## fractal geometry

**-**

## Fractal

# Fractal

A fractal is a geometric figure, often characterized as being self-similar; that is, irregular, fractured, fragmented, or loosely connected in appearance. Fractals were being used hundreds, maybe thousands, of years ago, but were not called fractals. They were designs contained within art and crafts, on designs of carpets and painted floors and ceilings, and within many objects used in everyday life. German mathematician Gottfried Wilhelm Leibniz (1646–1716) developed concepts that helped to eventually define fractals. German mathematician Karl Weierstrass (1815–1897) analyzed and graphed a function in 1872 that today is considered a fractal. Later, in 1904, Swedish mathematician Niels Fabian Helge von Koch (1870–1924) refined Weierstrass’s work and defined another function, what is now called the Koch snowflake.

Polish-French mathematician Benoit B. Mandelbrot (1924–) coined the term fractal to describe such figures, deriving the word from the Latin fractus meaning broken, fragmented, or irregular. He also pointed out amazing similarities in appearance between some fractal sets and many natural geometric patterns. Thus, the term natural fractal refers to natural phenomena that are

similar to fractal sets, such as the path followed by a dust particle as it bounces about in the air.

Another good example of a natural phenomenon that is similar to a fractal is a coastline, because it exhibits three important properties that are typical of fractals. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. Second, the irregularity is basically the same at all levels of magnification. Whether viewed from orbit high above Earth, from a helicopter, or from land, whether viewed with the naked eye, or a magnifying glass, every coastline is similar to itself. While the patterns are not precisely the same at each level of magnification, the essential features of a coastline are observed at each level. Third, the length of a coastline depends on the magnification at which it is measured. Measuring the length of a coastline on a photograph taken from space will only give an estimate of the length, because many small bays and peninsulas will not appear, and the lengths of their perimeters will be excluded from the estimate.

A better estimate can be obtained using a photograph taken from a helicopter. Some detail will still be missing, but many of the features missing in the space photo will be included, so the estimate will be longer and closer to what might be termed the actual length of the coastline. This estimate can be improved further by walking the coastline wearing a pedometer. Again, a longer measurement will result, perhaps more nearly equal to the actual length, but still an estimate, because many parts of a coastline are made up of rocks and pebbles that are smaller than the length of an average stride. Successively better estimates can be made by increasing the level of magnification, and each successive measurement will find the coastline longer. Eventually, the level of magnification must achieve atomic or even nuclear resolution to allow measurement of the irregularities in each grain of sand, each clump of dirt, and each tiny pebble, until the length appears to become infinite. This problematic result suggests the length of every coastline is the same.

The resolution of the problem lies in the fact that fractals are properly characterized in terms of their dimension, rather than their length, area, or volume, with typical fractals described as having a dimension that is not an integer. To explain how this can happen, it is necessary to consider the meaning of dimension. The notion of dimension dates from the ancient Greeks, perhaps as early as Pythagoras (582–500 BC) but at least from Greek mathematician Euclid of Alexandra (c. 325–c. 265 BC) and his books on geometry. Intuitively, mathematicians think of dimension as being equal to the number of coordinates required to describe an object. For instance, a line has dimension 1, a square has dimension 2, and a cube has dimension 3. This is called the topological dimension.

However, between the years 1875 and 1925, mathematicians realized that a more rigorous definition of dimension was needed in order to understand extremely irregular and fragmented sets. They found that no single definition of dimension was complete and useful under all circumstances. Thus, several definitions of dimension remain today. Among them, the Hausdorf dimension, proposed by German mathematician Felix Hausdorff (1868–1942), results in fractional dimensions when an object is a fractal, but is the same as the topological value of dimension for regular geometric shapes. It is based on the increase in length, area, or volume that is measured when a fractal object is magnified by a fixed scale factor. For example, the Hausdorf dimension of a coastline is defined as D = log(Length Increase)/log(scale factor). If the length of a coastline increases by a factor of four whenever it is magnified by a factor of three, then its Hausdorf dimension is given by log(Length Increase)/ log(scale factor) = log(4)/log(3) = 1.26. Thus, it is not

## KEY TERMS

**Dimension—** Intuitively, a figure’s dimension is the number of coordinates required to describe it, such as a line (one), square (two), or cube (three). However, there are other definitions of dimension, based on rigorous definitions of the measure of a set. One such dimension is the Hausdorf dimension, which applies to fractal sets.

