Fractals

Nature is full of shapes that are alike to themselves on different scales. A boulder looks like the mountain to which it was once attached. The structure of a twig is a lot like that of the tree from which it has fallen.  A coastline has the same irregular shape when viewed from various altitudes. 

The surfaces of certain cheeses and the random distribution of the stars in the sky display the property known as statistical self-similarity.  These phenomena and many others, such as the scattering of nuclear particles, are examples of fractals that happen in nature.

Many of nature's irregular and fragmented patterns exhibit a much greater level of complexity than can easily be explained with standard Euclidean geometry.  Such features have escaped the application of classical mathematics for a long time.

But now, due primarily to the work of Benoit Mandelbrot, this is quickly changing Mandelbrot is an IBM Fellow at the Thomas J. Watson Research Center.  His essay The Fractal Geometry of Nature is commonly received as the definitive work on the subject of fractals.

This Polish-born French mathematician has developed a new geometry and demonstrated its effectiveness as a model in a number of diverse fields.

Mandelbrot derived the word fractal from the Latin fractus, which means "fragmented" or "irregular." Frangere, the corresponding Latin verb, means "to break" or "to create irregular fragments." Fractals come in a broad assortment of visually fascinating patterns, many of which have practical scientific applications.

Some are referred to as "dragon curves," while others look precisely like mountain ranges. Fractals can mimic the ups and downs of the stock market, the erratic wanderings of molecular particles, or the growth of plants.

They have found applications in fields as different as physics, biology, sociology, and motion-picture simulation.  Mandelbrot has even used fractal geometry in the study of auditory noise transmission and galactic clustering.

Fractal geometry is, without a doubt, "one of the chief developments of twentieth century mathematics." While fractal geometry is relatively new, dating from about 1975, it builds upon the geometric measure theory for sets of integral and fractional dimension that was developed by pure mathematicians early in this century.

Number theory and the study of non-linear differential equations also give examples of fractal sets.  In addition, infinitely many fractal curves can be discovered in the complex plane. Julia sets, named after the mathematician Gaston Julia (1893-1978), and the Mandelbrot set are the chief examples.

Many fractals, especially those that copy natural phenomena, are generated with random numbers. The peaks and valleys of fractal mountain ranges are determined erratically, for instance; but uncertainty alone does not qualify a surface or curve as a fractal.

Many fractals, such as the Koch snowflake or the Harter-Heighway dragon, are not random at all. The necessary and sufficient property that distinguishes fractals is their fractional dimensionality. Fractals also exhibit self-similarity in one way or another, the smaller parts resembling the bigger, but this is not a mathematical requirement.

Mandelbrot has found order in places where others before him saw only chaos. In 1961 he established the importance of fractal geometry to economics. Next Mandelbrot recognized the central role that fractals play in many areas of physical science. He then discovered that the Hausdorff-Besicovitch dimension of certain sets has an essential application to fractals.

Mandelbrot suggested that the irregularity of a coastline could be measured by its Hausdorff dimension. In 1967 he posed the question, "How long is the coast of Britain?" The correct answer is, in his words, "It all depends." More specifically, it depends on the size of the instrument that a person is measuring with.

As the measurement becomes more and more precise, the measured length approaches infinity. But some coastline lengths are more infinite than others. Application of the Hausdorff dimension resolves this dilemma.  Euclidean geometry is sadly insufficient for the task.

Fractal graphics are almost impossible to generate without the aid of computers. The formulas that produce fractals are often fairly simple, but they must be calculated repeatedly, each iteration using the result of the previous one. Precise results are also best achieved by way of computers.

Computer graphics also facilitate comparisons between natural shapes and their computer imitations. Theories can be eliminated solely on the basis of the improper shapes they generate.

Depicting natural things such as clouds and mountain ranges has been a challenge for computer graphics systems based on everyday geometry. Modeling mountains with hyperboloids and clouds with ellipsoids is not very effective.

But the complexity of natural surfaces can be effectively modeled with fractal geometry methods. Fractal curves generated with probabilistic functions are particularly well suited for modeling nature's geological features.

