# 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 rule of a single side is:

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:

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

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

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

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.