Fractals are geometrically self-similar shapes that are
often fascinating to behold, such as the one below. They are created by
iterating a mathematical function or geometric transformation, but because
of certain qualities they can sometimes be more useful for describing natural
phenomena such as leaves and coastlines than the combinations of regular
geometric shapes such as squares and triangles.
To understand fractals, you must begin with understanding
the principles of iteration.
For instance, take a certain function, like the square function, f(x) = x^2.
What happens as you iterate any function is that the result most often does one
of two things: It will either
1. spiral to infinity, or
2. approach a fixed point.
For example, iterate the squared function, f(x) = x2. If you choose x = 4,
4^2 = 16
16^2 = 256
256^2 = 65536… it will spiral to infinity.
But, begin with the number 1, 1^2 = 1, and the function remains at 1. This makes
1 a fixed point. Similarly, begin with a number between one and zero, such as
x = .5,
.5^2 = .25
.25^2 = .0625
.625^2 = .00390625...
nd the result will eventually approach zero. Therefore, for
the squared function, the points 1 and 0 are fixed points. The definition
of a fixed point is a point where F(x) = x.
For some functions, points will oscillate between
different fixed points rather than going towards a single fixed point. This is
known as an orbit.
For example, in the function f(x) = -x, zero is a single fixed point but all
other points lie on a orbit of period 2. To demonstrate,
f(2) = -2
f(-2) = 2
so the function bounces from one point to another. Orbits can have any number
of fixed points.
In the quadratic equation, f(x) = x2 + c , certain ranges
of c values display a behavior where the number of fixed points increases exponentially,
for instance one range of c values will have 2 fixed points, the next range 4,
and then 8, and so on. This behavior swiftly diverges towards infinity, so certain
c values produce an infinite number of fixed points. This is bifurcating, chaotic
diagram plots the number of fixed points for each c-value. As you can
see, some c-values have one fixed point, some have two, and then the graph
swfitly diverges towards an infinite number of fixed points.
Points that are attracted to a cycle of fixed points,
regardless of how many fixed points are in the cycle, are in the basin of attraction.
A Julia set is defined as the set of points right on the boundary between the
points in the basin of attraction and the points that escape towards infinity.
It is plotted by coloring a point black if it is attracted to a fixed point,
and white if it is not.
The Mandlebrot set
The Mandlebrot set is the grandaddy of all fractals in the
sense that it connects chaotic behavior with the behavior of Julia
sets. To understand a Julia set, one can analyze the behavior of the critical
point, zero . Zero is the ciritcal point because it is where the slope
is zero for the quadratic equation. If, within the iterated function, an
input of zero diverges to infinity, the Julia set is called fractal dust
. If the orbit of zero is attracted, the Julia set is connected . The Mandlebrot
set is defined as the set of c-values for the function f(x) = x^2 +
c for which the critical orbit does not diverge to infinity. Each point
that is painted black corresponds to a Julia set that is connected.
A note about coloring: rainbow fractals are colored so that while each
black point corresponds to a set that is connected or attracted, the color
of the rainbow area depends on how swiftly the input diverges to infinity.
Whereas in black and white fractals this area is left white.
To see how each point on the Mandelbrot Set correlates to a Julia Set,
check out The
Mandelbrot and Julia Set Explorer. This awesome site depicts the entire
range of beautiful fractals possible within the Mandelbrot set.
Fractals are geometrical shapes that follow this same iterative process, on a
geometrical level rather than algebraic. They are defined as having two special
properties: 1. They are self- similar. 2. They have fractal dimension.
instance, one of the more famous, the Serpinski triangle, is generated by
removing the middle equilateral triangle from a larger equilateral triangle.
If you iterate this function for every equilateral section in the diagram,
you emerge with a fractal that is self-similar, because any magnified portions
yields the exact same properties as the whole.
The Koch snowflake is an example of the same, but with successive
additions of an equilateral triangle on every side.
One amazing property of fractals like this is that they have
a finite area but their perimeter is infinite . This had lead many observers
to surmise, as Mandelbrot did, that fractals could form an excellent description
of such natural phenomena as coastlines, ferns, leaves, and the bifurcation
of blood veins, that all have self-similar qualities as one descends through
scale. Especially in the development of organisms, it makes sense for purposes
of diffusion to have a nearly infinite surface perimeter in a finite area,
such as the development of circulatory systems.
For examples of fractal images in nature, look at The
Center for Polymer Studies' Image Galleries.
The classic example of fractals in plant-life is that of the
fern, where each smaller leaf resembles the larger one. Examining Microcladia
coulteri is reminiscent of examining ferns, as each smaller bifurcation
appears to retain the same angle of branching and alternate pattern. If
one were to describe its surface with a function, it is easy to imagine
an iterative function forming the basis for all of the development from
the tiny tips to the main thalli.