The inclusion of the right amount of randomness in the generating algorithm can yield very realistic results.  Mandelbrot proposes Brownian motion as the basis for random fractals. He has demonstrated that this model provides for both the self-similarity and irregularity that fractals exhibit.

Fractal geometry provides scientists with a mathematical model that embraces the irregularities found in nature. Natural phenomena such as fluid turbulence can be described with the concepts of fractal geometry.

Consequently, fractals have become increasingly important. What began as a purely mathematical concept has now found many applications in the sciences.

The ability of fractals to mimic nature has led to the widespread acceptance of fractal techniques. Chemists, biologists, physicists, and statisticians have used fractals to model a wide variety of phenomena.

The vast number of fractals occurring in nature is enough to justify the study of fractals. Recognition of an object as a fractal can improve one's understanding of its behavior.  The growing interest in fractal graphics has also been affected by the proliferation of more powerful microcomputers.

Numerous articles on fractals have appeared in microcomputer magazines recently. Part of this interest stems from the unpredictable nature of certain fractals; one can spend hour after hour exploring the variety of shapes that a single program can create.

In some sense, fractal geometry is first and foremost a new ‘language’ used to describe, model and analyze the complex forms found in nature. 

 

But while the elements of the ‘traditional language’, the familiar Euclidean geometry, are basic visible forms such as lines, circles and spheres, those of the new language do not lend themselves to direct observation.

 

They are, namely, algorithms, which can be transformed into shapes and structures only with the help of computers.  In addition, the supply of these algorithmic elements is inexhaustibly large; and they are capable of providing us with a powerful descriptive tool.

 

Once this new language has been mastered, we can describe the form of a cloud as easily and precisely as an architect can describe a house using the language of traditional geometry.

 

When people think about fractals as images, forms or structures people usually perceive them as static objects.  This is a legitimate initial standpoint in many cases.

 

But this point of view tells people little about the evolution or generation of a given structure.  Often, as for example in botany, people like to discuss more than just the complexity of a ripe plant.  In fact, any geometric model of a plant, which does not also incorporate its dynamic growth plan for the plant, will not lead very far.

 

The same is actually true for mountains, whose geometry is a result of past tectonic activity as well as erosion processes which still and will forever shape what we see as a mountain.  We can also say the same for the deposit of zinc in an electrolytic experiment.

 

In other words, to talk about fractals while ignoring the dynamic processes which created them would be inadequate.  But in accepting this point of view we seem to enter very difficult waters.

 

What are these processes and what is the common mathematical thread in them?  Aren’t we proposing that the complexity of forms, which we see in nature, is a result of equally complicated processes?  This is true in many cases, but at the same time the long-standing paradigm is far from being true in general.

 

Rather, it seems – and this is one of the major surprising impacts of fractal geometry and chaos theory – that in the presence of complex pattern there is a good chance that a very simple process is responsible for it.

 

In other words, the simplicity of a process should not mislead us into concluding that it will be easy to understand its consequences.

 

As mentioned, Mandelbrot is often characterized as the father of fractal geometry.  Some people, however, remark that many of the fractal and their descriptions go back to classical mathematics and mathematicians of the past like George Cantor (1872), Giuseppe Peano (1890), David Hilbert (1891), Helge von Koch (1904), Waclaw Sierpinski (1916), Gaston Julia (1918), or Felix Hausdorff (1919), to just name a few.

 

Yes, indeed, it is true that the creations of these mathematicians played a key role in Mandelbrot’s conceptual steps towards a new perception or new geometry of nature.

Mandelbrot Set

The correlation of chaos and geometry is anything but coincidental.  Rather it is a witness to their deep kinship.  This kinship can best be seen in the Mandelbrot set.  It has been discovered by some scientist as the most complex, and possibly the most beautiful, object ever seen in mathematics. 

Its most fascinating characteristic, however, has only just recently been discovered: namely, that it can be interpreted as am illustrated encyclopedia of an infinite number of algorithms.

It is fantastically efficiently organized storehouse of images, and as such it is the example par excellence of order in chaos.  Here is an image of the magnificent Mandelbrot set.

The Mandelbrot set is probably the most widely recognized fractal. Mandelbrot's discovery resulted from his research in the area of iteration theory, also known as complex analytic dynamics.

