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:

Click to read more ...

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

Click to read more ...

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

Click to read more ...

Update to Reflections of Chaos

Originally Posted by Jeremiah Noll

Click on image to enlarge

At the beginning of this course I was much more certain of things! I have learned quite a bit, but with this learning came new mysteries. Studying chaos has expanded my mind and taught me things I never would have dreamed. When you sit and think about the concept and the principles which founded chaos it just makes sense. Butterflies can start El Niño's and the stock market is predictable to a small degree. But sensitive dependence on initial conditions is not the only principle of chaos, there is also the fact that extremely simple rules can produce infinitely complex results. Who would have thought a couple lines of computer code could produce the Serpinsky Triangle or model the population growth of a small nation. But the coolest things I have ever seen are fractals. Through this course I have developed a great understanding of the image, mapping, and product of fractal code. Simple functions and an imaginary plane can model infinity in the most accurate and beautiful way. I enjoyed fractals so much I decided to make them my semester project. The value of this course is a tremendous amount of insight into how the world works and how things thought to be random can really be very simple. I feel as if I have been shown a piece of how God put this universe together and how he made it so complex in only six days. Like deciphering the human genome code chaos makes us more able to predict, heal, and understand the way things work and why events happen.

I still use fractal software today and it is still interesting and novel even though I have been exposed to it for some time. I still have my own fractals on my website so that I can, hopefully, get others interested in math and science too. My friends never thought Math could be so intriguing. Since the Fall semester I have progressed in the ability to create lifelike landscapes from fractal software.

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.

Extending the Chaos Game: Determinism and DNA

HIV Color Game

I recently came across some interesting variations and applications of the Chaos Game.

One version is located at the Boston U web site and is credited to Johanna Voolich and Robert L. Devaney. Their version is very unique twist on the Game - instead of the randomness that forms the basis of the Game, they have replaced the randomness with a strictly deterministic game. In their version, the player tries to get a random starting point into a specific target area of the Sierpinski triangle by choosing a series of half-way jumps to specific vertices. The "winner" of the gme is the one who reaches the target area in the fewest number of jumps. (Click here to play the game on-line.)

An even more fascinating Chaos Game comes from Dan Ashlock at Iowa State University, where he uses a 4-cornered Chaos Game to display the sequence of bases in DNA molecules. (The image at the top of this post is from a segment of HIV DNA.) In this Chaotic Game of Life, points are drawn half-way to vertices depending on the next base in the DNA strand. The pictures look remarkably similar to playing the Chaos game randomly! See Ashlock's website for a set of images from different organisms, as well as some Markov chain models that help explain and interpret the images.

The Chaos Game: 3rd Grade vs. University Faculty

I just finished two separate presentations of the Chaos Game exercise that I typically use to start the Honors class to students at Arcdia University. Most of the students were education majors - and most of them were elementary ed majors. I am always pleased at the reactions to the game - especially from students who believe, or claim to believe, that they are not good at mathematics. The Chaos Game is such a rich exercise, especially given the approach of Understanding vs. Prediction, that most get caught up in the inadequacies of prediction, and the frustration of not being able to get a real comprehension of how the patterns (and colors) arise.

But the best part of the exercise, for me, is the ability to run it in almost any setting - from a 3rd grade classroom to a university mathematics department. Each group will get something different from the exercise. Personally, I prefer the response of the 3rd-graders, who , unlike mathematics faculty, don't try to understand what is happening at a deep mathematical level. Instead, they react purely to the shapes and colors, and their inquisitiveness makes for a wonderful teaching moment.

Education majors, and especially elementary education majors, are also a terrific audience for the Game as we play it. I believe that most of them could see how the exercise, or something similar, can be transported to their own future classrooms!

Melt-O-Waves

The fractal image appearing in the site header was produced using Ultra Fractal 3.05. The original image was designed by Damien M. Jones in 1999, and published to the Ultra Fractal Formula Database

Click to enlargeThe image is titled Melt-O-Waves. I varied the colors, zoom factor, layer blending method, and starting values for the underlying Mandlebrot set.

Please visit fractalus - this is an amazing site of fractal art that is managed by Damien Jones.

Fractal Image Generation, Part 1

The fractal images generated for this blog have been generated using UltraFractal 3.05 - an incredibly robust package that is effectively the PhotoShop of the fractal kingdom.

In addition to all possible types of mathematical settings, UltraFractal allows layering (as in PhotoShop), with different blending styles adjustable (e.g. addition, difference, hue, etc.) The interface allows the designer to create or edit fractal formulas, set up color palettes, and add mathematical image transformations. There are many pre-loaded fractal types loaded - even with these you will have days of exploration as you vary coloring schemes and learn layering with fractals.

There is also a very active user group that uploads new fractal parameter files and color templates - all available for download. You can find appropriate links, and much more at Janet Parke's Ultra Fractal Resource Page.

The result is a degree of image control that is state of the art.

UltraFractal also contains a rendering engine with anti-aliasing for producing high-quality images for web and print applications.

You can try UltraFractal free for a brief period.

Note that ChaosPro is almost as powerful as UltraFractal and is freeware! Check it out!

For a good source of chaos and fractal links check out third.apex.to.fractovia by Juan Luis. This is a very nice site, and Juan is clearly interested in the fractal aesthetic:

Have in mind that this site, third.apex.to.fractovia, focuses on the artistic side of fractals, not in their mathematical aspects, so these applications are primarily intended for picture or music creation, not for studying fractals as mathematical models or for in-depth experimentation (what most of them can do in any event).

Note that some of the software generates fractal music.

Is Fractal Art Really Art?

Originally Posted by Jeremiah Noll

"grassyknoll"
"orangefla"
"hazyire" Fractal art is a controversial matter, but what in the art world isn't controversial? Contrary the opinions of some dissenters fractals are not easy to make, there are infinite possibilities, it takes a great deal of talent, and it is in no way cheating. Fractal art is an innovative art form and it will likely be the next art fad. Some of the greatest art was at some time controversial.

The fractal software on the web that I recommend is Sterling Fractal Generator, Chaos Pro, and Terragen.

These are also free and can be downloaded with little trouble. You can also check out the fractal portion of my website.

By the way they are both unreal.