Introduction (top)

Figure 1.1 A collage of fractals.

            Did you know that flowers, trees, lightning, cliffs and clouds all have fractal structures? In fact, it is hard to look around and not see something that exhibits a fractal pattern. Fractal shapes are not new; their repetitive patterns have always been with us. What is new is our awareness of them. It's as if we have opened our eyes for the first time to see that the world is full of rich textures and patterns. Ones that can be seen at all levels of magnification.

            Until very recently these natural structures seemed like they must be outside the scope of mathematics. Unlike the world of mathematics, the world of nature appeared random, haphazard, imprecise. We could recognize patterns in nature, as in the constellations of stars in the night sky, but we did not have a way of understanding why, or how, such patterns existed. We could not explain their geometry  as we could the geometry of a square, or a cone.

           

            Fractal geometry represents the end of these limitations, by helping us understand the 'how' and the 'why' of natural structures.  In this chapter, we'll introduce some of the fractals in the world around us--in nature, in science and in the arts.  Using your Macintosh and the software included in this book, you'll be able to see for yourself what a fractal is and how it is created.  Whether you're a first-time fractal adventurer or have a Ph.D. in Mathematics, we invite you to join this exploration.

Fractals In Nature (top)

 

" Why is geometry often described as 'cold' or ' dry' ? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. 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."

           

-Benoit Mandelbrot's opening to The Fractal Geometry of Nature, 1982

           

Figure 1.2 Johannes Kepler’s 1597 version of an Euclidean based solar system.

            In "classical" geometry, often referred to as Euclidean geometry, we are used to working in an environment with points, lines, triangles, squares, circles, spheres, cubes and a host of other shapes that are easy to construct. Euclidean shapes are useful in describing things people create, but are not helpful in explaining the shapes found in nature.

            A wonderful place to view the contrast between the smooth world of Euclid and the textured world of nature is the Golden Gate Bridge in California. If you look out to one side, you see the city of San Francisco, with its rectangular office buildings, a pyramid building, rows of cubical houses and an occasional cylindrical tower. On the other side of the bridge, you see the Marin Headlands, a coastal reserve with jagged cliffs, rugged mountainous terrain and an assortment of wind swept trees.  Color Plate 1 shows these two worlds.

Figure 1.3 Shapes found in Euclidean geometry.

 

            Euclidean geometry consists of rules, called axioms, that were thought up by men.  In traditional geometry, we learn axioms like these: the shortest distance between two points is a straight line; triangles are made by connecting three line segments; the area of a circle is given by multiplying the constant π  with the square of its radius. Traditional geometry doesn't teach us, however, how to describe the structures of nature. With traditional geometry, how can we describe a cloud--as a circle with fuzzy edges? What can we call a mountain--a triangle with bumps?

            Fractal geometry derives its rules from natural objects.  Clouds, mountains and trees do all have a distinctive geometry, even though we are not used to thinking of them as geometrical shapes. Fractal geometry teaches us to look at natural objects as shapes made up of distinctive self-similar  patterns. For example, look closely at a tree and you will see that it is made up of continually smaller branches that form a geometric pattern.

Figure 1.4 Fractal Generated Tree Shapes.

In the book we use icons to indicate areas with a special focus, as seen in the next using section using the program FractaSketch™. For a complete listing of the icons used and their color representation see Color Plate 2.

 

  Using FractaSketch™ to look at fractal trees

 

Figure 1.6 FractaSketch ‘Nature’ folder.

Figure 1.7 FractaSketch ‘Tree 1’ icon.

To begin, double-click on the FractaSketch™ ‘Nature’ folder found in the FractaSketch program folder (see figure 1.5). Then, double-click on 'Tree 1' icon in the ‘Nature’ folder to open up a sketch of a tree. Notice that the sketch is a very simple figure, made up of four line segments. This sketch is the basic figure--the seed--of the tree.  The seed is the first level of the fractal.
 