This field dates back to the investigations of P. Fatou and G. Julia in the early part of this century.  A one-to-one correspondence exists between the complex numbers and the points in the complex plane.

Repeated application of a simple function causes some of these points to flee toward infinity, while others never wander far from the origin. The latter points form the Mandelbrot set, seen here.

Mandelbrot Set and Miniature

The boundary of this set is an infinitely complex and strangely beautiful fractal.  Although, the iterative function that produces the Mandelbrot set is quite simple, the complexity of the set itself is mind-boggling.

To fully appreciate this, one must explore the region of the complex plane near the boundary of the Mandelbrot set.  Examining any portion of the boundary in greater detail reveals new complexities. This property of the Mandelbrot set makes it an endless source of fascinating computer art.

Iterating the function x_n+1 = f(x_n) produces a dynamical system.  There are several possible outcomes for the sequence of points produced by such a system.

The sequence may diverge to infinity, converge to a finite limit, or repeat a cycle of points. In the case of the Mandelbrot set, the resulting sequence is determined by the initial value of z.

To decide whether a given point lies within the Mandelbrot set, the sequence f(z), f(f(z)), f(f(f(z))), ... must be evaluated as many as 1000 times or more, testing each new result for membership in the set.

The function f(z) = z² + c is the dynamic that generates the Mandelbrot set. z and c both represent complex numbers, where z is allowed to vary and c is kept constant.  Initializing z to 0 yields f(z) = c.

This result is then substituted for z in the next iteration. The iterative process continues in this fashion, the output of each step becoming the input for the next.

The Mandelbrot set is defined as the set of values of c for which z = 0 fails to iterate to infinity under f.  The boundary surrounding this set of points is jagged and nondifferentiable.

The computer is an invaluable tool for studying complex dynamical systems. One uses the computer as a sort of microscope to examine the Mandelbrot set's boundary.

Zooming in for closer looks at higher levels of magnification reveals the amazing similarities and differences that exist within the set.  The magnification that is attainable depends on the machine representation used for floating point values.

If the algorithm is to be programmed in a language that does not directly support complex numbers in the way that FORTRAN does, one must recall that i² equals -1, and therefore (a + bi)² equals a² + 2abi - b².

In order to write a feasible algorithm based upon the Mandelbrot set's definition, the circle of radius 2 can be used as a suitable neighborhood of infinity.

Once the iterative process yields a result whose size exceeds 2, the sequence will always iterate to infinity.  The size, or norm, of a complex number is simply its distance from the origin in the complex plane

The real and imaginary parts of c are plotted on the x and y axes, respectively. The algorithm assigns a value to c for each pixel and then counts the number of iterations required before the norm of z exceeds 2.

When color graphics are available, the iteration count can be used to determine each pixel's color.  Smaller pixel sizes will improve the resolution, but at the cost of increased computation times.

The maximum number of iterations to be allowed, typically anywhere from 100 to 1,000, must also be determined.  Higher values will yield more accurate results if longer computation times are acceptable.

A region of the complex plane must be specified.

The Mandelbrot set includes filaments that reach out in all directions and even miniature versions of the set itself.  Yet none of these miniatures are exactly like the parent set.

Some of these smaller Mandelbrots appear to float freely in the complex plane. But A. Douady and J. Hubbard have proven that the Mandelbrot set is connected. Thus these miniatures are actually attached to the rest of the set by fine filaments.

Hubbard has called the Mandelbrot set "the most complicated object in mathematics."  Mandelbrot has invented a descriptive terminology for discussing the set that bears his name.

The main "continental molecule" is surrounded by infinitely many "island molecules." Each molecule is comprised of infinitely many "atoms," any two of which may share a common point known as a "bond." The entire set forms a branched "polymer" that is without closed loops.

Mandelbrot has conjectured that the boundary of the set, which he modestly refers to as the M-set, is a curve whose fractal dimension is D = 2.

This figure examines a small portion of the Mandelbrot set at increasing magnifications. The second view is very similar to the first, but the strips of white outside the set are narrower. Sets whose fractal dimensions are known to be D = 2 exhibit this same characteristic.

