The 1995 Fractals/1996 The Art of Fractals calendar is edited by Daryl H. Hepting of Simon Fraser University. It features an essay by Hepting and contributed images from Danielle Bercel, Michael Field, Martin Golubitsky, John C. Hart, F. Kenton Musgrave, and Lewis N. Siegel.
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by Daryl H. Hepting
von Koch's snowflake curve, an early mathematical monster, is depicted here in friendlier surroundings. The blue visible amidst the spheres is what remains after an algorithm to cover the complement of the fractal is applied. It works by finding the minimum distance of d of a point x from the fractal then drawing a disk of radius (maybe less than) d centered at x. Since any point already inside a disk cannot be part of the fractal, the algorithm progresses quickly as the complement is covered by disks. The spheres are the three-dimensional extensions of those disks. A fractal texture is applied to the spheres to create snowballs ‐ the making of which is a common pastime during long Canadian winters. The snowflake curve is based on an equilateral triangle, of which each side is divided into three parts then replaced by four parts, of the the same length, and so on.
Although chaos would seem to be the antithesis of symmetry, it is possible to find elaborate symmetric patterns in chaotic processes. Shown here is a hexagonal pattern similar to those that sometimes appear on quilts. This image is produced by iterating a map with hexagonal symmetry. Although the iteration possesses all the features of chaos, a surprisingly detailed and intricate symmetric pattern appears when we take a very large number of iterations -- in this case about six billion -- and color each pixel according to the number of times it is visited during iteration. In this sense we are computing an averaged picture. The colors themselves were chosen for their artistic effect. Note that the detail inside each of the hexagons happens to resemble a flower. The fractal quality of the picture is to some extent represented by the reappearance of the hexagonal symmetry on ever-decreasing scales.
In the first picture below [not shown], we show the center of one of the flowers magnified, in area, by a factor of about 400. (The colors used are different from those in the main picture). In the second image, the magnification is about 10,000 and hundreds of hours of cpu time on several high performance workstations were required to compute it.
This fern dune is one of the natural wonders on this distant world. The fern leaf is defined by a set of linear transformations, sometimes called an iterated function system. The sand dune which surrounds the leaf was created by evaluating a so-called escape-time function for the fern and subsequently interpreting the values as heights. The escape-time function works by determining whether a point escapes membership in the fern, and if so then how quickly. The fern shape at the top of the dune reveals the accuracy with which that question was answered here. More points are included than might otherwise be expected since it is easier to exclude points than verify their membership. The scene is made more believable by the use of fractal textures for the clouds, moon and sand. The technique for interpreting function values as heights is effective for adding a third dimension to two-dimensional data, and has been employed to great advantage in many earlier fractal images.
Shown below [not shown] are some other examples of natural landscapes created using this approach. The first shows an interpretation of the dragon curve. The bottom right image shows the interesting results when a distance function other than the usual Euclidean is used in computations. The bottom left image shows a view of the same fern dune, on a brighter day.
This image is the result of a two-year metamorphosis from the image shown below [not shown], entitled Rare, Long-Stemmed, Speckled Gigantic Flowers Slowly Advancing. The title of these images is a partial quote from Karen Blixen's novel Out of Africa. From inspection, one may see that both pictures depict the same fractal and that they share virtually all the same coordinates and parameters. The roles of color and black (or monotones) between the two images are reversed. Areas black in Variation #1 were colored according to logarithmic function in the original and monotone areas in the original received an algorithmic coloring. The form of the original is still visible in Variation #1, but it has been de-emphasized. The images are based on a variation of the usual formula describing Julia sets: zn+1 = zn2 + c, where one may adjust the value of c (a complex number) as a parameter to obtain different images. In the variation, a function is added to the mix and becomes another parameter for manipulation: zn+1 = f(zn) + zn2 + c. For these images, the conjugate of a cosine function was used. While I still prefer the original picture, as a parent might feel for a child, it appears that almost everyone who has seen both of these pictures prefers Variation #1. I present them to you for your inspection.
It has been observed, by Mandelbrot and others,that fractal forms are found abundantly in nature. In this spirit, Mandelbloom shows natural forms occurring in a fractal. The image was created with a method used to show how the Mandelbrot set evolves over time. To create an image of the Mandelbrot set, points in the complex plane are iterated according to the well known function Zn+1 = Zn2 + C. The structure in Mandelbloom comes from an attempt to capture the movement of these points at a particular moment in time. In traditional renderings of the Mandelbrot set, each point in the plane is iterated until it escapes from the neighborhood of the origin, or reaches an upper limit. The number of iterations is then utilized to generate a color for the pixel which represents that point. In Mandelbloom, each point is instead iterated exactly the same number of times, and each point's location in the plane is used to color the pixel.