Now ask FractaSketch to draw the fractal at the second level by clicking on box number 2 at the bottom of the screen.  Notice that the Level 2 sketch is made up 4 copies of the seed,  which FractaSketch has placed on the original line segments except for the trunk. Can you see how one branch of the tree resembles the whole tree?  Because sections of a fractal resemble the overall structure,  we call them self-similar. 

Click on box 3.  Find the smallest copy of the seed that you can.  How many copies of the seed can you count?  There are 16  small copies, placed on the line segments of the Level 3 sketch.  There are also 4 medium copies, placed on the line segment of the Level 2 sketch. Click on boxes 4 and 5 to watch the tree fractal grow into an increasingly more complex self-similar figure.

When through with the program select "Quit" from the "File" menu. The program will automatically ask if want to save any changes, this is unnecessary so select "No" from the displayed dialog box. The FractaSketch™ program is discussed in greater detail in Chapter 3 and in the FractaSketch manual section.

 

            The word "fractal"² was derived to describe these irregular shapes of nature. Its origins are from the Latin root word fractus ³ meaning "to break". Unlike squares and circles, fractals are broken into irregular fragments.  This new form of a non-Euclidean geometry enables us to measure fragmented structures.  In fact, when we link the word fractal with the word geometry-- derived form the Greek words  for "earth" and  "to measure"--we have a somewhat literal translation of "to measure the cracks of the earth". 

            With fractal geometry, we can "break" out of the limitations of traditional geometry.  Exploring fractals is like putting on a new pair of glasses that allow you to see the structures underlying the complex and beautiful objects of nature.

            Not only natural objects but also natural processes have fractal structure.  A river, for instance, is created from smaller tributaries that flow together: brooks flow into creeks, which flow into streams, which flow into rivers.  The same process takes place at different scales.  Brooks flow into creeks at the smallest scale in the process, and streams flow into rivers at the largest scale.  Thus, the process has a fractal structure, because it shows similar structures at different scales.  See Figure 1.8.

Figure 1.9 Woodcut of Fruit Trees with Varying Yields.

            In the 15th Century, a group of French monks were experimenting with orchard formations in order to get the most fruit from their trees.  At first, they planted the trees in equally distant rows. This allowed them to plant the maximum number of trees, but it also made it easy for fruit-eating insects to travel from one tree to the next, and from one row to the next. Insect could infest the entire orchard this way. The monks attempted to fix the problem by planting neighboring trees further apart from each other. But this meant that the orchards would contain fewer trees, and produce less fruit. How, then, could they best arrange the trees for maximum use of space, with minimum susceptibility to insect infestation?

            The answer turned out to be a pattern with fractal form.  The monks experimented by rearranging the groves in different patterns. Then, by observing various configurations over the decades, they were able to discover which arrangements would bring forth the greatest yields of fruit. In one successful pattern, they grouped trees in regions, with clusters of trees separated evenly from neighboring clusters. This way, if one cluster became infested it wouldn't infect the others. If a region had not been infected for some time, they narrowed the gap with its neighbors.  Conversely, if a region was prone to attack, they increased the gap from its neighbors.  What resulted was an intricate pattern of clusters and gaps--with no apparent uniformity in structure.  Centuries later,  fractal geometry would explain that this pattern was a fractal with 59.27...% density--the most effective density for maintaining natural growth in an orchard.

Figure 1.10 Percolated orchard patterns.

The Game of Go

            The Chinese game of Go, developed around 2200 B.C., also demonstrates the percolated structures of fractal geometry at scales of greater then a few squares. Using a 19 x 19 checkerboard and black and white stones, Go players try  to capture as much territory and stones as possible until they can make no further advances. The side with the most area and stones is the winner.  The patterns that the players create with their moves turn out to be percolated structures. Just as the Monks' orchard revealed a natural pattern of tree growth and decline, the game of Go models the territorial conquest in the real world. 

Figure 1.11 Percolated fractal structure found in the game of Go ^[2] .