Mandelbrot Set at Increasing Magnifications

 

Rather, what we know so well as the Cantor set, the Koch curve, the Peano curve, the Hilbert curve and the Sierpinski gasket, were regarded as exceptional objects, as counter examples

 

The most important example of a simple process with a very complicated behavior is the process determined by quadratic expressions, like x^2 + c, where c is considered to be a fixed constant, or p + rp(1 – p), where r is a constant.

 

Now, we're not done yet. The work shown above represents one iteration. We continue to run each new set of coordinates through the function until we can

 

prove that the point will a) leave the graph (example: on a ten by ten graph, the

 new coordinates are (-234, 97)) or b) never leave the graph (the rule is after 200

 

iterations, if the point is still on the graph, it will never leave.) This is how a color

is selected. If the point leaves after one iteration, it is assigned a color. Every

 

point after, that leaves the graph after one iteration, is that same color. All points

that leave after two iterations will be assigned a different color, and so on. Every

 

point that never leaves the screen is assigned one color, usually black. After

doing this process for each and every point of the graph, the result could look something like this Julian set.

 

 

As you can see, in many cases, 200 iterations are needed to assign only one point. On most PC's, a common number of points for a fractal is 303,200. This is why computers are needed to calculate the huge amount of iterations and to be precise.

Fractals do have a real-life purpose. Computers can take a normal shape, and run it through many iterations giving it a surrealistic look. A fractal equation can be made to make the seemingly randomness of clouds. Many movies use fractal landscapes to use as backdrops.

Here are some more images of fractals.

 

 

THE LORENZIAN WATERWHEEL

 

Most casual armchair scientists have no access to uniformly smooth boxes and elemental gases, much less instruments to measure the speed of the moving gases.

A metaphor for the gas chamber is found in the Lorenzian waterwheel. This is a thought experiment. Imagine a waterwheel, with a set number of buckets, usually more than seven, spaced equally around its rim.

The buckets are mounted on swivels, much like Ferris-wheel seats, so that the buckets will always open upwards. At the bottom of each bucket is a small hole. The entire waterwheel system is than mounted under a waterspout.

Begin pouring water from the waterspout. At low speeds, the water will trickle into the top bucket, and immediately trickle out through the hole in the bottom. Nothing happens.

Increase the flow a bit, however, and the waterwheel will begin to revolve as the buckets fill up faster than they can empty. The heavier buckets containing more water let water out as they descend, and when the water is gone, the now-light buckets ascend on the other side, ultimately, to be refilled.

The system is in a steady state; the wheel will, like a waterwheel mounted on a stream and hooked to grindstone, continue to spin at a fairly constant rate. But even this simple system, sans boxes or heated gases, exhibits chaotic motion. Increase the flow of water, and strange things will happen.

The waterwheel will revolve in one direction as before, and then suddenly jerk about and revolve in the other direction. The conditions of the buckets filling and emptying will no longer be so synchronous as to facilitate just simple rotation; chaos has taken over.

The explanation for the irregular movement of the gas lies at the molecular level. While the box sides may seem smooth and thus the flow of the should always be regular, at molecular levels the sides of the box are quite irregular due to the motion of atoms and molecules.

After all, in any solid not at absolute zero, total entropy is positive and there must be some irregularity in the molecular structure of the sides of the box. Molecular interactions are tiny, however.

How would such tiny things like slightly misplaced molecules affect the flow of the gas in such a profound way as to cause seemingly random motion? The theory behind how small deviations can lead to large deviations lies at the heart of chaos theory.

The explanation is simple, and in retrospect, obvious explanation commonly known as sensitive dependence on initial conditions.

Fractal Surfaces

The concept of fractal dimension can be extended to surfaces.  Most current methods for representing three-dimensional shapes are based on Plato's ideal forms, such as spheres, cylinders and cubes.

These methods are excellent for depicting man-made objects, but imitating complex natural surfaces presents a problem. Natural objects, such as mountains and bushes, contain too much variety and detail to be easily described by conventional means.

But since the roughness of a surface corresponds very well with its fractal dimension, fractal models are capable of describing such surfaces qualitatively.

