How do you make a fractal?
A fractal is simply a graph of a function. For example, a fractal function:
f(n) = f(n) * f(n) + c or f(n)2 + c. (Known as the Recursion Law).
This specific equation will form a fractal known as the Julia set. In this equation, c is a
complex number. Complex numbers are explained below along with iteration.
n represents the co-ordinates of the point.
In this situation, where the co-ordinates are (x, y), in fractal geometry, it would be
represented as x + iy. x is the real part whilst y is the imaginary number. In fractal geometry,
the x-axis represents the real numbers, and the y-axis the imaginary numbers.
The result of the function is a point (not a line), which can be infinitely small, (which
explains why it is possible to enlarge a section of a fractal to reveal a new fractal pattern).
The colours of fractals are created as follows: By first selecting a starting point to colour,
for example (2 + i) and a value of c, for example (1 + i), and putting them through the function
f(n),
f(n) = f(2 + i)2 + (1+i)
f(n) = 2*2 + 2i + 2i + i2 + 1 + i
f(n) = 4 + 5i.
The resulting value is the new set of co-ordinates. The process is iterated (see iteration
below). The colour is determined by the number of iterations it takes for the point to leave
the co-ordinates available on the graph. (This means that if the x co-ordinate is 124, and
the graph has a maximum x value of 10, that the point has left the graph).
If the point leaves after one iteration, it takes a certain colour, and every other point that
also leaves the graph after one iteration, takes the same colour. All the points that leave
the graph after two iterations will be given a different colour, and so on. Every point that
never leaves the screen is assigned one colour (which is usually black). A point is considered
to never leave if it is still there after 200 iterations. When this function is iterated for all
the values on the graph, the fractal is formed.
Next: Julia Sets
About Iteration and Complex Numbers
Iteration involves taking a starting value of x and putting it through a function. The resulting x is then put through exactly the same function to give the next x (an nth term of x). This process is repeated to find the required value, (at a reasonable level of accuracy). For example, it may give a result of 0.1 followed by 0.0000002 which if it was followed by 2*10-40 would be said to be tending to zero.
When iterating polynomials, the more interesting fractals are found using quadratics (with x2 terms) or higher polynomials. Linear functions yield less striking results. The example below in the Julia Set section shows the formation of a fractal with a quadratic function and a complex number. Complex numbers are used in expressing the roots of a negative quantity, for example, the square root of -36 = 6i. The use of imaginary numbers such as the one above combined with the real numbers, has lead to the formation of complex numbers which are used in the formation of Julia Sets like those in the gallery.
