{cov95.gif} The 1994 Calendar

The 1994 calendar was edited by F. Kenton Musgrave, and published by Universe. It features an essay by Musgrave and contributions from John C. Hart, Daryl H. Hepting, Henry Kaufman, Craig Kolb, Przemyslaw Prusinkiewicz, Dietmar Saupe, Karl Sims, and Kevin Suffern.

Light Lace

by Kevin Suffern


One of the most striking aspects of fractal geometry is the underlying simplicity which generates the visually-complex fractal forms. In this case, an approximation of the Sierpinski Gasket (as seen on the title page of this calendar) is generated by the interreflection of light between four spheres! The reflection model employed is not like the "real world": the light gains energy in each reflection, as opposed to losing energy.

Here we see the four spheres from the outside. The point of view for the main image is inside of the tetrahedron formed by the four spheres. Blueglass [not shown] is formed by 100 interreflections of light from a shiny blue sphere placed inside a shiny chrome sphere. The complexity here is also fractal (i.e., forms repeat over a variety of scales), but more subtly so. It is remarkable that such simple systems can give rise to fractal complexity!

The Five Non-Platonic Non-Solids

by John C. Hart

The five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Here they are transformed into fractals by recursively removing polyhedron-volumes from every solid part, generating Sierpinski's tetrahedron, octahedron and icosahedron, Menger's sponge (in green), and the (reddish) von Koch snowflake-a-hedron. As every solid portion has been reduced ad infinitum, the result has zero volume, and infinite surface area! Thus the moniker "non-Platonic non-solids".

The non-Platonic non-solids and this fractal forest are modelled using iterated function systems in which simple transformations are repeatedly applied, to place particles in space. The sum of these particles is one of the many forms seen here. Thus these forms are composed of vast numbers of very small particles. In true fractal form, the shapes we see are repeated over a range of different scales or sizes. The pulled-taffy shape of the quaternion Julia set is the result of a different type of simple iteration, performed in an extension of the complex numbers called the quaternions. The result is a four-dimensional fractal, which is projected down into three dimensions, then to the two of the image plane.

Anything but Blue

by F. Kenton Musgrave


A problem with computer models of landscapes is that they are entirely abstract ‐ just a bunch of numbers ‐ and have no intrinsic scale. If one simply projects them onto the image plane, the result will appear to be no bigger than the image itself. Landscape painters have known for hundreds of years that a critical distance cue for large scales is aerial perspective, or the bluing and loss of contrast over large distances. In Nature, this is due to complex effects of scattering of light by the atmosphere. Simplified models of such atmospheric scattering can impart a sense scale to terrain renderings, as Anything but Blue demonstrates.

In the tradition of painters, we often render the same scene in different light, to evoke different moods and visual effects. Both Lethe [not shown] and Cool Afternoon [not shown] are color studies of the mountain seen above, informed by the color use of both past and contemporary schools of landscape painting. To achieve greatness in its own right, computer art will have to stand on the shoulders of the giants who have come before. Interestingly, though, computer graphics reopens the artistic problem of representationalism, which has been regarded as a solved problem in the visual arts since the invention of the camera. Setting out to synthesize images of arbitrary realism from first principles we face a whole new set of problems, most of which yet await satisfactory solution.

Tea for 66,430

by Craig Kolb (with thanks to Eric Haines)


Let it never be said that fractalists are without a sense of humor! This ray-traced image was carefully constructed to contain as many computer graphics cliches as possible. A swarm of 66,430 mirrored versions of the famous-to-computer-graphics Newell teapot floats in a fractal arrangement. The large, central teapot is surrounded by nine smaller teapots, each of which is surrounded by nine smaller teapots, and so on, to five levels of recursion. The sum total of teapots is composed of over half a billion triangles (ray tracing an awful lot of triangles is a common effort in computer graphics). The teapot, shininess, shadows, fractal arrangement, the procedural fractal texture on the floor, even the scene itself are cast as cliches of computer graphics in this inside-joke-to-the-trade. This image was created in response to the "call for teapots" for the 1989 SIGGRAPH computer graphics conference, which chose the venerable Newell teapot for its logo.