Fractals are being used more and more in applications that require realistic simulation of natural phenomena.  Fractal models can describe a continuous range of surface textures, from perfectly smooth to extremely rough.

Simple random motion does not imitate nature accurately. But an image generation process that incorporates a scaling factor can create images that mimic nature quite well.

This technique has been especially successful in the generation of artificial landscape images.  Mountains, clouds, water, and plants have all been realistically portrayed using fractal techniques.

Fractal functions can accurately model natural surfaces because many physical processes produce fractal shapes.  Most forms that occur in nature are fractals. Any physical process that randomly modifies the shape of some surface through local action will usually create a fractal surface.

Different physical processes act over different ranges. Thus, the fractal dimension of a natural surface will depend on the dominant process at any particular scale.

Real surfaces cannot be true mathematical fractals. The size of a surface's basic particles prevents the infinite regression of detail that true fractals exhibit. But a surface can be called fractal if its fractal dimension is consistent over a wide range of scales.  This property is known as scale invariance.

Generating a fractal surface begins with the assignment of a real value between 0 and 1 to the fractal ratio r. This ratio determines the fractal dimension D of the surface; D = D_T + r, where D_T is the topological dimension of the surface (D_T = 2 for a plane).

Next n² large bumps are randomly placed on the plane, for some positive integer n. The altitudes of these bumps should have a Gaussian (normal) distribution with variance s². Then 4n² bumps are added, these being half the size of the first and having altitude variance s²r².

This is followed by 16n² bumps of one fourth the original size and variance s²r⁴, then 64n² bumps one eighth size with variance s²r⁶, and so on. The end result is a true Brownian fractal surface.

It is also possible to specify the location of the bumps down to a certain size, and then continue the process in a random fashion, specifying only the fractal ratio to determine the roughness of the surface.

In this way specific mountains or clouds, for example, may be described. The transition from a quantitative model to a qualitative one is quite effortless.  This technique imitates the method our brains seem to use when we store visual information mentally.

Creating landscapes in exact accordance with the above algorithm is computationally expensive.  Shortcuts have been devised by companies such as Pixar, which was formerly the Lucasfilm Computer Graphics Laboratory.

The most well known approach is the midpoint displacement algorithm. This method starts with a square and then increases or decreases the altitude of each side's midpoint by some random amount in proportion to the length of the side.

The point at the center of the square is also displaced vertically. The square is then divided into four smaller squares, and the process is repeated for each of the new squares until the desired resolution is achieved.

Using a triangle as the initial figure makes the construction process even easier. After displacing the midpoints of the three sides, they can be connected to form four new triangles.

Regardless of which initial configuration is used, this simple algorithm can produce a variety of complicated polygonal surfaces.

While the fractal ratio determines the dimension of the generated surface, the seed for the random number generator actually determines location of the peaks and valleys.

And a simple change of sign inverts the landscape, turning mountains into sinkholes, and vice versa.  The algorithm checks each segment's endpoints against the water level and makes the appropriate adjustments.

The actual display is created by plotting cross sections of the surface.  The fact that this technique cannot represent small details causes the landscape's apparent dimension to be somewhat smaller than D_T + r.

This approximation to the mathematically pure algorithm produces fairly realistic results, but it does not generate truly self-similar fractal shapes. Mandelbrot argues that the full algorithm, which is based on Brownian motion, must be used in order to obtain the most accurate imitations of natural landscapes.

Koch Snowflake

In 1904 the Swedish mathematician Helge von Koch proposed a method for constructing a "snowflake" curve.  Like the Sierpinski curve, the Koch curve is a closed limit curve of infinite length that bounds a region of finite area.  But unlike the Hilbert and Sierpinski curves, the Koch snowflake is a fractal curve that is not plane filling.

The method of construction that Koch proposed begins with an equilateral triangle with sides of unit length. Each side is then trisected, and each middle segment is replaced by a smaller equilateral triangle whose sides measure 1/3. The middle segments are deleted, resulting in a Star of David.

This process of trisecting the resulting sides and replacing them with smaller triangles is repeated ad infinitum.   Cesaro devised an alternate method for constructing the Koch snowflake in 1905.