Fractals in Science (top)

            Without knowing it, the monks and the Go players had demonstrated a principle that would be used centuries later by: inventors to improve the efficiency of the light bulb, doctors to monitor epidemics, foresters to model the spread of fire and physicists to describe lightening.

Improving the Efficiency of the Electric Light Bulb

Figure 1.12 Thomas A. Edison original patent application for the electric light bulb.

            A light bulb's illumination is directly related to the resistance of a wire filament for a given current ^[3] .  In order to expose as much of the filament as possible while keeping the current's outer surface small, filaments are often wound into spiral loops.  This way of providing greater light emission, typically can be extended to three levels of repetition before the filament, generally made from tungsten, becomes too fragile for the required need.

Monitoring Epidemics

            Doctors monitor epidemics by studying the factors that influence the growth and containment of a disease.  For example, a disease is more likely to spread in a densely-populated city, and is more likely to be contained in a rural setting with acres between houses. By identifying infection patterns, doctors can anticipate where and how fast a disease will spread, and can control the spread by enforcing quarantines and giving vaccinations.  Eventually, an epidemic will reach a percolation threshold--the point at which the disease can spread no further.  For example, a small pox epidemic has reached a percolation threshold when all people are either quarantined, vaccinated or have built up an immunity to contact with the infection.  

                       

Modeling Forest Fires

            Growth and containment are also key factors in the spread of forest fires.  Left alone, the forest achieves a balance of growth and containment that prevents vast destruction.  Small fires leave gaps in the forest that act as fire breaks.  Without these gaps, the forest becomes too dense, fueling large rapidly-moving fires that can rage out of control.  You can see how fractals structures evolve from percolation in the Forest Fire Game.   

Figure 1.13 Fire spreading through two forest densities.

       Forest Fire Game:

 

            See how percolated structure helps to control forest fires.  With the forest fire game, you explore the spread of fires through two forests, each with different densities.  (The densities  representing different fractal dimensions.) By starting identical fires in each forest, you compare the rate at which the fires spread.  National  Parks often let naturally-occurring fires burn to help clear out the under brush and  prevent larger fires from occurring.

            To begin, you need to make 40 slips of paper labeled 1-20 and A-T.  By drawing these slips out of two paper bags, you will randomly determine where lightning strikes (the fire staring points) on our forest grid. You also  need a pencil to mark the fire's path. You might want to make a copy of the two forests so you won't have to mark up the book.

Figure 1.14 Forest grids.

Step 1:  Draw a number and a letter.  On both grids, find the square with that row number and column letter.

Step 2:  If you land on a clear square, stop.  Otherwise, mark an 'X' through the tree in that square and in all squares that touch this one--on the sides, above, below.  Continue the chain of fire by marking out trees in all connected squares until you reach a clear square or the edge of the forest.

Step 3:  Compare the destruction in the two forests.

           

Step 4:  Repeat steps 1 and 2 several times--marking out trees until you reach a clear square, the forest edge, or a previously burned region. If one forest fire stops, continue the procedure with the other one until its fire also stops.  As you continue, you will see an percolated structure emerge.

Figure 1.15 Fire Path

How Lightening Travels

           Lightening also forms connecting paths referred to as aggregate growth patterns.  Lightening is an electrical charge that jumps between atoms forming crooked patterns.  To find its connection, the charge takes a path, jumping to the nearest available atom. Interestingly, this electronic arcing formed by a chain of charged atoms is similar to regions in the Mandelbrot set, a fractal we'll discuss in Chapter 5. The jaggedness of a lightening strike see Figure 1.16 and Color Plate 3 is calculate to have a dimension of roughly 1.3, similar to many of the fractal patterns we will create in Chapter 3.

Figure 1.16 Similar fractal patterns between lightning and the Mandelbrot set.

 

  Using MandelMovie™ to look at Lightening

Lets use MandelMovie to briefly to see how a lightning's structure resembles parts of the Mandelbrot set.

Figure 1.17 MandelMovie program folder.

Figure 1.18 MandelMovie icon.