The overall image is a send-up of image Sphereflake by Eric Haines. That image was designed as a standard benchmark image for comparing the speed of various ray tracing algorithms and programs, thus it in itself it something of a cliches. The Incandescent Teapot [not shown], by F. Kenton Musgrave, was created in collusion with Craig Kolb in response to the same "call for teapots"; both it and Kolb's image were rendered in stereo pairs to add a dimension of cheap 3-D horror.


by F. Kenton Musgrave (with thanks to Lionel Woog and Benoit Mandelbrot)


An area of current scientific research in fractals is the study by physicists of diffusion-limited aggregations or DLA's: the structures that result when particles moving about on random walks stick to one another when they make contact. This image represents the use of state-of-the-art computer graphics effects in service of scientific visualization. Today's powerful computers can generate such vast quantities of information that scientists can only deal with it visually. (Of all our senses, vision provides by far the greatest bandwidth for getting information to the cerebral cortex.) In this case, a computer simulation has generated a shape -- a three-dimensional DLA -- which is so complex that it is hard to render a way which clearly conveys its spatial structure. An atmospheric function (the same one which surrounds the earth in Gaea & Selene) has been centered on the DLA, the shiny spheres of which reflect its glow. The color of the spheres reflects the time at which they joined the cluster, telling us something about the growth history of the cluster.

Here we see the same scene rendered in a different way: the emphasis here is not on scientific visualization, but rather on aesthetics. Again, the atmosphere function is used to disambiguate three-dimensional structure after projection to a two-dimensional image, and the atmosphere is modulated to emulate a range of incandescent black-body temperatures. Understanding the growth of DLA's is a prerequisite to understanding phenomena as diverse as the efficient extraction of oil from porous rocks underground and the formation and structure of both soot and interstellar dust particles. Below is a rendering of the Laplacian potential around a portion of a 2-D DLA cluster. The grey scale represents the value of the potential, and each cycle of grey represents a drop by a factor of 10 in the value potential. The source at the top is at potential 1.0 and the DLA boundary is at 0.0. The arborescent structure of the fjords is still a matter of controversy among physicists: it is not yet decided whether they are exactly self-similar, or whether the fjords are parallel, or wedge-shaped. The green lines are field lines emanating from points on the DLA boundary, ascending to the source, while the red lines start along a horizontal line and descend towards the boundary by steepest descent of the potential. Note that ascent is convergent, but descent is unstable in the fashion of deterministic chaos: it exhibits sensitivity to initial conditions (in this case the starting point of the path).


by F. Kenton Musgrave (with thanks to Karl Sims and Robert Alec Cook)


A genetic algorithm developed by Karl Sims was used to generate this image on a massively parallel supercomputer In the genetic algorithm, expressions which generate surface textures from the LISP programming language are randomly generated and "mutated" by the computer, and "selected" and cross-bred by the user. This powerful random process rapidly creates striking images which are entirely procedural in nature; that is, they represent the unmodified output of the computer program. Splash features a pattern "painted" on a fractal surface. The coloring, the pattern, and the fractal surface texture all arose spontaneously in the process of "aesthetic selection".

Splash and these other fractal images were all created over the course of several hours of "work" by a previously-inexperienced user. The power of the process which created these images lies in its automaticity: in principle, and person using the system could have come up with these images (or better), as it is driven by simply choosing "what you like" and having the computer "evolve" that. Its appeal lies in the parallel to Darwinian evolution which, presumably, created all life on Earth through a similarly arbitrary process of mutation and selection.


by F. Kenton Musgrave (with thanks to Craig Kolb)


One of the fundamental problems with making computer graphics pictures of fractals is that their high spatial-frequency content, when combined with the point-sampling techniques used in image synthesis, can lead to aliasing or "noisy" images. Another problem with fractal landscapes is that a very detailed model requires a lot of storage space, yet it can have too little detail in the foreground, and too much in the distance (leading to visual noise there). A solution to these problems is procedural modelling, in which the terrain model does not exist a priori, but rather is calculated the appropriate level of detail -- on-the-fly, as the image is made. Slickrock is an example of such a landscape. Note that it features sharp ridge lines, which would appear to have saw-teeth if the triangles tessellating the surface were much bigger than pixel-sized. Slickrock uses the aerial perspective model first tested in Anything but Blue (see March) to impart a sense of very large scale.