His construction begins with a larger regular hexagon and proceeds by displacing the midpoints of each side inward. The limit snowflake curve lies between these outer approximations and the inner approximations of the original method.

Each iteration of Koch's algorithm increases the length of the curve by a factor of 4/3. Thus it is easy to see that the curve's length approaches infinity as the order of the curve increases without bound. The nth approximation, C_n, has four times as many sides as C_n-1.

Therefore the number of triangles to be added at each stage quadruples, while the triangles added to C_n have 1/9 the area of those added to C_n-1. Thus the areas enclosed by successive approximations to the Koch snowflake form an infinite geometric series with common ratio 4/9.

As n approaches infinity, the area of C_n approaches 8/5 that of the original triangle.  Thus the Koch snowflake is an infinitely long curve that encloses a finite area.

The Koch snowflake clearly illustrates the concept of fractal dimension. Each side is split into four new sides; N = 4. The length of each of these sides is 1/3 the length of the replaced side; r = 1/3.

Thus D = log 4/log 3, which is about 1.26181.  In addition, the curve is everywhere continuous but nowhere differentiable.  The absence of tangents and the curve's infinite length suggest that the Koch snowflake is comprised of "infinitely small deviations which one could not dream of tracing."

The Koch curve can be generalized in a number of ways. Any regular polygon can be used initially, and many methods for sectioning the sides exist in addition to trisection. The basic algorithm has also been applied in dimensions other than two.  This type of recursive procedure can generate an enormous variety of non-intersecting fractal curves and surfaces.

Applications

Fractals have been used to describe many aspects of nature. Mathematicians have used fractals to simulate the effect of shoreline decay on fisheries. Biochemists have investigated the influence of irregular protein surfaces on molecular interactions with fractals.

Climate and other apparently chaotic phenomena can be modeled and even predicted with fractal methods.  Studies of topics as diverse as fluid turbulence and bone structure have benefited from the use of fractal structures.

Fractal geometry has also provided the computer graphics artist with an exciting new palette of intriguing shapes and surfaces.

In 1974 Jerry P. Gollub and Harry L. Swinney performed an experiment in fluid turbulence, examining the flow patterns in a fluid contained between two concentric rotating cylinders.

Their findings supported the theory that chaotic attractors cause fluid turbulence. Chaotic attractors were also discovered by Robert S. Shaw in an experimental study that measured the time intervals between the drips of an ordinary faucet.

Another test, conducted by Mitchell J. Feigenbaum, Mogens H. Jensen and Itamar Procaccia, simultaneously applied heat and electricity to a chamber of mercury. The fluctuating temperatures that resulted revealed a fractal pattern.

Fractals have also been used to describe the irregularity of protein surfaces. Certain protein surfaces have a fractal dimension D of approximately 2.4, but this value varies considerably. Experiments suggest that variations in surface texture may affect molecular interactions.

The diffusion-limited aggregation model of Witten and Sander has generated a great deal of research activity during the past few years.  In 1984 Brady and Ball used this model to measure the electrode position of solid copper limited by the diffusion of Cu²⁺ ions.

In a similar study, Matsushita, Sano, Hayakawa, Honjo and Sawada measured the dimension of metallic zinc deposits to be 1.7, which agreed with the dimension of the computer-generated fractal.

Another phenomenon that exhibits fractal qualities was first observed by Henry S. Hele-Shaw in the 19th-century. When a less viscous fluid is injected into glycerin or oil, a finger-like pattern results.

Nittman, Daccord and Stanley realized the detrimental effect that viscous fingering might have on enhanced oil recovery, where water is pumped into porous rock to displace oil.

The patterns formed by viscous fingering strongly resemble those generated by the DLA method. Lincoln Paterson has explained the similarities between the principles underlying both processes.

Electrical discharges also produce forked, lightning-like patterns that bear a resemblance to the DLA model. These patterns are called Lichtenberg figures in honor of Georg Christoph Lichtenberg,

the 18th-century German physicist.  Niemeyer, Pietronero and Wiesmann have verified that the DLA model is appropriate when the ionized region surrounding a central electrode is at equipotential.

