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Microcladia and Fractals
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.
Iteration:
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.
Orbits:
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.
Chaotic Quadratics:
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 behavior.
This 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.
Julia Sets:
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
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.
For 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.
What remains self-similar across scale:
- angle of branching
- alternate nature of branching
- number of branches per thallus
A note about Fractal Dimension:
The fractal dimension of any shape can be calculated by this formula:
D = log (number of self-similar pieces)
log (amount by which each piece would have to be magnified to become the whole)
so for instance, in the Serpinski triangle, the main triangle has three separate self-similar parts that each would have to be magnified by a factor of 2 to become the whole. So its fractal dimension would be
D = log 3 = 1.585…
log 2
For the Koch snowflake, each side is composed of four smaller sides that could be magnified by a dimension of 3 to be the main side. Therefore, the fractal dimension is
D = log 4 = 1.262…
log 3
The fractal dimension of a line is 1, and the fractal dimension of a square is 2, a cube 3. So calculating the fractal dimension of the above fractals really only indicates that the Sierpinski triangle is more two-dimensional than the Koch snowflake, which makes sense; it appears to have more area. In that sense, examining the fractal dimension of Microcladia is more interesting on a theoretical level than on a computational level.