**Scale factor—** A scale factor is a constant equal to the ratio of two proportional lengths, areas, or volumes. It is also called the constant of proportionality.

**Similarity—** Two geometric figures are said to be similar if their corresponding shapes are the same but corresponding measurements are in the same proportion, or have the same scale factor.

the length that properly characterizes a coastline but its Hausdorf dimension. Finally, then, a fractal set is defined as a set of points on a line, in a plane, or in space, having a fragmented or irregular appearance at all levels of magnification, with a Hausdorf dimension that is strictly greater than its topological dimension.

Great interest in fractal sets stems from the fact that most natural objects look more like fractals than they do like regular geometric figures. For example, clouds, trees, and mountains look more like fractal figures than they do like circles, triangles, or pyramids. Thus, fractal sets are used by geologists to model the meandering paths of rivers and the rock formations of mountains, by botanists to model the branching patterns of trees and shrubs, by astronomers to model the distribution of mass in the universe, by physiologists to model the human circulatory system, by physicists and engineers to model turbulence in fluids, and by economists to model the stock market and world economics. Often times, fractal sets can be generated by rather simple rules. For instance, a fractal dust is obtained by starting with a line segment and removing the center one-third, then removing the center one-third of the remaining two segments, then the center one-third of those remaining segments and so on.

Rules of generation such as this are easily implemented and displayed graphically on computers. Because some fractal sets resemble mountains, islands, or coastlines, while others appear to be clouds or snowflakes, fractals have become important in graphic art and the production of special effects. For example, fake worlds, generated by computer, are used in science fiction movies and television series, on DVDs (digital versatile discs), and in video games, because they are easily generated from a set of instructions that occupy relatively little computer memory.

## Resources

### BOOKS

Dimri, V.P., ed. *Fractal Behaviour of the Earth*. Berlin, Germany: Springer-Verlag, 2005.

Falconer, Kenneth J. *Fractal Geometry: Mathematical Foundations and Applications*. Chichester, UK, and Hoboken, NJ: Wiley, 2003.

Stevens, Roger T. *Creating Fractals*. Hingham, MA: Charles River Media, 2005.

J. R. Maddocks

## Fractal

# Fractal

A fractal is a geometric figure, often characterized as being self-similar; that is, irregular, fractured, fragmented, or loosely connected in appearance. Benoit Mandelbrot coined the term fractal to describe such figures, deriving the word from the Latin "fractus" meaning broken, fragmented, or irregular. He also pointed out amazing similarities in appearance between some fractal sets and many natural geometric patterns. Thus, the term "natural fractal" refers to natural phenomena that are similar to fractal sets, such as the path followed by a dust particle as it bounces about in the air.

Another good example of a natural phenomenon that is similar to a fractal is a coastline, because it exhibits three important properties that are typical of fractals. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. Second, the irregularity is basically the same at all levels of magnification. Whether viewed from **orbit** high above **Earth** , from a helicopter, or from land,