The sharp tips on the structure have large electric fields, which stimulate the growth rate at these points.  These diverse examples demonstrate the many forms that are governed by the DLA process.

Fractal geometry has even been applied to computer-aided music composition. Choosing a "generating motif" for the slowest moving line, adding faster repetitions of the motif, and then repeating the process results in a time-filling musical equivalent of a plane-filling curve.

Altering the motif changes the nature of the resulting music. The musical fractal sounds less mechanical when a certain degree of randomness is incorporated into the algorithm.

Dodge and Bahn suggest the addition of random Brownian offsets to the pitch levels of the higher layers. These offsets should be chosen from a range of six semitones above or below the original pitch. This technique is closely related to the method used to make fractal mountain ranges.

An unusual application for fractals has been found in acoustical research. Converting the Weierstrass function into audible sound has led to an interesting paradox.

Manfred R. Schroeder has devised a Weierstrass function that, when reproduced at twice the speed, creates a chord that is one semitone lower rather than an octave higher. This effect is in exact agreement with modern theories of pitch perception.

Alan Norton, a computer scientist at the Watson Research Center, has been using parallel-processing super-computers to investigate three- and even four-dimensional fractals, called quaternions.

Rather than evaluating every point in a large three-dimensional matrix, Norton has introduced a boundary-tracking algorithm that reduces both time and memory space requirements.

A line segment that crosses from the interior to the exterior of a given shape must contain at least one boundary point. This set of points then serves as a starting list for the determination of the surface.

Generating Mandelbrot set images is becoming a recognized benchmark for parallel-processing computer systems. An array of floating point "transputers" can generate a 512 x 512 pixel Mandelbrot image in a matter of seconds.  But the fractals that lie between one and two dimensions can be explored with even modest computer graphics hardware.

The opportunity to discover objects of abstract geometry that have never been seen before is very inviting. Computer graphics have made it possible for the non-mathematician to see the beauty of mathematics.

Mandelbrot suggests that fractals should be introduced to students when the idea of the derivative is first presented. The knowledge that continuous functions are not necessarily differentiable would strengthen the student's grasp of calculus.

Mandelbrot also supports the inclusion of fractals in the physics and geophysics curricula, where fractals play an important role. In the field of computer science, fractals should be taught in courses on computer graphics. Fractals might also be used as examples when teaching rational iteration theory.  According to physicist John A. Wheeler,

no one is considered scientifically literate today who does not know what a Gaussian distribution is, or the meaning and scope of the concept of entropy. It is possible to believe that no one will be considered scientifically literate tomorrow who is not equally familiar with fractals.

More Fractal Info

The combination of fractal geometry and computer graphics gives scientists a powerful tool for modeling the fragmentary aspects of nature. Many natural phenomena exhibit fractal dimension, the quality that all fractals have in common.

The fractal dimensions of many curves can be determined from the curves' definitions with the equation (1/r)^D = N. Each step in the production of the Hilbert and Sierpinski curves, for example, replaces one segment with four half-sized segments.

Therefore, the similarity ratio r = 1/2 and N = 4. Solving for the fractal dimension D gives D = 2. For the Koch snowflake, r = 1/3 and N = 4, which results in the fractal dimension D = log 4/log 3. In the case of the Harter-Heighway dragon, the fractal ratio is half the square root of two and N is two, so its fractal dimension is also two.

Because these fractals are determined by generator shapes, their fractal dimensions can also be derived from the recursive algorithms that generate them. For each increment in the order of the Hilbert and Sierpinski curves, the SET_UP function divides the length of each segment by two, giving r = 1/2. The PLOT functions that draw each of these two curves make four self-recursive calls, which means that N = 4.

In the Koch snowflake program, the SET_UP function divides the segment length by three every time the order is incremented, and the PLOT function calls itself recursively four times; therefore r = 1/3 and N = 4. The PLOT function in the dragon program calls itself only twice; thus N = 2.

Since the segments in this curve are alternately on- or off-axis, depending on the order of the curve being drawn, the similarity ratio is not as apparent as in the other examples. But the SET_UP and MOVE functions work together to produce the correct value.

