The Mandelbrot Set

The Mandelbrot set is the picture of a kind of repetitive calculation.

The Mandelbrot set is a collection of points on a plane. The points form an irregular area inside a single closed curve, like the inside of a circle or an ellipse.

The horizontal dimension of the plane represents real numbers, positive and negative, extending in either direction from the origin 0. Real numbers create the arithmetic of the everyday world.

The vertical dimension of the plane represents imaginary numbers (positive and negative) from the origin 0. An imaginary number is any real number multiplied by i, the square root of -1. Imaginary numbers are useful in certain algebraic problems.

The real and imaginary number dimensions form the complex plane, the geometrical equivalent of complex numbers.

Complex numbers are the sum of a real number and an imaginary number:
z = x + iy.

Real numbers (such as -7.324 or 3.14159...) can be greater than or less than other real numbers. So each number corresponds to a point on a line, like a mark on a ruler. A line is the geometrical equivalent of real numbers.

Complex numbers cannot be greater than or less than other complex numbers. So they cannot be ordered on a line. But the pair of numbers (x,iy) can be mapped as the points on a plane. This complex plane is the geometrical equivalent of complex numbers.

The repetitive process that defines the Mandelbrot set on the complex plane is the mathematical calculation called a Verhulst process:

$$z_n+1 --> z_n² + c.

It's a simple feedback mechanism: an endlessly repeated calculation performed at each step on the result of the previous step.

It works like this: start with any number $$z₀.

Square this number.

Add to $$z₀² the constant c. Call this number $$z₁.

Square $$z₁ and add the constant c to create $$z₂.

Square $$z₂ and add the constant c again to create $$z₃,

and again to create $$z₄,

and so on without limit.

What happens as you repeat these calculations? Depending on the values of z and c you start with, one of three things: z increases rapidly toward infinity; z comes to rest at a constant value, or a repeated series of values; or z jumps randomly and unpredictably in an endless series of unique values.

So the Verhulst process classifies the starting values of z or c into three groups, depending on the outcome of the repeated calculations:

All the starting values of z that eventually fly off to infinity form one group of numbers, called a set. They are the escape set.

The values of z that come to rest at one constant value or series of repeated values form the prisoner set.

The values of z that jump unpredictably in an endless series of unique values form the boundary set.

The Mandelbrot set contains all the points that are in the prisoner set and the boundary set for the behavior of z in a Verhulst process where both z and c are complex numbers:

z = x + iy

c = a + ib.

And this new Verhulst equation looks like this:

$$z_n+1 => [ x_n + iy_n ]² + [ a + ib ]

This equation involves a more complicated set of repeated calculations, because both real and imaginary numbers are involved.

The equation can be algebraically rearranged so that only real numbers remain. In this version, you must calculate separately at each step the values of x and y, then square these and add them together to get the value of z:

$$x_n+1 = x²_n - y²_n + a

y_n+1 = 2x_ny_n + b

z_n+1 = x²_n+1 + y²_n+1

Each of the complex numbers in this equation, z and c, can be represented as a point on the (real, imaginary) dimensions of the complex plane: z by the point (x,y) representing the numbers x and iy, and c by the point (a,b) representing the numbers a and ib.

This means there are two ways to represent graphically on a plane the total range of outcomes of this Verhulst process.

You can use a constant starting value for c and many different starting values for x and iy, and plot the results of these iterations on the (x,y) plane. This will create an image of the Verhulst process called a Julia set.

Or you can use a constant starting value for z and many different starting values for a and ib, and plot the results of these iterations on the (a,b) plane. This will create the Mandelbrot set.

To create the Mandelbrot set, you start with $$x₀ = 0 and $$y₀ = 0 and watch z to see what it does for the constant values of a and ib you have chosen.

Because you always start with the same values of $$x₀ and $$y₀, all differences in the way the repeated calculations unfold are due entirely to the values of a and ib you choose.

The Mandelbrot set is thus an image of the effects on the Verhulst process behavior of variations in the value of the constant c. The set is a pictorial encyclopedia of all these variations.

Every point on the plane will be in the escape set, the prisoner set, or the boundary set of the Verhulst process.

If you perform the repeated calculations for a large number of different values of c, you can show the results by coloring each corresponding point (a,b) in the complex plane according to the behavior of the value of z.

Here are typical coloring rules:

If z comes to rest at a constant value or repeated series of values, color the point (a,b) black. Black represents the prisoner set.

If z increases to infinity, color the point using a contrasting color to black that is not white. Color represents the escape set.

If z never goes either to infinity or to a stable value or repeated series of values, color the point white. White represents the boundary set.

In practice, it's not possible to decide among these three possibilities for many values of (x,y) or (a,b) you might choose, because thousands or millions of iterations may be required before z repeats a numerical orbit or escapes to infinity. This is particularly true for values of (a,b) that are on or near the boundary set.

At the boundary between the prisoner and escape sets, the Verhulst process is chaotic, unpredictable. Even after a million repetitions of the calculations, we can't be sure that a value of z will not escape to infinity, or begin exactly repeating the whole series of numbers again, if we were to continue the computations just a little longer.

But you can reliably abbreviate the process by repeating the calculations until you reach arbitrary but diagnostic limits.

If z increases above some minimum value (such as 10), it is mathematically certain that z will escape to infinity. So when any value of z goes above 10 you stop the process, and color the point (a,b) from blue to red to yellow to show how many repetitions were required for it to reach this point. Blue means a few iterations, red means many.

To prevent the process from repeating forever, if you exceed some arbitrary number of repetitions (such as 1000) and z is still less than 100, the probability is high that it is in the prisoner set. You stop the process and color the point black.

If you record in this way the results for every unique pixel combination of points a and b in the complex plane, the result will be similar to the image above.

These coloring rules create graded contours brightening to the edge of a black silhouette, contrasting clearly the Mandelbrot set and the escape set outside it. But this image of the Mandelbrot set is approximate. More detail will always appear if we increase the number of iterations and the precision used to define the points a and b. The depth of detail is too complex and intricate to permit us to create its exact image.

In the image above, a bead of light shows the zero point where the imaginary and real axes intersect. The Mandelbrot set lies inside a circle of radius 2 inches [96 pixels per inch on a PC] around the origin; it can be proven that every point in the Mandelbrot set must be on or inside this circle. The quadrants of the circle are clouded if they lie in the negative side of either the real or imaginary number dimensions.

Credit to Adam Smith's Floating Fractals 5.1 for the figures shown here. This program is available for free at shareware.com and other shareware sites on the Web.