical.

I first read Lem in 1983, when my best friend, Eric Törnqvist, gave me a copy of Solaris as a birthday gift and demanded that I read it. To this day it remains not just the greatest "science fiction" that I have ever read, but one of the best books I have ever read. It is a book in which there is no action of the type usually associated with a sci-fi stories. Instead, Solaris chronicles centuries of observation of a liquid planet and its seemingly non-descript moons, a planet that may be sentient, and may be malicious. With this simple idea , an idea that seems to present little opportunity for

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Splattery Will Get You Somewhere: Fractal Forgery

"Convergence" Click to EnlargeIn a bit of fractal irony, the latest Jackson Pollock controversy concerns whether or not recent paintings that have turned up are really his, or whether they are forgeries.

On one side are art experts who claim that the paintings are real. On the other side is physicist Richard Taylor, who was the first to publish definitive studies of Pollock's works that found a fractal nature that increased during his career. (For an excellent intro. to Taylor's findings, see J. Ouelette's 2001 piece in Discover Magazine)

Taylor suggests that the newly-discovered Pollocks may be fake because they display a different fractal character than what he has measured in works that are definitely Pollock's. (Click here for the story.)

So for all of you on the side that fractals can't be art, here we have paintings denied authenticity because they aren't fractal enough.

But this raises a very interesting question, and contradiction. If the paintings are forgeries, were they done by a computer? If so, why didn't they match the fractal properties of Pollock's works as measured by Taylor (fractal dimensions in the 1.5-1.7 range)? It would seem to be a simple matter to turn the dial on the fractal-generating software, choosing the appropriate fractal dimension of the counterfeiters desired Pollock period. However, if the paintings are forgeries not done by computer, then whoever painted them was a first-rate counterfeiter.

There is an even deeper issue here, one that may be the most crucial because it touches on society's ability to bestow the label of genius. As written by Don Foster in his NYT piece of Feb 19, 2006:

At the heart of the controversy lie critical questions about artistic meaning and value that have vexed literary scholars no less than art historians. Would the exposure of a hitherto successful forgery diminish Jackson Pollock's reputation as a unique creative genius, by demonstrating that his work is replicable?

Appropriately, Foster does not stop at the artist, and asks the fundamental question: is something that has passed for the real deal really worth less as art? Is the art in the piece, or is it an impossible-to-deconvolute amalgam of the piece, the artist, and the context of the times in which it was created?

Ultimately, Foster's answer may be viewed as a too-clever attempt to use the context of Pollock's times to remove the question:

Meanwhile, Jackson Pollock may be chuckling in his grave: if the object of Abstract Expressionist work is to embody the rebellious, the anarchic, the highly idiosyncratic - if we embrace Pollock's work for its anti-figurative aesthetic - may faux-Pollock not be quintessential Pollock? May not a Pollock forgery that passes for authentic be the best Pollock of all?

(Read Foster's full article here )

The Fractal-Pollock fracas has inspired a large number of inspired blog posts. One interesting counter can be found on the New-Art blog of "VVoi."

A more detailed, and ultimately more damning rebuttal appears on John Haber's The Fractal Geometry of Vision: Pollock's Patterns and Rembrandt's Eyes . Haber expands on the debate by comparing the complex, contextual aspects of attribution in both art and literature, ultimately asserting that Foster asks the wrong questions.

I can only thank all the students of Chaos and Fractals, Fall 2005 edition, for asking the right questions about the validity of fractal art.

Grim Modeling: Overpopulation, Disease, and War

Battle of TrafalgarI often wonder how appropriate it is in an Intro. Diff Eq. course to discuss the darker side of modeling - the business of modeling death and destruction via war, terrorist activity, or plain, old-fashioned man-made disaster. (I have no qualms about this in our Chaos and Fractals course - after all, one of the goals of the course is to consider the ramifications of the pervasiveness of mathematical modeling in all aspects of society, and especially in areas where religion, philosophy, and politics intersect.)

But back to Differential Equations - basically a mathematics course where students get their first real concentrated exposure to applied mathematics and modeling via basic calculus.

It seems OK to discuss population models such as the Logistic equation, where growth rates go negative for population values exceeding the carrying capacity of the ecological niche. The logistic equations is, after all, a sterile shorthand for what is an accepted "law of the jungle," at least for animals: if there ain't enough food, you die.

