Paul Santoleri: A Fractal Muralist

Philadelphia-area artist Paul Santoleri draws/paints amazing images that are evocative of fractals.  Not the Mandelbrot-class fractals that are the staple of most  fractalologists and fractal software, but reminiscent of  the fiendishly delicate and organic-looking fractals created by  Clifford Pickover.

Only Pickover does his with a computer, while Santoleri paints his.  The image at the top of this post is titled Red Spider, and is a 2' by 2'  painting using acrylic on canvas.  (Santoleri also paints on a very large scale:  he is also a muralist, and has done 70+  large murals around the world, with a few in Philadelphia.

Here's a fairly well-known Pickover creation titled From the X-Files.  See more of Picover by clicking here.

While Pickover has done an enormous amount of spreading the fractal word, and has been a great spokesman for the melding of art and mathematics, I find Santoleri's work much more interesting because its organic-like nature is truly evident, painted without any dependence on computers.  In Santoleri's own words:

In my works I erase the borders between the visible and invisible matter and create a new medium generating object and beings whose meaning, gender, and whereabouts is unclear and not important. My works should be viewed just as a part of the moving beyond their bounds whole that knows no spatial or temporal limitations.

Pickover certainly captures an artistic view of infinity with his images, but they are more of space alone and can't compete with Santoleri's temporal infinity.

Copenhagen, Quantum Mechanics, and a Shot of Glen Livet

The tag line for this blog, indeed the overarching theme of the Chaos and Fractals course is the fuzzy three-legged monster of Modeling, Understanding, and Prediction. Fuzzy because the boundaries are never clear; they are themselves fractal-like. Suffice it to say that the non-linear dynamic modeling of most systems is used for both the understanding the models provide about why something happens the way that it does as well as the prediction of future states.

Which brings me to quantum mechanics - a field where models routinely predict experimental results with extraordinary accuracy, yet there is still debate on what it all means. Taking this to an extreme, if there is disagreement among physicists on the meaning of quantum mechanics (specifically, the meaning of the quantum mechanical wave function and the nature of observation) then there is a lack of understanding. Whether one considers this "good prediction, no understanding" scenario unsatisfying or not comes down to one's proclivity for philosophizing.

Actually, for me I developed a "proclivity for philosophizing" because of Quantum Mechanics.

I remember as a physics student being totally mystified at and angry with quantum mechanics. Sure I could do the mathematics, but I really had no clue as to what a stationary state was, or what it meant for a wavefunction to collapse. Even after getting very good grades in both semesters of quantum, I really couldn't articulate the connection between the mathematics and reality in a reasonable way, or at least not in the tangible way that I could describe the (apparent) reality of Newtonian physics.

Click to enlarge the Quantum CatGrad school and a post-doc in solid state physics finally did bring some aha! quantum moments; I could finally talk-the-talk of quantum mechanics interpretation as well as theory. Like most students, I was taught the Copenhagen Interpretation of Quantum Mechanics as promulgated by Bohr and Heisenberg in the 1920's. I adopted it whole heartedly, and soon Schrödinger Cats and Wave-Particle Duality were common topics of late-night sessions, often fueled by single-malt ...

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Does Art Make the Scientist?

Underwater silica streamers in New Zealand. Click to enlarge.

I recently came across an interesting quote by Robert Root-Bernstein, a MacArthur Fellowship "genius" teaching physiology at Michigan State, and the controversial author of Rethinking AIDS.

The quote is from a letter to the editor in the July 2006 Physics Today written by Kent Eschenberg

Most eminent scientists agree that nonverbal forms of thought are much more important in their work than verbal ones. This observation leads me to propound the following hypothesis. The most influential scientists have always nonverbally imagined a simple, new reality before they have proven its existence through complex logic or produced evidence through complicated experiments.

...I suggest that this ability to imagine new realities is correlated with what are traditionally thought to be nonscientific skills—skills such as playing, modeling, abstracting, idealizing, harmonizing, analogizing, pattern forming, approximating, extrapolating, and imagining the as yet unseen—in short, skills usually associated with the arts, music, and literature.  (Click here for full quote.)

Root-Bernstein investigates creativity and is a champion of the essential nature of the art-science interface.  (He himself is something of a digital artist.) In his Music, Creativity and Scientific Thinking, he goes even further than the above quote, clearly putting scientists and artists at the same level:

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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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To Drill or Not to Drill: Modeling Oil Production in the Arctic National Wildlife Refuge

Click to enlarge mapThe Arctic National Wildlife Refuge is a flash point for environmental and energy issues, combined in a volatile atmosphere of anti-terrorism. If only the U.S. could start drilling for oil there, the up to 8 billion estimated recoverable barrels of oil would surely lower our dependance on foreign oil, especially from the Middle East.

Or so proclaim those in favor of opening the refuge for drilling. (See ANWR.org - with a tag line of "Jobs and Energy For America" - for a collection of pro-drilling arguments.)

Currently, ANWR drilling is not permitted by federal statute. Over the past few years there have been several attempts in Congress to allow drilling - on some occasions from the House, only to be turned down by the Senate, and then from the Senate, shot down by the house.  (Click here for more details.)

Even though it appears as if those against drilling are currently holding their own, the no-end-in-sight Iraqi situation combined with the explosion of jihadist sentiment and activity could tip things enough so that both the Senate and House finally agree to open up the slopes for drilling. (Which is just one of the reasons that the upcoming mid-term elections, and possilbe change in majority party in both the Senate and the House, are extremely significant for the ANWR.)

The Sierra Club is one of the more public groups leading the charge against ANWR drilling.

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On the Increase in Greenland Ice Loss

Iceberg in Jacobshavn Isfjord, Greenland. Photo by Evert Wesker.Greenland holds an important place on the frontier - both of the habitable world, and of global warming. Greenland is where sudden climate changes have been mapped, a phenomenon that was one of the first markers of global warming to be widely accepted. (Click here for a previous post.)

Greenland is back in the global warming news: the rate at which ice is leaving Greenland (through meltwater and ice shearing) is apparently accelerating.

The study was done by Eric Rignot (NASA's Jet Propulsion Laboratory) and and Pannir Kanagaratnam (Univ. of Kansas Center for Remote Sensing of Ice Sheets.)

Ten years ago Greenland was losing ice at a rate of 22 cubic miles/year. It has now increased to 53 cubic miles/year. The size of this number is staggering. (22 cubic miles/year was already incomprehensible.) 53 cubic miles has a weight of approximately 2.4 trillion tons. To put this number in perspective, an asteroid that is totally iron would be over a mile in diameter to have the same weight!

Glaciers at the edge of Greenland have

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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.

Scientific Methodology vs. Prayer: And the winner is...

A fractal-like prayer rugMy post on March 19, 2006, described the Templeton Foundation, whose mission is to

"pursue new insights at the boundary between theology and science through a rigorous, open-minded and empirically focused methodology."

The Templeton Foundation appears to do a creditable job in following their mission. But what happens when scientific methodology meets faith-based belief and science wins (or appears to win)? The clash between religion and science, most recently in full view during the Dover evolution vs. intelligent design imbroglio, is much more subtle, but still inescapably present.

A case in point is the recent "rigorous, open-minded and empirically focused study" funded by the Foundation attempting to find the effect of prayer on healing. Titled STEP (Study of the Therapeutic Effects of Intercessory Prayer) , researchers studied the effects of prayer on 1,800 patients recovering from heart bypass surgery. The results, published in March 2006 and published in the American Heart Journal did not find any statistically significant effects.

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