To begin, double-click on the MandelMovie™ icon. Notice the Mandelbrot set in the "Mandelbrot Set" window. Now look for regions that resemble lightening.  To help you in your search you can look at the Mandelbrot set through a cycle of different colors. This is done by choosing "Animate Colors In" and "Animate Colors Out" from the "Spectrum" menu. To stop the cycling, click once on the "Mandelbrot Set" window. If you wish to choose a different  cycle speed select "Select Animated Speed..." from the "Spectrum" menu. To quit MandelMovie just select "Quit" from the "File" menu. The MandelMovie program is discussed in greater detail in Chapter 5 and in the MandelMovie manual section.

Fractals in Our Bodies

            Fractal patterns are observed in many parts of the human body.  The nervous system for instance, exhibits fractal patterns seen at both visible and microscopic levels. Looking at the cerebellum, a structure located at the base of the brain, one can see a series of continually smaller nerve branches forming a network that sends sensory information throughout the body. This branching structure is called the arbor vitae which literally translated means the “tree of life”.  On a cellular level, these nerve branches are comprised of cells with similar structure, neurons and astrocytes, that also display these fractal patterns.  Neurons, which are the cells primarily responsible for the transfer and processing of information within the nervous system, contain cell extensions called dendrites.  Dendrites receive information in the form of electrical impulses from a variety of sources.  This network of dendrites, which is often called the “dendritic tree” can be seen in Purkinje cells of the cerebellum, which are organized into short, asymmetric branches that break off into smaller branches and so on. The same branching pattern are also observed in the cell extensions of the astrocytes, one type of support cell in the nervous system see Figure 1.19.

Figure 1.19 Neurons Exhibiting Branching Structure.

            Another system which demonstrates fractal structure, is the cardiovascular system, which is composed of the heart, arteries, and veins see Figure 1.20.  The heart’s main function is pumping blood to and from the organs of the body.  Typically, blood flows from the heart and travels in arteries, which eventually branches into smaller arterioles, which in turn branch into smaller arterioles, and so on, until finally reaching the capillary network.  The thin walls of the delicate, fine capillaries allow the blood to supply organs and tissues with vital nutrients and oxygen.  Then, the process is reversed - capillaries feed into larger tributaries called venules, which feed into larger venules, etc. until the veins finally carry the blood back to the heart.  The redundancy coupled with the irregularity of the fractal structure of the blood vessels ensure that tissues and organs receive an adequate supply of blood even in the event of injury. This structure is well illustrated by the blood vessels surrounding the heart see Figure 1.20. Here the cardiac muscle fibers are shown with their connective tissue network of the chordae tendinae encircling the heart valves.

Figure 1.20 Branching of the Cardiovascular System.

Figure 1.21 Close-up Region of the Heart Showing Redundancy, © 1993 Dynamic Software by Mauren Carey .

In mammography fractal structures are being used to detect lumps found in malignant cell growth. By comparing the fractal dimension of calcium clustering associated with normal regions -low clustering density to cancerous regions -higher clustering density doctors can use this technique to aid in the detection of breast cancer. 

            Fractals are also observed in the bronchi and bronchioles, the air passages of the lungs.  Computer simulations of lung development have shown that fractal modeling is a more accurate depiction of the structure of an actual lung than conventional modeling, which limits itself to merely a few levels of branching. An understanding of these new principles derived from fractal geometry has provided us with greater insight into the human body.

Figure 1.22 A lung exhibiting a self-similar branching pattern.

Figure 1.23 A self-similar fractal pattern with a close resemblance to a lung ^[6] .

Figure 1.24 Andromeda Galaxy with a Fractal Dimension 2.2 - 2.4 also see Color Plate 4.

We've looked at only a few of the fractal patterns found in science. Other uses of these patterns range from modeling crystal growth used in manufacturing semiconductors, to calculating the density of our galaxy. Now lets see how we have incorporated fractals in other areas.