In addition to having fractal dimension, fractals usually exhibit some form of self-similarity. This property is readily observed in the Mandelbrot set, as well as in other fractals. The next two figures demonstrate the reoccurrence of similar, but not identical, shapes in the Mandelbrot set. These four images show consecutively closer looks at the same portion of the complex plane.

Similar Mandelbrot Set Shapes

Similar Mandelbrot Set Shapes

Many shortcuts are often taken when fractal landscape scenes are generated. The midpoint displacement algorithm is itself an approximation. This algorithm is often further simplified by fixing the fractal dimension at D = 2.5 and using a random number generator with a uniform distribution.

The figure on top contains landscapes generated by this simplified algorithm. A qualitative comparison can be made between these surfaces.  The average displacement can be varied in either case.

The effect of increasing this parameter is seen in the4 figure above. While the fractal ratio determines the roughness of the surface, the average displacement controls the heights of the mountains. The midpoint displacement algorithm can generate landscapes that range from the realistic to the surrealistic.

Computation times for the different types of fractals vary a great deal. For fractal curves that are based on a generator shape, such as the Koch snowflake, computation times are proportional to N^m, where N is the number of parts in the generator and m is the order of curve being drawn.

The time required to generate a Julia set or some portion of the Mandelbrot set depends on a number of factors, such as the resolution of the output device, the maximum number of iterations permitted, and the portion of the complex plane being graphed.

The value chosen for the complex number c will also have an effect when generating a Julia set. The inverse iteration method for creating Julia sets is an infinite loop that may be terminated by the user at any time.

The original algorithm takes anywhere from one to five hours to generate a Julia set image on the equipment used here; with the inverse iteration method, little improvement in the image is observed after twenty or thirty minutes.

The computation time for the bifurcation diagram is proportional to the desired output resolution. It will also depend on the complexity of the dynamic that defines the system.

For the midpoint displacement algorithm, the number of elements in the surface matrix will determine the computation time. This matrix will always contain (2ⁿ + 1)² elements, for some positive integer n.

The diffusion-limited aggregation model is very expensive in terms of computer time. The maximum distance from the cluster that a particle may travel before being destroyed, the manner in which the step length is varied, and the desired cluster size all affect the computation time. Even with the large step sizes used here, this program took over nineteen hours to execute.

The number of possible applications for fractal computer graphics seems almost endless because so many natural phenomena have fractal characteristics. This is precisely the reason that the study of fractals is important.

Besides the many applications described, fractal landscapes could possibly be used in flight simulators. Pilots training in conventional simulators often memorize the terrain after a number of sessions. The creation of different and realistic fractal landscapes for different training sessions would eliminate this problem.

There are many ways in which this research could be expanded. The number of fractal curves based on different generator shapes is limitless. A program that would graph the fractals resulting from different user-defined generators would be an excellent tool for experimentation in this area.

Designing Mandelbrot and Julia set programs for direct output to the printer would improve the image resolution. A program that applied solid modeling and appropriate shading techniques to fractal landscapes would be a rewarding project for a computer graphics student. Such a program might allow the user to input the location of the observer and the direction of the sunlight.

The abundance of fractals in nature makes the study of fractals as important as it is fascinating. Fractal computer graphics continue to prove their value as a scientific modeling tool. Students of mathematics, natural science and computer science should all be introduced to this new field of study.

Conclusion

Fractals and modern chaos theory are also linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.

 

From the perspective of our inherited understanding of mathematics, this is a challenge, which is felt by some to be a powerful renewal and liberation and by others to be a degeneration.

 

However this dispute over the ‘right’ mathematics is decided, it is already clear that the history of the sciences has been enriched by an indispensable chapter.  Only superficially is the issue one of beautiful pictures or of perils of deterministic laws.

 

In essence, chaos theory and fractal geometry radically question our understanding of equilibria, and therefore of harmony and order, in nature as well as in other contexts.

 

They offer new holistic and integral model, which can encompass a part of the true complexity of nature for the first time.  It is highly probable that the new methods and terminologies will allow us, for example, a much more adequate understanding of ecology and climate developments, and thus they could contribute to our more effectively tackling our gigantic global problems.