The landscape seen in Slickrock is the destination in an animation currently under production, a short sort of "Powers of Ten" zoom from deep space. The planets seen in Gaea & Selene will grow from points in the far distance, as we zoom up to it. Then we will fly down through the atmosphere (as seen in the detail below) and up to the mountains seen in Slickrock. The procedural approach allows the models to be generated with appropriate spatial frequency content for its distance from the eye point (i.e., without too many tiny details, which lead to aliasing or visual "noise" in the image). Furthermore, the nature of procedural fractal models allows us to investigate the scene as closely as we like -- the program just keeps adding as much detail as is required. All features, both large and small, arise from the underlying randomness of the fractal functions, therefore the artist is not responsible for explicitly specifying any of them. Thus the artist who creates the scene may be as surprised by what is to be found there, as anyone else! When implemented in virtual reality, this promises exciting explorations of a "virtual" universe filled with such "virtual" planets.

Desert Ball

by F. Kenton Musgrave


In all of the landscape images seen in this calendar, procedural textures are used to color the surfaces and to add visual detail. A procedural texture is a mathematical function which is defined throughout three-space, and evaluated at the points where the rays emanating from the eye (in the ray tracing rendering method for generating synthetic images) intersect surfaces in the scene. In the course of development of one of these functions, it is generally rendered on the surface of a sphere, to see how it looks. A sphere is used for two reasons: it is one of the quickest objects to ray trace (ray tracing being a notoriously slow procedure), and it shows what the texture will look like at all angles of illumination. While such images are simply tests along the way to a specific visual goal, sometimes they represent striking images in their own right. Desert Ball came about while developing the texture used on the terrain in Slickrock (see July). All that was added to the test image is the burgundy background and the faint light source coming from the right.

Desert Ball and Earth Sequence [not shown] both employ two different uses of a vector-valued fBm (fractional Brownian motion) function. The vector-valued function simply returns three separate values for independent fBm functions. This 3-vector can be interpreted however we please. Interpreted as a spatial vector, it can be added to the surface normal (the vector sticking straight up from the surface) before calculating the lighting there, which of course depends on the surface's orientation to the light source. This method is called bump-mapping. Applied to the large burgundy sphere in the background of Desert Ball, it generates the rough-looking highlight to the left. In Earth Sequence [not shown] it is used to generate oceans and continents. Another interpretation of the fBm vector is as an rgb (red, green, blue) color value, which can be added to the basic surface color. This perturbation gives rise to the subtle color variations in Desert Ball and in the last sphere of Earth Sequence [not shown]. Fire Ball [not shown]is another test image from the development of an fBm-based fractal flame texture.

Spirit Lake

by F. Kenton Musgrave


Fractional Brownian motion, or fBm for short, is (loosely) defined as the integral through time of the increments of a random walk with Gaussian increments. It is a random fractal function which resembles many things in Nature: the flood levels of the Nile through time, stock and commodity price fluctuations, the transparency of a cloud along a line, and the skyline of mountains, to name a few. Some years ago, Benoit Mandelbrot noted the latter resemblance and reasoned that fBm extended to 2-D to form a surface would resemble a mountain range. Thus were born fractal mountains for computer graphics. But the resemblance of fBm to mountains has no known causal basis; fBm simply looks like mountains. A research area, then, has been to improve and augment the descriptive power of fBm-based terrain models. Spirit Lake is an early illustration of a terrain model in which the fractal dimension and crossover scales varies with altitude. Thus the mountain is smoother at the lake's edge, and rougher near the peak. The triangles visible near the peak show the scale of the geometric detail; smaller details are visual effects generated by a procedural texture, which does not vary roughness with altitude.