whether viewed with the naked **eye** , or a magnifying **glass** , every coastline is similar to itself. While the patterns are not precisely the same at each level of magnification, the essential features of a coastline are observed at each level. Third, the length of a coastline depends on the magnification at which it is measured. Measuring the length of a coastline on a photograph taken from space will only give an estimate of the length, because many small bays and peninsulas will not appear, and the lengths of their perimeters will be excluded from the estimate. A better estimate can be obtained using a photograph taken from a helicopter. Some detail will still be missing, but many of the features missing in the space photo will be included, so the estimate will be longer and closer to what might be termed the "actual" length of the coastline. This estimate can be improved further by walking the coastline wearing a pedometer. Again, a longer measurement will result, perhaps more nearly equal to the "actual" length, but still an estimate, because many parts of a coastline are made up of **rocks** and pebbles that are smaller than the length of an average stride. Successively better estimates can be made by increasing the level of magnification, and each successive measurement will find the coastline longer. Eventually, the level of magnification must achieve atomic or even nuclear resolution to allow measurement of the irregularities in each grain of **sand** , each clump of dirt, and each tiny pebble, until the length appears to become infinite. This problematic result suggests the length of every coastline is the same.

The resolution of the problem lies in the fact that fractals are properly characterized in terms of their dimension, rather than their length, area, or **volume** , with typical fractals described as having a dimension that is not an integer. To explain how this can happen, it is necessary to consider the meaning of dimension. The notion of dimension dates from the ancient Greeks, perhaps as early as Pythagoras (582-507 b.c.) but at least from Euclid (c. 300 b.c.) and his books on **geometry** . Intuitively, we think of dimension as being equal to the number of coordinates required to describe an object. For instance, a line has dimension 1, a **square** has dimension 2, and a cube has dimension 3. This is called the topological dimension. However, between the years 1875 and 1925, mathematicians realized that a more rigorous definition of dimension was needed in order to understand extremely irregular and fragmented sets. They found that no single definition of dimension was complete and useful under all circumstances. Thus, several definitions of dimension remain today. Among them, the Hausdorf dimension, proposed by Felix Hausdorf, results in fractional dimensions when an object is a fractal, but is the same as the topological value of dimension for regular geometric shapes. It is based on the increase in length, area, or volume that is measured when a fractal object is magnified by a fixed scale **factor** . For example, the Hausdorf dimension of a coastline is defined as D = log(Length Increase)/log(scale factor). If the length of a coastline increases by a factor of four whenever it is magnified by a factor of three, then its Hausdorf dimension is given by log(Length Increase)/log(scale factor) = log(4)/log(3) = 1.26. Thus, it is not the length that properly characterizes a coastline but its Hausdorf dimension. Finally, then, a fractal set is defined as a set of points on a line, in a **plane** , or in space, having a fragmented or irregular appearance at all levels of magnification, with a Hausdorf dimension that is strictly greater than its topological dimension.

Great interest in fractal sets stems from the fact that most natural objects look more like fractals than they do like regular geometric figures. For example, **clouds** , trees, and **mountains** look more like fractal figures than they do like circles, triangles, or pyramids. Thus, fractal sets are used by geologists to model the meandering paths of **rivers** and the rock formations of mountains, by botanists to model the branching patterns of trees and shrubs, by astronomers to model the distribution of **mass** in the universe, by physiologists to model the human **circulatory system** , by physicists and engineers to model turbulence in fluids, and by economists to model the stock market and world economics. Often times, fractal sets can be generated by rather simple rules. For instance, a fractal dust is obtained by starting with a line segment and removing the center one-third, then removing the center one-third of the remaining two segments, then the center one-third of those remaining segments and so on.

Rules of generation such as this are easily implemented and displayed graphically on computers. Because some fractal sets resemble mountains, islands, or coastlines, while others appear to be clouds or snowflakes, fractals have become important in graphic art and the production of special effects. For example, "fake" worlds, generated by computer, are used in science fiction movies and **television** series, on CD-ROMs, and in video games, because they are easily generated from a set of instructions that occupy relatively little computer memory.

## Resources

### books

Peterson, Ivars. *Islands of Truth, A Mathematical Mystery**Cruise.* New York: W. H. Freeman, 1990.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Dimension**—Intuitively, a figure's dimension is the number of coordinates required to describe it, such as a line (one), square (two), or cube (three). However, there are other definitions of dimension, based on rigorous definitions of the measure of a set. One such dimension is the Hausdorf dimension, which applies to fractal sets.

