I keep hearing a lot about fractals. Even the kids in high school are working with them. Can you tell me what fractals really are all about?
Response from Cliff Pickover:
That's an excellent question. These days computer-generated fractal patterns are everywhere. From squiggly designs on computer art posters to illustrations in the most serious of physics journals, interest continues to grow among scientists and, rather surprisingly, artists and designers. The word "fractal" was coined in 1975 by mathematician and IBM fellow Benoit Mandelbrot to describe an intricate-looking set of curves, many of which were never seen before the advent of computers with their ability to quickly perform massive calculations. Fractals often exhibit self-similarity which means that various copies of an object can be found in the original object at smaller size scales. The detail continues for many magnifications -- like an endless nesting of Russian dolls within dolls. Some of these shapes exist only in abstract geometric space, but others can be used as models for complex natural objects such as coastlines and blood vessel branching. Interestingly, fractals provide a useful framework for understanding chaotic processes and for performing image compression. As I mentioned in my interview on fractals, the dazzling computer-generated images can be intoxicating, motivating students' interest in math more than any other mathematical discovery in the last century.
I've played several adventure games on the computer and many of them are very in depth. Traditional computer games, however, become very uninteresting after you have played them because of their lack of variety. I've often wondered how fractals could be implemented into a computer game to give it infinite variety and therefore make the game of continual interest to the player. I know that this forum is designed more for discussion on the theory of fractals, but I can't help but think that one could find a very interesting use for fractals in the entertainment industry.
Response from Cliff Pickover:
Perhaps you might be asking about using fractals to create infinite, stimulating environments for adventure games. Of course, the idea of creating virtual-reality structures for human exploration is not new. In fact, in my books and articles I have discussed a variety of virtual reality journeys: computer-generated lava lamps decorating living room walls in the 21st Century, virtual vacations on Mars, electronic ant farms, and so forth. These examples not only please the eye but confound the mind with their complexity derived from simple rules.
Fractals have been used with great success for creating artificial yet natural-looking terrain. Take a look at the mountains on my website. How would you like to scale those Martian mountains described in my book Computers, Pattern, Chaos and Beauty!
Also take a look at my virtual caverns. The future of electronic spelunking is equally bright. Just as today we play 3-D interactive computer games like Doom or Quake, in the future we should look forward to exploring virtual caverns such as the ones I am beginning to explore. Who knows what odd geological formations we will encounter? If my simple algorithms generate lifelike and intricate formations, slightly more complex computational recipes will no doubt produce formations like those found in the Lechuguilla cavern: delicate helicite tendrils, calcite pearls, and gypsum beards. Like a submarine pilot exploring coral formations in the Sargasso sea, modern computers allow one to explore the strange and colorful caverns using a mouse or joystick.
How can you tell if something is a fractal, or if it is just random? Is everything that looks random a fractal?
Response from Cliff Pickover:
Kurt, that's an excellent question and gets right to the heart of fractals. There are a few definitions of fractals with subtle differences. Generally speaking, fractals include objects (or sets of points, or curves, or signals, or patterns) which exhibit increasing detail ("bumpiness") with increasing magnification. Many interesting fractals are self-similar which means that structures are repeated at different scales of size. For example, if you examine the image "The Fractal Wave" below, you'll see the same pattern repeated at different sizes. Although the pattern is quite irregular, almost random-looking in places, the self-similarity is evident.
B. Mandelbrot informally defines fractals as "shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps only after slight limited deformation."
You mentioned randomness. Randomness -- sometimes referred to as "noise" in various statistical, audio, and physics applications -- comes in many flavors including brown noise (such as traced out by a particle undergoing a random walk) pink noise, and white noise. Many kinds of noise distributions exhibit fractal characteristics.
Again, From Kurt Anderson:
According to your description of fractals then, wouldn't a completely random pattern be, in a way, the ultimate fractal? I mean, it looks completely similar on any length scale you choose to look at it. I admit, it's a kind of boring fractal, though.