Carolina [not shown] is a different use of the same heterogeneous terrain model, this time used to emulate ancient, heavily eroded mountains such as the Blue Ridge Mountains in the American Southeast. Medicine Lake [not shown] features the same mountain seen in Spirit Lake, but with a different procedural texture applied to the surface. Both Spirit Lake and Medicine Lake [not shown] feature a model of the rainbow which, unlike the mountain models, is based on the laws and measurements of physics and geometric optics. It was created by recreating with a computer, a simulation of the propagation of light through an idealized raindrop, first published by Rene Descartes in 1637. This was coupled with a model of dispersion (discovered by Sir Isaac Newton a few decades later) to generate an accurate model of both the position and colors of the rainbow.

Dragon Mesa

by P. Prusinkiewicz, D. Saupe, and Daryl H. Hepting (with thanks to Allan Snider)


This landscape, perhaps from some distant planet, illustrates a continuous escape-time function evaluation for an iterated function system (IFS) of the famous dragon curve. This image may seem similar to some of the familiar images created for Julia sets, and this is not by accident. In fact, this picture was created as part of a study which developed rendering functions for iterated function systems analogous to those which had previously existed for Julia and Mandelbrot sets. In seeking out similarities between Julia sets and those defined by iterated function systems, differences were observed as a product of the manner in which the fractals are defined (with quadratic or linear transformations, for example). The IFS escape-time function does not have the same interpretation as does the function for Julia and Mandelbrot sets. The values in the IFS case can interpreted as indications of membership in the fractal: the points with highest elevation are still candidates for membership in the attractor, whereas lower elevations have already been discounted. The discrete escape-time function values indicate boundaries between successive transformation applications over some region of interest.

The left column of images [not shown] illustrate the continuous version of the escape-time function over a small initial region of interest, performing calculations with the Manhattan metric (top) and the more familiar Euclidean metric (bottom). The right column [not shown] shows a two-dimensional view of the continuous escape-time values used in the Dragon Mesa picture (top), an example of an image using the discrete escape-time function (middle), which is much simpler to compute. The bottom right image shows an image of a height field derived from a distance function, and it serves to illustrate some of the differences between the two functions.

Other State

by F. Kenton Musgrave


We can use the descriptive power of fractals to create visions of a Nature-that-never-was. Other State is an example of such an image. It was composed from the well-known image Blessed State [not shown], which appears below, and a fractal model of a Saturn-like ringed planet (also seen below). The scene is rendered from fully three- dimensional models, the likes of which we will be able explore in virtual reality, when the capability to render such scenes at interactive rates has been developed. Currently, it takes about an hour of supercomputer time to synthesize such an image.

Here we see the components from which Other State was composed. The image Blessed State features an early fractal model of a moon: a simple grey bump-mapped sphere. This moon has been replaced by the more sophisticated model of a Saturn-like planet, and a more complex model of atmospheric scattering has been employed.. The planet features a fractal model of the bands of clouds and a fractal ring structure. The structure of the rings is simply fBm (fractional Brownian motion) applied as a transparency map to a disk around the planet. No attempt to model features of Saturn's rings -- such as Cassini's division -- was made; they simply fall out of the fractal model.


by F. Kenton Musgrave (with thanks to John Amanatides)


For the holiday season we return to the theme we started with in January: ray tracing shiny spheres, which look like Christmas-tree ornaments. Orbit uses the same subtle color and surface modulations seen in Desert Ball (see August). The burgundy sphere is exactly the same one seen in the background of Desert Ball, while the two other spheres are a shiny and copper-colored version of the foreground sphere seen in that image. Beneath is a very large sphere with a simple fBm texture modulating lightness -- close inspection of this texture will provide a good sense of the character of the fBm function which underlies most of the models of natural phenomena seen in this calendar. The intent of Orbit was to tackle the artistic challenge of creating a "sense of place", using only four ray-traced spheres with sophisticated procedural textures. (There is actually a fifth sphere in the scene, a white one representing the Sun, seen only in reflection.) It, like Desert Ball, is a subtle color study, which will be lost if the printer doesn't get these images exactly right.