**Scale factor**—A scale factor is a constant equal to the ratio of two proportional lengths, areas, or volumes. Also called the constant of proportionality.

**Similarity**—Two geometric figures are said to be similar if their corresponding shapes are the same but corresponding measurements are in the same proportion, or have the same scale factor.

## Fractal

# Fractal

A fractal is a geometric figure with two special properties. First, it is irregular, fractured, fragmented, or loosely connected in appearance. Second, it is self-similar; that is, the figure looks much the same no matter how far away or how close up it is viewed.

The term fractal was invented by Polish French mathematician Benoit Mandelbrot (1924– ) in 1975. He took the word from the Latin word *fractus,* which means "broken."

The idea behind fractals is fairly simple and obvious when explained. But the mathematics used to develop those ideas is not so simple.

## Natural fractals

Most objects in nature do not have simple geometric shapes. Clouds, trees, and mountains, for example, usually do not look like circles, triangles, or pyramids. Instead, they can best be described as fractals. Natural objects that can be described as fractals are called natural fractals.

One of the natural objects most often used to explain fractals is a coastline. A coastline has the three properties typical of any fractal figure. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. By definition, any fractal must be irregular in shape.

Second, the irregularity is basically the same at all levels of magnification. Whether viewed from orbit high above Earth, from a helicopter, or from land, whether viewed with the naked eye, or a magnifying glass, every coastline is similar to itself. While the patterns are not precisely the same at each level of magnification, the essential features of a coastline are observed at each level. This property is the self-similar property that also is basic to all fractals.

Third, the length of a coastline depends on the magnification at which it is measured. Measuring the length of a coastline on a photograph taken from space will give only an estimate of its length. Many small bays and peninsulas will not appear, and the lengths of their perimeters will be excluded from the estimate. A better estimate can be obtained using a photograph taken from a helicopter. Some detail will still be missing, but many of the features missing in the space photo will be included. Thus, the estimate will be longer and closer to what might be termed the actual length of the coastline.

This estimate can be improved further by walking the coastline wearing a pedometer. Again, a longer measurement will result, perhaps more nearly equal to the actual length. But the result is still an estimate because many parts of a coastline are made up of rocks and pebbles that are smaller than the length of an average stride. Successively better estimates can be made by increasing the level of magnification, and each successive measurement will find the coastline longer. Eventually, the level of magnification must achieve atomic or even nuclear resolution to allow measurement of the irregularities in each grain of sand, each clump of dirt, and each tiny pebble, until the length appears to become infinite. This problematic result suggests the length of every coastline is the same.

The resolution of this problem requires that we rethink the way space is described. Standard one-dimensional space (such as a point), twodimensional space (such as a line), and three-dimensional space (such as a sphere) are not adequate for the analysis of fractals. Instead, nonintegral dimensions (1½; 2⅓; etc.) are needed.

## Constructions

Fractals can often be drawn by rather simple rules, as shown in the accompanying illustration. This drawing shows how the pathway taken by a dust particle in air can be modeled by using fractals. We begin in Step A with a straight line. First, the center one-third of the line is removed and broken in half, as shown in Step B. Next, the center one-third of each of the three remaining line segments is removed and broken in half, as shown in Step C. This process is repeated over and over again until a fractal figure is formed that looks like the path followed by a dust particle in air.

## Applications

The similarity between fractals and natural objects has resulted in a number of important applications

## Words to Know

**Dimension:** The number of coordinates required to describe a figure, such as a point (one), line (two), or sphere (three).

**Self-similar:** Having an appearance that does not change no matter how far away or how close up an object is viewed.

**Similarity:** Having corresponding shapes.

for this field of mathematics. Fractals are used by geologists to model the meandering paths of rivers and the rock formations of mountains; by botanists to model the branching patterns of trees and shrubs; by astronomers to model the distribution of mass in the universe; by physiologists to model the human circulatory system; by physicists and engineers to model turbulence in fluids; and by economists to model the stock market and world economics.