Fractals in The Arts (top)

            Whether we are conscious of it or not, fractal shapes have become part of our thought patterns. Perception scientists have carried out psychology experiments that shown we are most comfortable with objects whose dimensions range between 1.2 - 1.4 and 2.2 - 2.4, these coincidentally are the dimensions most commonly found in trees, mountains and clouds. Perhaps as we coexist harmoniously with the fractal patterns in nature, our visual systems process thoughts that themselves have become fractalized, subconsciously in tune with the world around us. In so doing, we have inadvertently added this familiarity to our surroundings by letting fractals permeated our art, architecture, music, and even popular culture.

Fractal Art

            Fractals appear in our culture as an art form. They are show up in traditional forms such as patterns in quilt patches and make modern statements in futuristic art see Color Plate 5 of a fractal quilt, Color Plate 6 of modern art, and Color Plate 7 for adaptation of a Rockwell print.  In Figure 1.25 we compare a 1920 Mondrain abstract composition with its use of rectangles to a Longone-Birni settlement in Cameroon to the first two stages of a fractal generated piece. This geometric style referred to as Neo- Plasticism, influenced many of architectural and sculptural designs that we now see in the modern world.

 

 

Figure 1.25 Modern art vs a Longone-Birni settlement vs fractal generated art.

      

Fractals themselves are often the objects of admiration. Look, for example, at the Koch snowflake in Figure 1.26.

Figure 1.26 The Koch Snowflake ( 7-line sweep variety ) admired for its grace in construction.

Fractal Architecture

            Architecture has many examples of self-similar structure. Fractal patterns show up in the architectural design of Roman bridges, Gothic churches, and on a larger scale in African settlements (even in Cairo street maps) and the urban sprawl of modern cities.

Figure 1.27  Roman Bridge, Pont du Gard, with Self-Similar Arches.

Roman bridges embodying the first stages of fractal design have lasted over 2000 years. This is clearly evident in the Pont du Gard bridge of the Gardon Valley near Nîmes, France [Figure 1.27]. Incorporated into its design are considerations for passage of high winds experienced at different levels of the gorge, along with efficient use of materials. This accounts for the top stages having smaller arches, ones that do not experience as much wind force or need to support so much weight. In the Gothic churches you will find this self-similarity principle repeated at many levels, with arches dependent on larger arches dependent on still larger arches dependent on still larger arches. Many chandeliers of this time also exhibit these patterns. These chandeliers are often held by a rod or chain that branches into smaller parts that in turn branch into smaller parts. Both these forms of architecture help distribute weight evenly. Large cathedrals used this type of construction because stone, the primary building material of the time, has a strength insufficient to support a structure with only a frame. This self-similar structure can be seen in the construction of  the Church of the Saint Michael in Northern Germany in Color Plate 8.

Figure 1.28 Computer Generated Fractal Chandelier Similar to those found in Gothic Churches.

            In Africa, you will find self-similar structure incorporated in community design. Figure 1.29 compares  a Songhai village in Mali to a circular fractal.  How would you account for the similarity.  Some say that the fractal pattern was adopted because it makes efficient use of space.  Others suggest that the village is an conscious expression of a fractal structure that became a cultural construct of many regions in Africa.

Figure 1.29 The Songhai settlement compared to circular fractals.

Courtesy of Duly⁴ © 1979 and Mandelbrot⁵ © 1982

            In contrast to the circular structure of the Mali village, notice the branching fractal structure of the 1898 street map of Cairo, shown in Figure 1.30.

Figure 1.30 Saharan village, streets of Cairo, branching fractal all with a comparable structure.

            Figure 1.31 compares Egyptian architectural designs to the Cantor set, a fractal discussed in Chapter 4.  The Egyptian design is based on the symbol for biological recursion of the lotus flower.  Perhaps this design influenced Cantor, who was fascinated with ancient Egypt.

Figure 1.31 Cantor set found in Egyptian capitals⁶.

            The urban sprawl of modern cities often takes on a circular fractal pattern similar to African settlements.  Look at several different maps you own to see if you can identify any fractal patterns.

Fractal Music