It is remarkable what we can do with ray-traced spheres which are perfectly smooth, geometrically. Moonrise [not shown] demonstrates that two spheres -- one white and one chrome -- with two different fractal fBm bump-maps, can alone provide a model of a natural scene. The cover image is composed of two spheres, one for the planet surface and one for the clouds just above it, an atmosphere function and a background image of the Mandelbrot set. The visual complexity and fidelity to Nature of these images belies the conceptual simplicity behind their generation; therein lies the primary wonder of fractal geometry.


by F. Kenton Musgrave
Yale University


A basic tenet of Western science is that the behavior of systems in nature can be accurately described with mathematics. In 1623 Galileo wrote: Philosophy is written in this grand book -- I mean universe -- which stands continuously open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.

Scientists today still regard this statement as essentially correct. However, Galileo's language of nature -- that attributed to Euclid of ancient Greece, and known today as Euclidean geometry -- was a bit short in vocabulary. Geometry is a language of shapes, but many of the shapes in nature lie in Galileo's vague category of "other geometrical figures", not within the scope of Euclid's shapes. They require another kind of geometry, one unimagined in Euclid's or Galileo's time.

Recently the mathematician and natural philosopher Benoit Mandelbrot observed: Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line... This observation heralded his introduction in 1975 of the new branch of mathematics known as fractal geometry.

Fractal geometry can be thought of as a new dialect of the mathematical language of shapes (geometry). Unlike Euclidean geometry, fractal geometry deals with the very complex shapes found in nature: things like trees, river networks, and billows of smoke. The difference between Euclidean shapes, such as planes, spheres and cones, and fractal shapes is that Euclidean shapes are all locally flat: if you look at them closely enough, they become flat, planar, and boring, while fractal shapes can be complex at every scale, that is, they may possess an infinite wealth of detail.

Where does this complexity come from? It could come from two places: there could be a vast quantity of essentially independent information -- such as the nicks, scratches and stains on an old pair of tennis shoes -- or there could simply be a repetition of an underlying form, at a variety of scales. The latter type of complexity is the secret of fractal geometry: self-similarity, or the repetition of form over a variety of scales. Note that the former kind of complexity is hard to conceive of except as the result of an accumulated history of independent events over a substantial period of time, while the latter -- fractal complexity -- can often be succinctly described by a simple specification of the underlying shape, plus the relation of its manifestation to the scale at which it is manifest.

Many forms in nature are a combination of both types of complexity; the addition of the fractal vocabulary of shape to our scientific language of nature has revolutionized our ability to describe, in the internally-consistent, formal and deterministic language of mathematics, the forms and order we find in nature. The images in this calendar -- particularly the fractal "forgeries" of natural scenes -- attest to the success of fractal geometry as a language of nature. Even the wholly abstract fractal images have a "natural" feel to them; they echo the complexity of form so common in nature.

It is interesting to contrast the places where we find Euclidean versus fractal shapes: Except for the near-perfect spheres and ellipses of soap bubbles, astronomical bodies (planets and suns) and their orbits, it is a bit unusual to find Euclidean shapes in nature -- they generally appear in the domain of man-made objects, such as buildings and machines. Fractal objects are generally too complex to be explicitly constructed by man; nature, on the other hand is full of them: mountains, trees, clouds, and fractures (indeed, the word "fractal" intentionally shares the Latin root of "fracture"), to name but a few. Hence it is appropriate to think of fractal geometry as a "geometry of nature".