## fractal

**fractal**
•**battle**, cattle, chattel, embattle, prattle, rattle, Seattle, tattle
•fractal
•**cantle**, covenantal, mantel, mantle, Prandtl
•pastel • Fremantle • tittle-tattle
•**startle**, stratal
•Nahuatl
•**fettle**, kettle, metal, mettle, nettle, petal, Popocatépetl, settle
•**dialectal**, rectal
•**dental**, gentle, mental, Oriental, parental, rental
•transeptal
•**festal**, vestal
•gunmetal
•**antenatal**, fatal, hiatal, natal, neonatal, ratel
•**beetle**, betel, chital, decretal, fetal
•blackbeetle
•**acquittal**, belittle, brittle, committal, embrittle, it'll, kittle, little, remittal, skittle, spittle, tittle, victual, whittle
•**edictal**, rictal
•**lintel**, pintle, quintal
•**Bristol**, Chrystal, crystal, pistol
•varietal • coital • phenobarbital
•orbital • pedestal • sagittal • vegetal
•digital • skeletal • Doolittle
•**congenital**, genital, primogenital, urogenital
•capital • lickspittle • hospital • marital
•**entitle**, mistitle, recital, requital, title, vital
•subtitle • surtitle
•**axolotl**, bottle, dottle, glottal, mottle, pottle, throttle, wattle
•**fontal**, horizontal
•**hostel**, intercostal, Pentecostal
•greenbottle • bluebottle • Aristotle
•**chortle**, immortal, mortal, portal
•Borstal
•**anecdotal**, sacerdotal, teetotal, total
•**coastal**, postal
•subtotal
•**brutal**, footle, pootle, refutal, rootle, tootle
•**buttle**, cuttle, rebuttal, scuttle, shuttle, subtle, surrebuttal
•**buntal**, contrapuntal, frontal
•crustal • societal • pivotal
•**hurtle**, kirtle, myrtle, turtle

## fractal geometry

fractal geometry, branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry. Unlike conventional geometry, which is concerned with regular shapes and whole-number dimensions, such as lines (one-dimensional) and cones (three-dimensional), fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions—linelike rivers with a fractal dimension of about 1.2 and conelike mountains with a fractal dimension between 2 and 3.

Fractal geometry developed from Benoit Mandelbrot's study of complexity and chaos (see chaos theory). Beginning in 1961, he published a series of studies on fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.

See B. B. Mandelbrot, *The Fractal Geometry of Nature* (1983); K. J. Falconer, *Fractal Geometry: Mathematical Foundations and Applications* (1990); H.-O. Peitgen, H. Jurgens, and D. Saupe, *Chaos and Fractals: New Frontiers of Science* (1992).

## fractal

**fractal** Geometrical figure in which an identical motif is repeated on a reducing scale; the figure is ‘self-similar’. Coined by Benoit Mandelbrot, fractal geometry is closely associated with chaos theory. Fractal objects in nature include shells, cauliflowers, mountains and clouds. Fractals are also produced mathematically in computer graphics.

## fractal

**fractal** A set whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension. Intuitively, a fractal is a set which at all magnifications reveals a set that is exactly the same (self-similar). Such sets can be generated by the repeated application of some collection of maps. The term is generally associated with Benoit Mandelbrot and appeared in the literature in the late 1970s. Many naturally occurring objects, such as trees, coastlines, and clouds, are considered to have fractal properties, hence their interest to computer graphics.

## fractal

**fractal** A geometric entity which has a basic pattern that is repeated at ever decreasing sizes. Fractal patterns are not able to fill spaces and are hence described as having fractal dimension. Fractals occur frequently in nature, such as in forked lightning or in chaotic systems (see CHAOS).