From Cliff Pickover:
Kurt, I think you're on the right track when you say "it's kind of boring" when referring to randomness. The total absence of structure, such as in the visual noise produced by a TV set turned to an unused channel, is not very appealing from an artistic standpoint. Similarly, the vertical-bar test pattern on the TV set is also aesthetically uninteresting. In fact, the precise balance of randomness and order in a picture can often control the degree to which the human eye considers the pattern "beautiful". The computer is a tool which allows us to explore and produce art by playing with this mix of randomness and order.
Let's get back to your question about fractals. It depends somewhat on what you mean by "completely random." A particle undergoing a random walk leaves a fractal trail. This is called Brownian motion and it has a certain spectral characteristics that suggest it is a fractal. This is probably more complicated than we want to address here, but it is possible to compute a spectral density that gives an estimate of the mean square fluctuations at frequency f. There is a relationship between the slope of these curves and fractal dimensions. For Brownian noise, there are many more slow (low frequency) fluctuations than (high frequency) fluctuations, and its spectral density curve is quite steep.
When you used the words "completely random", you may be referring to pure white noise such as produced by a random number generator; here there is no correlation from point to point. Its spectral density is a flat line, representing equal amounts at all frequencies. Because of these considerations, many of the experts I've consulted do not wish to call white noise a fractal.
If you wish to delve further into this issue, you might enjoy books that discuss random data in the context of fractals, and how to characterize such data using power spectra and the concept of fractal dimension. One such book is Manfred Schroeder's Fractals, Chaos, Power Laws . Another is Heinz-Otto Peitgen and Dietmar Saupe's The Science of Fractal Images.
Can you make a fractal without using a computer program? Why area all of the picture of fractals that I see made on computers?
Response From Cliff Pickover:
Your questions about fractals and computers is a good one. You might enjoy the recent article in Odyssey titled "Three Historic Fractals". ("Historic" means B.C. -- before computers!) The article is by Francis Scheid, and it appears in volume 6, number 7, page 24-28.
To answer your question, "Can you make a fractal without using a computer program?", the answer is "Sure!" Nature makes them all the time. However, you're probably talking about making fractals with the human hand. Here, too, the answer is "sure." However, fractals display details at many size scales, so your hand would get awfully tired. But let me give you a few examples on how you might start. You can find the books mentioned below in your local library.
1. You can create the Sierpinski gasket simply by hand-coloring even values in Pascal's triangle. (See, for example, Chapter 7 of my book Computers, Pattern, Chaos and Beauty for all sorts of examples you could do by hand.)
2. You can draw Cantor dusts and Cantor Cheese as in Chapter 24 of my book Computers and the Imagination.
3. You can draw Koch curves by hand, as in Chapter 1 of my book Mazes for the Mind.
4. You can draw anabiotic Ana fractals as in chapter 5 of my book Mazes for the Mind.
Lots of artists, such as Roger Shepard, draw wild fractal forms -- such as embryos within embryos and faces with faces. (These are in chapter 5 of my book Mazes for the Mind.)
The point is that you can draw fractals by hand, but it is so much quicker and easier to capture the details using computers that never tire and that never make drawing mistakes. The computer is like a microscope opening a portal to a vast, unexplored and unpredictable universe. With many fractals, like the Mandelbrot and Julia sets, who would have thought that such beauty could be hidden within what was initially so lumpy looking? How could such a simple formula produce an inexhaustible reservoir of magnificent shapes and forms? The complexity exhibited by simple formulas correspond to behavior that even mathematicians could not fully appreciate before computers could display the results.
Arthur C. Clarke in the The Ghost from the Grand Banks notes:
"In principle the Mandelbrot Set could have been discovered as soon as men learned to count. But even if they never grew tired, and never made a mistake, all the human beings who have ever existed would not have sufficed to do the elementary arithmetic required to produce a Mandelbrot Set of quite modest magnification."
I hope this helps.
With organic-seeming forms emerging from very simple mathematic relationships, and ever increasing evidence (supported by HST results)that properties of self similarity apply to the universe as a whole, it seems that chaos theory almost proves that it is not only likely that life occurs throughout the universe, it is inevitable. Do you agree?