Some artificially-generated fractal shapes appear to imitate phenomena in nature such as mountains, clouds and trees. Others appear regular and man-made, such as the Sierpinski Gasket and the von Koch Snowflake. Yet others, such as the Mandelbrot and Julia sets, appear to be entirely abstract in form. Yet all fractals share the characteristic of self-similarity: they appear more or less the same at a variety of scales. This self-similarity may be exact, as in the case of the Sierpinski Gasket and the Menger Sponge; it may be statistical, as in the case of fractal mountains; or it may be more difficult to characterize, as with the Mandelbrot set. Although the character of the self-similarity differs among fractals, they all have in common the fact that when you look at them ever more closely, you see more of the same, and they all have in common a potentially-unlimited complexity which is a result of this repetition of form over a potentially-unlimited range of scales.

Fractals also share a common mathematical characteristic: the fractal dimension. The fractal dimension is an analytic measure of the "wigglyness" of a fractal line or the "roughness" of a fractal surface. It is a number which agrees with our everyday sense of dimension (three dimensions define space, two dimensions a plane, one dimension a line, and zero dimensions a point) but which has non-integer values: for the Sierpinski Gasket it is ???; for the von Koch Snowflake it is ???; and for fractal mountains it is usually between 2.1 and 2.3. The larger the fractal dimension, the more wiggly the shape, and as the fractal dimension of, for instance, a line approaches the integer value of the next higher dimension (in this case, going from 1.0 to 2.0), the fractal curve becomes space-filling, that is, it fills the entirety of some part of the next higher dimension (in this case, a plane). Not an immediately intuitive notion, this idea of fractal dimension, but it becomes so surprisingly quickly -- one comes to think of the fractal dimension as simply a numerical measure of just how convoluted the curve or surface is, and with some practice one can estimate its value fairly accurately just by eye. Cumulus clouds, for instance, have a fractal dimension that's usually about 2.2 to 2.3.

A Bit of History

Around the turn of the century, there occurred something of an upheaval in the world of mathematics. Mathematicians such as Weierstrass, Cantor and Peano conceived of bizarre constructions: strange "dusts" of unending complexity; functions with wildly unpredictable behavior; curves that could fill space or have no tangents. They boasted that these entities had no counterpart in nature and were somehow inherently intractable; having discerned what they could about them, the mathematicians labeled them "monsters", ill-begotten and unwanted children of mathematical speculations. Interestingly, while they had studied the beasts formally, they were never concerned by what they actually look like and pictured only the simplest among them. In general, the tedious and copious calculation required to make such a pictures would have to await the invention of the modern digital computer.

As a young man in the 1940's, Benoit Mandelbrot had an uncle (also of the name Mandelbrot) who was a well-known mathematician in his own right. One day he showed the young Benoit a mathematical paper by Fatou, telling him "there is a career to be made in this, for the person who can figure it out and pursue this work". Young Benoit had a look at the paper, and concluded "this is definitely not for me". Over the years, however, the ideas germinated in the back of his mind, and he came to be working on problems less far-removed than he himself realized, at the time. As is usually the case with scientific discovery, fractal geometry took form in his mind as if by a slow process of awakening, rather than an instantaneous "aha!"

Mandelbrot was employed by IBM as a research scientist in 1958. This gave him ready access, in the 1960's, to the then-exotic digital computer. Armed with the requisite mathematical insight, the computer provided the tool he needed to explore the latter-day "monsters" of mathematics and discern something more of their true nature. What he found is illustrated in the images in this calendar. Fractals and the computer are inextricably intertwined. While ordinary mathematics often takes the form of equations such as which may be solved explicitly, fractals are generally specified in terms of recursive procedures such as, that is, the mathematical operation is applied repeatedly to its previous result. This is a recipe for tedium, for a human calculator, but it is exactly the kind of thing a computer does best: vast quantities of relatively simple operations. The fractal equation is then:
[simple operations] times [a mind-numbing number of repetitions] = [mind-boggling complexity].

The "simple operations" term means that fractals are relatively easy to program into the computer. The "mind-numbing number of repetitions" term means that the computer will do it much faster, more accurately, and with less pain and complaint than a human. The result -- "mind-boggling complexity" -- was a complete surprise, and means that we need to invoke special measures to deal with the problem of making sense of the results of the computation.