With this method, a procedural texture is used to associate a color with every point in the complex plane. The left column of the image below shows a Julia set rendered with a texture that uses a point's distance from the origin to generate a color for the pixel; white near the origin, and black toward infinity. It shows iterations zero (with the points in their original locations), one, two and four. The same set rendered with a more complex "basket weave" pattern is shown in the right column of the image below. Each point's location in the pattern is used to calculate color values for the corresponding pixel. This same basket weave pattern was used to generate Mandelbloom, with an extra step at the end. Rather than calculating colors for the pattern, color values are read from bitmaps of either the stem or the rose.
by Daryl H. Hepting (with thanks to F. David Fracchia)
In the midst of the harsh surroundings, a lone tree struggles for life. Each element of the scene modeled here employs fractals to shift attention from the mathematical details of the model to the illusion which it creates. The terrain is created from instances of a height field derived from a planar linear fractal, [not] shown below. The density of the fog which covers this desolate land is modulated by a fractal function. The tree, though idealized, does incorporate some observations from nature which, along with a wood texture, add to its realism. Every branch in the tree is created by linearly transforming the trunk. Beginning with a very concise description, the model is expanded to include over 150,000 primitives for rendering. Further intuition about the connection between plants, trees and fractals is given by the other two images below -- shapes which look as if they may belong in nature, but come from a concise mathematical description.
This image takes a portion of the standard Mandelbrot set as its starting point. The portion chosen, shown at a relatively low magnification factor of 414 in Untitled #7 n-2 (top image below [not shown]), seems rather boring with little to recommend it. However, at a magnification factor of about 40,000, as seen in Untitled #7 n-1 (bottom image below), this same location reveals the familiar shape of the Mandelbrot set and its lake. Even at relatively low magnifications, images of fractals can be made more interesting by bringing out other details and the main image is the result applying one such technique. Points outside the Mandelbrot set will "escape" to infinity after some number of iterations, and this number is usually called the escape time (points which have not escaped after the maximum allowed number of iterations are considered to be part of the Mandelbrot set). Traditionally in images of the Mandelbrot set each point represented in the image is colored according to this value, but I have also experimented with several alternatives. One may determine the color by making use of the real component of the location at which the point was deemed to have escaped (the Mandelbrot set resides on the complex plane and so each point on the plane has both real and imaginary components). Another technique which can be used to determine color is based on the quadrant of the complex plane in which escaped points first land. The result is a sort of curved-space checkerboarding of color, as points belonging to a single contour may take on any one of four colors depending on the point of departure for the escaping point. To this, an additional logarithmic coloring algorithm is applied to produce smooth color transitions.
Fractals can be created to mimic the real world and they can also be used as a source of pattern generation. Combining fractal methods with symmetry leads to a method for creating many kinds of repeating patterns. The seventeen different symmetry types of planar repeating patterns were classified formally by Federov in 1891 -- but their existence has been known for centuries. Indeed, each of these patterns appears in the Alhambra, a group of buildings overlooking the city of Granada that was constructed by the Moors in the thirteenth and fourteenth centuries. Only more recently have mathematicians considered how many different ways there are to construct two-color planar repeating patterns -- there are forty-six possibilities. A two-color repeating pattern is one in which half the symmetries of the figure preserve the colors while half the symmetries interchange colors. Fractal methods have been used to create the large two-color fractal quilt on the opposite page. It is amusing to check whether you can, in fact, identify all of the symmetries in this figure. (Besides the translations, there are eight symmetries -- four preserve colors and four interchange them.)
A quilt pattern can have more than one two-coloring. We illustrate this below [not shown] by showing three different ways to two-color the same fractal quilt. Here, another two-color fractal quilt pattern serves as background.
This image demonstrates recent advances in computer graphics at modeling supernatural phenomena. This crop circle, that might have otherwise been the result of a UFO encounter, as actually created using a technique called procedural geometric instancing. It allows a new representation for procedural models, which are typically translated into simpler polygonal representations for rendering. The usual unorganized polygonal representations can be unmanageable due to size. Procedural geometric instancing encapsulates procedural components into the geometric representation, which are evaluated on demand during rendering. The crop circle, containing over 100 million primitives, was specified by a short description file which represents the field of wheat with a CSG (Constructive Solid Geometry) hierarchy of instances of a single wheat stalk (right image below) which may or may not bend over, depending on the location into which it is instanced. Given this position, a simple lookup in a low-resolution "crop map" of the Mandelbrot set determines whether the UFO should bend the stalk or not.
This technique can also trample a teapot into a field of cones (upper left image below [not shown]). Procedural geometric instancing can also be used to represent tropism in trees, which models the effect of gravity, sunlight and wind on branching patterns. The left tree was modeled without any tropism whereas the right tree feels the effect of gravity (lower left image below [not shown]).