And there doesn't seem to be any queasiness among students when a Lotka-Volterra model is used for a Predator-Prey system. This again may be explained because it is a fact of life on the other side of the human/animal divide - eat and be eaten, in an endless cycle of life and death. Or maybe it's because the use of cute function names - R(t) and F(t) for rabbits and foxes - provides a welcome relief from reality due to the abstractness of the notation: we don't feel the rabbits' pain, or wince at the growth of the Rabbit-Fox interaction term when dR/dt<0.

Epedemiology is a fertile source of models for Diff. Eq., but even here what is being modeled may be especially stark. (See Deterministic Modeling Of Infectious Diseases:Theory And Methods, a thorough review of disease modeling by Helen Trottier and P. Philippe of the Univ. of Montreal). In this case, an ambiguous use of English can go a long way towards not having to think about the grim reality of what the models often predict.<

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SWARM: Art Through Multiplication

Fractal art abounds...consider the show now appeariing at the Fabric Workshop in Philadelphia, where all sorts of mathematical chicanery is used to produce stunning images and sculpture.

The show is titled "Swarm." From the Fabric Workshop website:

Swarm theory is an idea animating contemporary art, science, design, digital media, and social theory. "Swarm logic" is seen in works that use vast numbers of small parts to create systems whose final behavior or effect cannot be wholly predicted. Artists working with computers and new media construct rules that draw together data and generate behaviors that evolve over time. Sculptors and painters create structures and patterns based on the interrelationships and inherent properties of individual elements. Swarm connects the social life of bees, birds, crowds, and cities to contemporary aesthetics, as seen in the fascination of artists and designers with how simple, discrete units accumulate into complex systems.

Sounds like the ultimate Chaos Game to me!

In his review article By Multiplication, Products of Art Inquirer art critic Edward Sozanski describes his ambivalence to mathematically-generated art:

I have long been ambivalent about art made by clumping. At a glance it
can appear simple-minded - take one "hand grenade" and multiply to infinity.
There's no denying, though, that in art as well as in nature simple multiplication can produce aesthetically powerful and beautiful results, especially when the individual building blocks aren't very prepossessing.

"Swarm" isn't entirely about creating elegant structure; process and how it's controlled are presented as equally important, especially in systems that exhibit communal decision-making.

Read the entire Sozanski piece - he did not like everything he saw. Nevertheless, isn't the idea of SWARM further fuel against those who discredit fractal art?

It's hard to argue with the fact that "simple multiplication can produce aesthetically powerful and beautiful results. "

To all fractal artists: keep on multiplying...to infinity and beyond...

"Swarm" continues at the Fabric Workshop and Museum, 1315 Cherry St., through March 18. http://www.fabricworkshop.org/

M.C. Escher - Art, Mathematics, Recursion, Infinity

M.C. Escher's name also came up in the fractal/art discussion. Escher (1898-1972) was a Dutch graphic artist whose most well-known work was highly mathematical, incredibly complex, and, unbelievably, hand-made. Making many of his works with woodcuts, his technical expertise was staggering - this is not someone using a computer program, but instead pencil, paper, wood, knives, and ink.

Many Escher prints suggest infinity , but in a somewhat different way from a basic fractal such as the Sierpinski triangle. The image at the top of this post - titled "Circle Limit III" - implies an infinity of interlocking fish, with the surface of the sphere as the "inhabitable" universe. The visualization of infinity is suggested by the diminishing views in perspective at the sphere boundary. While the creatures near the edge are self-similar to the middle-sphere images, the Sierpinski fractal's self-similarity is found by zooming Into a section, rather than moving to the boundaries. Thus the Escher image is more akin to the Mandelbrot bug, appearing at different sizes and orientations along tendrils emanating from the main bug. (For the mathematics behind the graphic, see The Trigonometry of Escher's Woodcut "Circle Limit III", by H. S. M. Coxeter.)


Ascending and DescendingEscher also encapsulated an infinity of time in some of his prints - most famously displayed in "Ascending and Descending," a print that was one of his set of "impossible structures." Are the figures in the picture walking up or down the stairs? And when do they reach the top (or bottom)?

Note that the Escher web site contains a free interactive puzzle in which your task is to build an impossible structure.

Read about more of the mathematics behind another of Escher's famous works - The Print Gallery - as well as the connection with Dutch chocolate at the web site Escher and the Droste Effect. (Be sure to check out the animations at this site.) Here you'll find ample mention of scaling - a necessary feature of fractals.