It turns out that the way we human beings are wired up, of all our senses, vision provides by far the greatest bandwidth (amount of data per unit time) for getting information to the cerebral cortex, to the higher brain centers. Then, rather than looking at the results as a bunch of numbers (which is how the computer deals with them), we can make pictures out of them. The original fractal computer graphics mark one of the earliest uses of the computer for the purposes of scientific visualization. In fact, fractal computer graphics are being used even now in the service of making scientific discoveries: the images make possible the recognition of patterns and relationships which would otherwise be lost in floods of computational data. Serendipitously, the complexity inherent in these synthetic fractal images often comes hand-in-hand with an astonishing beauty, making accessible to every person some of the much-touted abstract beauty of pure mathematics. Indeed, some fractal geometers would claim that the images provide a sort of intuitive, visual proof of the existence of such beauty.

Fractals in Science and Philosophy

What, one might ask, are fractals good for? Well, obviously, they generate some convincing models of natural phenomena such as mountains and clouds for use in computer graphics imagery, and they provide some very compelling abstract pictures. But recently, something like one third of all physics papers submitted to journals for publication at least mentioned fractals somewhere. The fact is, fractal geometry is so new on the scientific scene that its uses are still being puzzled out. The modern philosopher Martin Heidegger argued that language itself allows for, even generates a world. If this is true, we can expect a fundamentally new language with a hitherto-untouched domain of expression to generate a new world view. Fractal geometry is in the process of doing just that.

Perhaps the most profound impact of fractal geometry to date is in the new science of chaos. Scientists have recently discovered order in natural systems, where previously there had seemed to be none; the language of fractals provides the vocabulary with which they can speak of this order; without it, that order probably would have remained unrecognized. The science of chaos deals with the behavior of nonlinear dynamical systems, that is to say, "equations that model natural systems well, and how they evolve through time". Scientists have long used linear approximations to nonlinear systems as a matter of mathematical expedience. The nonlinear mathematics models nature more accurately, but is intractable in comparison to the linear approximations. When computers made it possible for scientists to begin to cope with these previously-intractable nonlinear systems, they discovered something very surprising: they call it deterministic chaos or sensitivity to initial conditions; it means that any perturbation to the initial state of the system, no matter how small or seemingly insignificant, will cause the system to diverge; i.e., to evolve into an arbitrarily different future state, within a finite period of time.

This is a counterintuitive notion: we would expect systems that started off in very nearly the same state to continue, forever, to evolve upon reasonably parallel paths. Not so, we find, and this has profound philosophical consequences. In the 1600's Descartes and Newton, as natural philosophers, fleshed out a world view so compelling that, if the average educated person in our society today stops and thinks about it, it seems to be "the obvious way that things are". In the Cartesian universe with Newtonian dynamics, if we knew 1) the position and velocity of every particle in a closed system and 2) the rules for their interactions, and we had sufficient power to compute all those interactions, we would have the power to predict the future, forever, for that system. Obviously correct, right? If our "closed system" were the entire universe, this would have profound philosophical implications: there could be no free will, it implies that we are all witless automatons, mere puppets in some sort of deterministic, already-written cosmic script. It would affirm the nihilistic philosophy of fatalism, and undermine the basis of human morality: that we have a choice in matters, and that what we choose to do -- and not to do -- makes some kind of a difference.

Perhaps fortunately, our century has seen a series of repudiations to this deterministic model of the universe. First, Einstein dealt a blow to Cartesian geometry with his theories of relativity: space/time is curved, and a line is straight or curved relative to the observer's frame of reference. Then came quantum mechanics, wherein the Heisenberg Uncertainty Principle states that, for particles on the subatomic scale, we can know their position or the velocity, but not both -- this torpedoes the first premise for computing the future in a deterministic universe. Very recently -- within the last 15 years or so -- the science of chaos has driven a second nail into the coffin of the deterministic universe: Suppose we did know the position, velocity and rules of interaction for all particles. Then any error, no matter how small, in the initial data, in its representation (or that of intermediate results), or in the computation, would lead our computation-of-the-future to be wrong by an arbitrary amount within a finite period of time.