Response From Cliff Pickover:
My personal belief is that life exists on other worlds, although I believe that chances for finding technologically-advanced, spacefaring creatures is very unlikely.
Any discussion of other forms of life does not address the question, "What is life?" In fact, the very contemplation of alien life forms begins with this question. One might identify life as anything that ingests, metabolizes and excretes, but this description might apply to a car, rust, or a candle flame. Other sophisticated definitions recognize life as a departure from thermodynamical equilibrium, but much of nature (like lightning and the ozone layer) is out of equilibrium. Biochemical definitions of life that require proteins or nucleic acids seem restrictive. For example, if we found an alien worm that could do everything a worm could do on Earth but was made of different molecules, certainly we would not declare it "lifeless." In the final analysis, most definitions may be impractical on other worlds.
We do have some ideas on how quickly life evolved on Earth. Some say the Earth formed through an assembly of ancient "planetesimals" around 5 km in radius. These chunks began crashing into one another, producing fragments that in some final "Great Bombardment" assembled into the planets of today. On Earth, primitive life originated very soon after the Great Bombardment which ended about 3.8 billion years ago. Numerous fossil evidence suggests that primitive life was already well established on Earth 3.5 billion years ago. From studying Earth's geological history, it seems much easier for primitive cells to evolve from organic chemicals than for multicelled creatures to evolve from single-celled creatures -- because multicelled creatures do not appear in the fossil record until less than 1 billion years ago.
If simple lifeforms exist on a planet, what are the chances they will evolve into higher organisms like humans? During the evolution of life on Earth various catastrophes have taken place, such as the one that caused the destruction of the dinosaurs or the one that killed 80% of the marine animals during the Middle Cambrian period (about 515 million years ago), each of which cleared the Earth for a burst of evolution in new directions. It's unlikely that these chance events were replicated anywhere, so life cannot evolve exactly as it has here on other worlds. However, once the spark of life is ignited, it will flame into whatever crack or niche that is available to it, over and over again, leading to a conflagration of different creatures.
Someday in the not-too-distant future we will find life on other worlds. The fact that life emerged on Earth suggests that it exists in other parts of the cosmos because the elements of which the entire universe is composed are remarkably uniform. If some of the elements have combined in ways that produce life on Earth, it is likely they have combined in similar ways elsewhere. We have every reason to believe that there are other water-rich worlds in the universe with complex organic molecules. This means that there should be many worlds in the Milky Way capable of supporting simple life forms. Even as you read these words, there must be planets in other galaxies on which life is just emerging or even flourishing. Just as you blink, some new lifeform is arising.
We have seen that life is built into the chemistry of the universe, poised to evolve wherever conditions are right. If we discover advanced lifeforms in the universe, far from demoting humanity to inferior creatures, this discovery would give us reason to believe that we are part of a grander process of cosmic organization and hope.
In determining the type of fractal to use to model some application, is there any natural phenomena or evidence that can be used in the determination? Or, has it just been coincidence that some fractal gives insight into a particular problem?
Response from Cliff Pickover:
It is not clear to me what you mean by "type of fractal." Various iterative procedures can be used to produce and model natural formations. "Whatever can be done once can always be repeated," begins Louise B. Young in The Mystery of Matter when describing the shapes and structures of nature. From the branching of rivers and blood vessels, to the highly convoluted surface of brains and bark, the physical world contains intricate patterns formed from simple shapes through the repeated application of dynamic procedures. Questions about the fundamental rules underlying the variety of nature have led to the search to identify, measure, and define these patterns in precise scientific terms. Our seemingly chaotic world is actually highly structured. From an evolutionary standpoint, biological themes, structures, and "solutions" are repeated when possible, and inanimate forms such as mountains and snowflakes are constrained by physical laws to a finite class of patterns. The apparently intricate fabric of nature and the universe is produced from a limited variety of threads which are, in turn, organized into a multitude of combinations.