A dragon curve is captured here in what may be its natural habitat. The brooding dragon curve remains after its complement has been covered by the blocks which form the walls which enclose it. The primitives are not spheres, as before, since the Manhattan distance function is used in place of the usual Euclidean. This means that distance is calculated x + y rather than ÷x2+y2 and so diamonds in the Manhattan metric fill the same role as circles in the Euclidean metric. To create the third dimension in this image, boxes with height proportional to the size of the original diamonds were used. The sharpness of these boxes and their shadows provide an interesting contrast to the apparent roughness of the dragon curve, created with a fractal texture.
Shown immediately below [not shown] is the two-dimensional view of the computations that created the model which was ray-traced for the main image. Further below is an image similar to the main which instead uses as primitives the Manhattan metric's equivalent to spheres.
A shortcoming of computer-generated images is often that they lack the
level of visual complexity of a photograph or a painting.
An impressionist painting, for instance,
can have a rich variety of color in each brush stroke;
furthermore, paintings always have a textured surface, and a painter is
likely to modulate the color of an otherwise flat field of color,
such as a blue sky.
In Barque a variety of fractal methods are used to obtain similar visual
we start with a fractal mountain and fractal water, as seen in the image
[not shown] below.
Then a fractal color perturbation, as seen in the second image below,
is applied to the mountain and to the sky.
Finally, a fractal post-processing filter is applied to the image to
create a pointillist effect.
This filter samples the color of the image at a particular location,
perturbs that color with a fractal distribution,
then replaces a small area with a spot of the perturbed color.
Floating in space, this fractal staircase leads toward a perception of infinity. Each step in the staircase is both a small copy of the whole shape and a collection of many smaller copies of itself, which illustrates the key concept of self-similarity. The fractal, called a dragon curve, is formed by iteration of just two linear transformations. Each step in the staircase can be identified by a common sequence of transformation applications. By labeling the transformations with 0 and 1 and recording how they are applied, one can associate a binary number with each step -- used here to assign heights. The staircase looks somehow natural but the shape is more regular than would be found in nature. Like many fractal objects, it possesses a sense of familiarity. The clouds in the sky are also fractal, but they represent an example of stochastic fractals -- in contrast to deterministic fractals like the staircase or the omnipresent Mandelbrot set.
Below [not shown] are two generations of the dragon curve, constructed in the same manner used to form the staircase. The left and right images show a low and medium number of divisions respectively. Further below is part of the Mandelbrot set, mapped onto a sphere which serves as the dome of the sky -- seen before other image elements are added.
Nature is filled with a beauty easily perceived at many levels. Trees and forests, rocks and mountains, streams and mighty rivers, wisps of fog and great billowing clouds all contribute to an immediate, undeniable impression upon our visual sense. Although the impression may be hard to quantify, it is no less real.
Artists often use poetry, paintings, or photographs in an attempt to communicate this impression of beauty. All these forms share in that they are patterns -- of words, pigment or light -- which gain permanence as they are connected with ideas. In this way, mathematicians also make patterns with permanence.
To understand nature, we must also understand the language with which it speaks to us. Galileo said that this language is mathematics "its characters are triangles, circles and other geometrical figures." The attempt to understand nature on the basis of mathematical patterns has a long tradition, which includes Pythagoras and Kepler. Though scientists since may have generally agreed with Galieo's statement, they found the triangles and circles of Euclid's geometry to be limiting. Mandelbrot acknowledged that:
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 ...
and he provided an alternative with fractal geometry. Mandelbrot coined the word fractal from the Latin root fractus to capture the rough, broken and irregular characteristics of the objects he intended to describe. In fact, this roughness can be present at all scales, which distinguishes fractals from Euclidean shapes.
In the twenty-odd years since its introduction, fractal geometry has found a wide purview. Fractals seem to be everywhere, as the word fractal is now used to describe the universe and some of the very smallest things in it. Fractals, which belong to Galileo's realm of "other geometrical figures," add a very rich vocabulary for describing nature concisely. Now, the ruggedness of a coastline and the roughness of a rock can be quantified.
This calendar celebrates the deep symbiotic relationship between fractal geometry and computer graphics, which has grown as each nascent field has developed. The use of computer graphics for the study of fractals represents one of the first applications in the now popular realm of scientific visualization. As mathematical objects of infinite complexity, the detail of fractals can only be appreciated with the help of computers and this dependence represents the epitomy of scientific visualization's goal to see the unseen. Apart from fractals with geometric constructions, the computer is also essential in the study of systems which exhibit fractal behavior over time. Here, computer-generated fractal images provide a glimpse of nature not so easily seen, one filled with deterministic chaos. Many physical systems are governed by deterministic laws, yet behave unpredictably. The unpredictability comes from sensitive dependence on initial conditions -- two states which begin arbitrarily close together may become arbitrarily far apart after a finite period of time. Though the rules which define them are exceedingly simple, the complexity of shape is derived from repetition of those rules at a level that only the computer can manage. Computer graphics allows the communication of this complexity, with an eloquence not possible through words alone. Surprisingly, even these abstract images can appear organic and natural.