We are left with a bizarre world view: The universe may indeed be deterministic (determinism means that the past absolutely determines the future, that there is no "choice" at any time, and therefore no true randomness and no free will) yet that observation is useless, it is meaningless, it does us no good whatsoever -- we may as well be living in a nondeterministic universe brimming over with free will! This is philosophically profound, and evidence that our fractal geometry has indeed "generated a new world", in the sense that it has fundamentally changed the way we see our universe, as well as the way we expect it to behave. Such is the character of scientific revolutions.

Fractals in Art and Music

Back on the more prosaic plane of everyday life, we find that fractal geometry is beginning to influence the visual arts. As a language of shape and form of unprecedented richness, it is fairly easy to see that it can provide a new language for art. Fractals images are most readily generated with computer graphics, but computer graphics as a medium for the fine arts is nascent: the medium and process need to be developed and refined, and artists with madness (in Plato's sense of the word) and understanding must come to work with it. Fractals provide a visual dialect of natural forms, couched in the formalisms of mathematics. The latter makes it challenging -- even alienating -- to artists, while the scientists and mathematicians who are prepared to deal with that are rarely trained and practiced in the discipline of visual aesthetics. There is beauty to be found in deterministic fractals such as the Mandelbrot set and in random fractals such as fractal mountains; indeed, that beauty often has the character of seeming to exist a priori, somehow inherent in the (in fact always-deterministic) procedures used to calculate them. The role of the artist is in exploring the fractal forms and interpreting them, visually, in a way that brings aesthetics to the forefront. Another fascinating association of fractals with the arts is in music. It turns out that music from all cultures is fractal in an essential way: there is a repetition of form over a variety of scales, "scale" in this instance being over time. This fractal character is somehow so essential that a sequence of random notes sounds quite "musical", if it is only constrained to have fractal changes. That is, if the random "score" is constrained, say, to look like the profile of a mountain range, we hear something that is tantalizingly close to a non-random, human-crafted musical score. Much as the fractional Brownian motion we use to create fractal mountains lacks some of the features of real mountains, yet nevertheless captures the essence of "mountainous", a random fractal musical score somehow has the essence of music, without the structure that a human composer would impart. Musical composers such as Wuorinen and Legeti are (consciously) using fractals in their compositions, yet this area of exploration of the applicability of fractals has also barely begun. An early synthesis of fractally-informed music and fractal imagery was the experimental performance by Mandelbrot and Wuorinen of "New York Notes" at the Guggenheim museum in the Spring of 1990, with a reprise performance at the Alice Tulley Hall of Lincoln Center in the Spring of 1991. This and other seminal fractal artworks make it clear that the association of fractals and the arts is a potentially rich field, with much exciting work yet to be done.

The Fractal Calendar

This calendar of fractal images was designed with a combination of emphasis on aesthetics and on variety of imagery. Some attempt has been made to represent the major types of fractals: deterministic fractals such as the Mandelbrot set and its variations, as well as the whimsical "non-Platonic solids"; and random fractals as manifest in landscapes with their fractal mountains, clouds, waves, trees, and surface textures, as well as the diffusion-limited aggregation or DLA. I have also striven to represent a wide variety of fractal generation methods: iterations on the complex plane and in the quaternion numbers; iterated function systems; graph grammars or L-systems; midpoint-displacement and spectral synthesis methods for terrains; procedural textures for surface detail and artistic image processing; simulation of physical processes; and genetic algorithms for image synthesis. Some of the images are intrinsically two-dimensional, others are of three- dimensional objects; some are abstract and some represent "fractal forgeries of nature" and are thus quite realistic. On the whole, they represent some the most interesting of recent fractal images, and I hope that they will offer both pleasure and stimulation, and perhaps even inspire some future fractalists.