To derive a formula which will work with all figures, lets first look
at how to calculate the dimensions for the figures which we already
know. A line can be divided into n = n1 seperate pieces. Each
of those pieces is 1/n the size of the whole line and each piece, if
magnified n times, would look exactly the same as the original.
Repeating the process for a square, we find that is can be divided into
n2 pieces. The same concept holds true for a cube, we need
n3 pieces to reassemble a cube. Each of the pieces would be
1/n the size of the whole figure. The exponent in each of these
examples is the dimension. For fractals, we need a generalized
formula, which can be derived from what we already know. The steps bellow
assume you have a working knowledge of logarithims and basic algebra.
Note: ln denotes loge and may be refered to as the
natural logarithim. Because of the way in which this formula ends up, it
is independant of the base used for the logarithims.
for a line: ln(number of divisions) = ln(n1)
for a square: ln(number of divisions) = ln(n2)
for a cube: ln(number of divisions) = ln(n3)
If you look back, the figure was divided into pieces that when zoomed
in on n times, revealed to starting figure. Because of this, we divide
the ln(number of divisions) by the natural logarithim of the
magnification facator. The resulting formula gives the dimension,
represented by D.
D=ln(number of divisions)/ln(magnification factor)
for a line: D = ln(n1)/ln(n) = 1
for a square: D = ln(n2)/ln(n) = 2
for a cube: D = ln(n3)/ln(n) = 3
Each of these examples was easy because the magnification factor was
always n. But for fractals, magnification factor will be a constant,
which varies for each fractal. Because you are unfamiliar with specific
fractals, we can not examine specific cases now. Under the section
the dimension of the individual fractals will be examined in more detail.
What Are Fractals?
For the most part, when the word fractal is mentioned, you
immediately think of the stunning pictures you have seen that were called
fractals. But just what exactly is a fractal? Basically, it is a rough geometric
figure that has two properties: First, most magnified images of fractals are
essentially indistinguishable from the unmagnified version. This property of
invariance under a change of scale if called self-similiarity. Second, fractals
have fractal dimensions, as were described above. The word fractal was
invented by Benoit Mandelbrot, "I coined fractal from the Latin adjective
fractus. The corresponding Latin verb frangere means to break to create
irregular fragments. It is therefore sensible and how appropriate for
our needs! - that, in addition to fragmented, fractus should also mean
irregular, both meanings being preserved in fragment."
Graphical Representation Of Fractals
Graphically, fractals are images created out of the process of a mathematical
exploration of the space in which they are plotted. For this page, a computer
screen will represent the space which is being explored. Each point in the area
is tested in some way, usually an equation iterating for a given period of time.
The equations used to test each point in the testing region are often extremely
simple. Each particular point in the testing region is used as a starting point to
test a given equation in a finite period of time. If the equation escapes, or
becomes very large, within the period of time, it is colored white. If if
doesn't escape, or stays within a given range through out the time period, it is
colored black. Hence, a fractal image is a graphical representation of the points
which diverge, or go out of control, and the points which converge, or stay
inside the set. To make fractal images more elablorate and interesting, color is
added to them. Rather than simply plotting a white point if it escapes, the point
is assigned a color relative to how quickly it escaped. The images produced
are very elaborate and possess non-Euclidean geometry. Fractals can
also be produced by following a set of instructions such as remove
the center third of a line segment. A more complete explanation
of how to generate fractal images, specific to individual fractals, follows.
The Cantor Set
The Koch Curve
The Koch Snowflake
The Sierpinski Triangle
The Lorenz Model
The Mandelbrot Set
The Julia Set
The Logistic Equation
Bellow are several sections, each dealing with an individual
fractal. Of course not all of the fractals in the world are listed
bellow, but only ones which are well known or show and important point
which everyone should know. With each fractal, there is a picture,
followed by some information about it. For many of the fractals, there
is also a link to a C/C++ or BASIC program which will generate a
picture of the fractal. For more working soure code, visit the
Appendix Of Source Code.
Even if you are not interested in these specific fractals, it is strongly
enouraged that you read through each one because many topics other than
the specific fractal are reviewed. For example, strange attractors and
several applications of fractals to real-life situations are discussed.
The Cantor set
Figure 3. The Cantor set
The Cantor set is a good example of an elementary fractal. The
object first used to demostrate fractal dimensions, figure 1c, is
actually the Cantor set. The process of generating this fractal is very
simple. The set is generated by the iteration of a single operation on a
line of unit lenght. With each iteration, the middle third from each lines
segment of the previous set is simply removed. As the number of
iterations increases, the number of seperate line segments tends to
infinity while the length of each segment approaches zero. Under
magnification, its structure is essentially indiguishable from the
whole, making it self-similiar.