Many of those who know of fractals were introduced to them through this application of computer graphics. With the increase of available computing power, the rewards and dangers of exploring the Mandelbrot set have become more evident. Arthur C. Clarke, in Ghost of the Grand Banks, speculates that in the future, Mandelmania will become a serious affliction for many.
A sense of fractals can be gained without the computer, as nature abounds with forms possessing fractal characteristics. Although the largest and smallest examples of these cannot be seen by the naked eye, many remain easily accessible. More formally, fractals can be seen as abstractions of the natural qualities found in mountains, trees, and clouds. This process of abstraction leads to algorithms which allow computer graphics to produce astounding forgeries of nature that capture our impressions.
One hopes that the result of the computer graphics process, approached from either of the above perspectives, is a pleasing picture. Each image may be created with a different purpose in mind, but ultimately the pictures are a means for effective communication of the material. And if the picture is pretty enough, it may even help to convince some of the beauty in mathematics. After seeing computer-generated pictures of fractals, some people will ask if there are any real uses of fractal geometry. In this case, it is important to emphasize the connection of the images to the reality just beyond the computer screen.
There are many examples of the struggle to evaluate and apply the new idea of fractals and acceptance has not been unanimous. Mandelbrot gave a tentative mathematical definition for his discovery, but many have been concerned by its lack of precision. Then again, who could argue with the beautiful images, produced from simple equations, like the famous:
Zn+1 = Zn2 + C
which defines the Mandelbrot set.
Natural objects possess detail at several levels but manufactured objects generally do not. In becoming focused on or bound to a particular scale, an important connection with nature is lost. Self-similarity is the key to much of the detail in natural forms. Similarity, in a mathematical sense, means that objects may differ only in size or position but not shape. Self-similarity means then that parts have the same shape as the whole. Fractals may also possess the self-similarity in a statistical or approximate sense, as many natural objects do. In its simplest form, fractal dimension can be said to measure the degree of this similarity.
Experience has shown that it is difficult to be precise when defining fractals. Mandelbrot's original definition based solely on dimension was considered too narrow, but Falconer has tried to capture the essence of what it is to be fractal. He has done this by adopting the approach of the biologist who would define life -- most living things possess certain properties, most of the time. In the case of fractals, these properties are:
Although one may appreciate equally the beauty of the Mandelbrot set and a mountainous landscape, it is not always easy to see a unifying theme if it even exists. One may understand the recipes for generating images of the Mandelbrot set, but it is quite another matter to see fractals when looking away from the computer screen. It is revolutionary to think that fractals really are much more than the pictures which represent them and it may be more reasonable to view fractals only as isolated curiosities. Is fractal geometry a set of techniques for computer graphics or a new way to look at the world? Many of those involved with fractal geometry advocate the latter, but in order to arrive there from the former position, a significant transition must take place. Several paths may be followed, for example:
Once the process has begun, it becomes possible to develop an eye for fractals.
Although fractals are used mainly in physics, they have appeared of late in such diverse areas as aesthetics and psychology, indicating again that fractals are a very powerful idea. As researchers have struggled with the implications of fractals and chaos, so too have philosophers and artists. Just as the flapping of a single butterfly's wings may affect global weather patterns, the actions of any individual may have far-reaching implications. These changes in world view are already being echoed in the arts. Historically, the introduction of a new geometry has been inspirational for artists. Although fractal geometry is significantly different from Euclid's, the challenge may be to understand how these different puzzle pieces fit together.
Fractal art is in a unique position since computers wait to become a widely accepted artistic medium. The procedural, or algorithmic nature of fractal art places it at a crossroads between mathematics and art -- a fertile ground waiting to be fully explored. But more than just in visual arts, fractals are being actively explored in books, television, motion pictures, dance, theatre and music. In fact, a great deal of music exhibits the fractal distribution of 1/f noise over time, a sort of middle ground between unstructured white noise and highly structured brown noise. It contains enough structure to be regular, and enough variation to be surprising. An early example of fractally-informed music with computer-generated fractal imagery was Mandelbrot and Wuorinen's experimental performance entitled New York Notes.
Fractals, as a language of nature, give a new understanding of the finite world around us and provide a glimpse of infinity. Represented in this calendar is a collection of exciting and diverse fractal images. The Mandelbrot set is depicted in some interesting new ways. Both deterministic and random fractals are represented -- some closely related to their mathematical roots. I hope you will be encouraged to explore fractals in new ways.