The Fractals' Math

How can Fractals be generated? And what's their characteristics?

How can they be generated?

Fractals can be generated by:

  • computer
  • mathematical formulae
  • iterations
  • rendering algorithms

The most important feature of a Fractal are iterations. To obtain spectacular result, a high number of iterations must be computed. This number can be infinite too.

Unfortunately, computers don't have unlimited computational capacity. If the number of iteration are high enough, flaws are not visibile by human eye anymore.

These imperfections remain hidden whether the Fractal is magnified. However, if the zoom is too high, these flaws are visible again.

To overcome this problem, the Fractal must be generated with increased number of iterations. Nevertheless, the computation will become more and more complex, therefore it will become slow.

Iteration

Iteration is a single computation applied recursively on a formula. A recursion is the calculation of a function which uses previous results as parameters.

For instance, we compute this formula:

\[a_{n+1}=a^2_n+1\]

If we set \(a_0=2\), then the first iteration is:

\[a_1=a^2_0+1=2^2+1=4+1=5\]

The second iteration is:

\[a_2=a^2_1+1=5^2+1=25+1=26\]

The third iteration is:

\[a_3=a^2_2+1=26^2+1=676+1=677\]

And so on.

Complex Fractals can be rendered with simple mathematical computation. However, complex rendering algorithms are required to generate and view these Fractals correctly and efficiently.

Fractals' rendering algorithms

Rendering algorithms classify each Fractal by a particular set. The most important Fractals sets are:

  • Lindenmayer system Fractals
  • Escape time (or orbit) Fractals
  • Flame Fractals
  • Chaotic attractors
  • Three and four dimensional Fractals
  • And more

The simpliest Fractal are generated by Lindenmayer algorithm. For each iteration, this algorithm transform every side to a repeated geometric structure.

The starting geometric shape is axiom. Rule states how each side must be repeated.

For example, we take the Von Koch snowflake. The axiom is:

The axiom of the Von Koch snowflake
The axiom of the Von Koch snowflake

The rule of a single side is:

The transformation rule of the Von Koch snowflake
The transformation rule of the Von Koch snowflake

The axiom is a triangle with three equal sides. This rule transform each side into a four sided shape. If we apply this rule for each side on the axiom, the shape will become a twelve sided star. Therefore, the transformed shape is the Von Koch snowflake at first iteration:

Von Koch snowflake, iteration 1
Von Koch snowflake, iteration 1

The second iteration transform the shape into \(12 \times 4=48\) sided one:

Von Koch snowflake, iteration 2
Von Koch snowflake, iteration 2

The third iteration transform the shape into \(48 \times 4=224\) sided one:

Von Koch snowflake, iteration 3
Von Koch snowflake, iteration 3

After a suffient number of iterations, the shape will be like a real snowflake:

Von Koch snowflake, iteration 10
Von Koch snowflake, iteration 10

It should be noted that this Fractal doesn't have got any visibile flaws, thanks to a high enough number of iterations.

Conclusion

Fractals can copy natural shapes, such as snowflakes, trees, plants (for example cauliflowers) and so on. These algorithms can also be used to render realistic natural environment, with little human efforts.

Mountain environment created by a fractal rendering algorithm
Mountain environment created by a fractal rendering algorithm