To calculate the dimension of the Cantor set, we first realize that
its magnification factor is three, or the fractal is self-similiar if
magnified three times. Then we notice that the line segments decompose into
two smaller units. Using the formula given in the section entitled
Calculating Fractal Dimensions,
D = ln(2) / ln(3)
The Cantor set has a dimension of 0.6309.
D = 0.6931 / 1.0986
D = 0.6309
The Koch Curve
Figure 4. The Koch Curve
So far, all of the examples in this document have delt with removing
pieces from various geometric figures. Fractals, and fractal dimensions
can also be defined by adding onto geometric figures. The Koch curve was
named after Helge von Koch in 1904. The generation of this fractal is
simple. We begin with a straight line of unit length and divide it into
three equally sized parts. The middle section is replaced with and
equilateral triangle and its base is removed. After one iterations, the
length is increased by four-thirds. As this process is repeated, the
length of the figure tends to infinity as the length of the side of each
new triangle goes to zero. Assuming this could be iterated an infinite
number of times, the result would be a figure which is infinitely wiggly,
having no straight lines whatsoever.
To calculate the dimension of the Koch Curve, we look at the image of the
fractal and realize that it has a magnification factor of three and with
each iteration, it is divided into four smaller pieces. Knowing this, we get
D = ln(4) / ln(3)
The Koch Curve has a dimension of 1.2619.
D = 1.3863 / 1.0986
D = 1.2619
Figure 5. The Koch Snowflake
As would be expected, the Koch Snowflake is generated in very much
the same way as the Koch Curve. The only variation is that, rather than
using a line of unit length as the intial figure, an equilater triangle
is used. It is iterated in the same way as the Koch Curve. The length
of the resulting figure tends to infinity as the length of the side of
each new triangle goes to zero. Iterated an infinite number of times, the
Koch Snowflake, like the Koch Curve, has absolutely no straight lines in it.
This fractal, if magnified three times in any area, also displays the
property of self-similiarity.
As mentioned above, the magnification factor of this fractal is
three, and as with the Koch Curve, the number of divisions in each
magnification is four. With this we get:
D = ln(4) / ln(3)
The Koch Snowflake has a dimension of 1.2619.
D = 1.3863 / 1.0986
D = 1.2619
The Sierpinski Triangle
Figure 6. The Sierpinski Triangle
Unlinke the Koch Snowflake, which is generated with infinite
additions, the Sierpinski triangle is created by infinite removals. Each
triangle is divided into four smaller, upside down triangles. The center
of the four triangles is removed. As this process is iterated an infinite
number of times, the total area of the set tends to infinity as the size
of each new triangle goes to zero.
After closer examinition of the process used to generate the
Sierpinski Triangle and the image produced by this process, we realize
that the magnification factor is two. With each magnification, there are
three divisions of the triangle. With this data, we get:
D = ln(3) / ln(2)
The Sierpinski Triangle has a dimension of 1.5850.
D = 1.0986 / 0.6931
D = 1.5850
The Sierpinski Triangle is one of the easiest fractals to generate
yourself. This particular program is written in C/C++ and uses the
Borland graphics routines, making it easy to port to other languages on
different platforms. The source code to this program is available
The Lorenz Model
Figure 7. The Lorenz Model
The Lorenz Model, named after E. N. Lorenz in 1963, is a model for the
convection of thermal energy. This model was the very first example of
another important point in chaos and fractals, dissipative dynamical
systems, otherwise know as strange attractors. Strange attractors are
covered more in depth
The Lorenz Model is not a particularly difficult fractal to generate
graphically. The source code for a program, written in C/C++, which will
generate an image of the Lorenz Model is available
The Mandelbrot set
Figure 8. The Mandelbrot set
Named after Benoit Mandelbrot, The Mandelbrot set is one of the most
famous fractals in existance. It was born when Mandelbrot was playing
with the simple quadratic equation z=z2+c. In this equation,
both z and c are complex numbers. In other words, the Mandelbrot set is
the set of all complex c such that iterating z=z2+c does not
To generated the Mandlebrot set graphically, the computer screen
becomes the complex plane. Each point on the plain is tested into the
equation z=z2+c. If the iterated z stayed withen a given
boundry forever, convergence, the point is inside the set and the point
is plotted black. If the iteration went of control, divergence, the
point was plotted in a color with respect to how quickly it escaped.
When testing a point in a plane to see if it is part of the set, the
initial value of z is always zero. This is so because zero is the
critical point of the equation used to generate the set. Critical points
are explained in depth
For a program, again written in C/C++, which will generate the
Mandlebrot set graphically and allow you to explore the set, go
For a program writting in BASIC which will generate the Mandelbrot set, go
The Julia set
Figure 9. The Julia set
The Julia set is another very famous fractal, which happens to be very
closely related to the Mandelbrot set. It was named after Gaston Julia,
who studied the iteration of polynomials and rational functions during
the early twentieth century, making the Julia set much older than the
The main difference between the Julia set and the Mandelbrot set is
the way in which the function is iterated. The Mandelbrot set iterates
z=z2+c with z always starting at 0 and varying the c value.
The Julia set iterates z=z2+c for a fixed c value and varying
z values. In other words, the Mandelbrot set is in the parameter space,
or the c-plane, while the Julia set is in the dynamical space, or the
A program writtin in C/C++ which will graphically generate the
Mandelbrot set can be found
Its BASIC counterpart is located
Figure 10. Diagram plotted for the logistic equation
This particular fractal has much more of a relevance to real-life
than many other fractals. The logisitc equation is a model for animal
populations. The actual equation used is t=c(1-t), where t is the
population, between 0 and 1, and c is a constant representing the growth
rate. Iteration of this equations results in the doubling route to
chaos. For c equaling a value between one and three, the population will
settle to a fixed value. In other words, one year the population is
high, causing the population the next year to be low, causing a high
population the year after. With this equation, the population cycle is
spread out over four years. The period doubles faster and faster,
resulting in the numbers 3.54, 3.564, 3.569, etc. When this value
reaches 3.57, chaos occurs and the population never settles to a fixed
point. For most values of c between 3.57 and 4, the population is
In a period doubling situation, such as the logistic equation, the
ratio of distances between the consecutive doubling parameter values as
the equation is iterated to infinity is call Feigenbaun's constant.
Based on computations by Jay Hill and Keith Briggs, the constant has a
value of 4.669201609102990671853... The interpretation of this equation
is that as you approach chaos, each periodic region is smaller then the
previous by a factor which approaches Feigenbaum's constant. This
constant is the same for any quadratic function of system that follows the
periodic doubling route which results in chaos. The constant varies for
funtions of a higher order.
Real-Life Relevance And Importance of Fractals and Fractal Geometry
Fractals have and are being used in many different ways. Both artist
and scientist are intrigued by the many values of fractals. Fractals are
being used in applications ranging form image compression to finance. We
are still only beginning to realize the full importance and usefullness
of fractal geometry.
One of the largest relationships with real-life is the similarity
between fractals and objects in nature. The resemblance many fractals and
their natural counter-parts is so large that it cannot be overlooked.
Mathematical formulas are used to model self similiar natural forms. The
pattern is repeated at a large scale and patterns evolve to mimic large
scale real world objects.
One of the most useful applications of fractals and fractal geometry
in in image compression. It is also one of the more controversial
ideas. The basic concept behind fractal image compression is to take an
image and express it as an iterated system of funtions. The image can be
quickly displayed, and at any magnification with infinite levels of
fractal detail. The largest problem behind this idea is deriving the
system of functions which describe an image.
One of the more trivial applications of fractals is their visual effect.
Not only do fractals have a stunning aesthic value, that is, they are
remarkably pleasing to the eye, but they also have a way to trick the mind.
Fractals have been used commercially in the film industry, in films such as
Star Wars and Star Trek. Fractal images are used as an
alternative to costly elaborate sets to produce fantasy landscapes.
Another seemingly unrelated application of fractals and chaos is in
music. Some music, including that of Back and Mozart, can be stripped
down so that is contains as little as 1/64th of its notes and still
retain the essence of the composer. Many new software applications are
and have been developed which contain chaotic filters, similiar to those
which change the speed, or the pitch of music.
Fractal geometry also has an application to biological analysis.
Fractal and chaos phenomena specific to non-linear systems are widely
observed in biological systems. A study has established an analytical
method based on fractals and chaos theory for two patterns: the dendrite
pattern of cells during development in the cerebellum and the firing
pattern of intercelluar potential. Variation in the development of the
dendrite stage was evaluated with a fractal dimention. The order in many
ion channels generating the firing pattern was also evaluated with a fractal
dimension, enabling the high order seen there to be quantized.
Return to the Chaos Theory, Dynamic Systems, And Fractal Geometry