Various types of fractal

There are various types of fractals.

Fractal which uses only iterations (IFS fractal or rather Iterated Function System)

These fractals uses a simple calculation. To finish the calculation requires an infinite number of iterations. This calculation is usually developed on a basis that can be a line too. Examples of these types of fractals are:

  • The Von Koch snowflake
  • The Von Koch curve
  • The Cantor set
  • The Sierpinsky carpet
  • The Sierpinsky triangle
  • The dragon curve
  • The Wolfram triangle
  • The Menger sponge (3D)
  • The Peano curve and so on.

Questi frattali rispettano la definizione “frattale” of Benoit Mandelbrot.

Tappeto di Sierpinsky

Sierpinsky Carpet

To represent its structure and its algorithm it uses the L system.

Escape-time fractals (or orbit fractals)

They are orbit fractals if, during the iterative calculation, it uses precisely an iterative function. This function is applied to a real horizontal plane and an imaginary vertical one. It is used the Bailout's condition to prevent to the points (that exceed this condition) the reaching of infinity.

mandelbrot

Mandelbrot set

The most famous representation is the Mandelbrot fractal one. But, there are various methods of calculation such as the Gaston-Julia fractal, too.

Insieme di Julia ingrandito

Julia set

The Julia fractals use a very special procedure: it use a seed to calculate the fractal. The seed is a complex number x iy. Unlike the Mandelbrot fractal, the Julia fractal is self-similar. The sum of n Julia fractals with 1/n width is a fractal like the Mandelbrot set.

The most common orbit fractals are:

  • All Mandelbrot and Julia fractals with any exponent
  • Newton Fractal
  • Nova Fractal (the Julia of Newton fractal)
  • Lyapunov fractal
  • Barnsley Fractals type 1, 2 and 3
  • Magnet Fractal
  • The conjugate of the Mandelbrot set (Mandelbar)
  • Various types of fractals such as the phoenix, the burning ship, the spider, the tricorn, the cat's eye fractals and so on.

Flame fractals (or Flame)

The flame fractals are IFS fractals but they have a peculiarity: they couldn't be self-similiar. Transformations and attractors are used to calculate these fractals. It is used an indefinite number of iterations. Indeed, si utilizza il fattore “peso” factor of each transformation for a clearer graphic and geometric representations.. Coordinates xy are used particularly. The program that calculates this fractal type is Apophysis.

Frattale fuoco tipo "calm"

Fractal fire

Quaternion set

This particular type use a complex number made up of a real part and three imaginary ones. These fractals use, then, four axes where an axis is not visible. The discovery of these fractals is very recent, so there are still many secrets to be solved.

Insieme di quaternioni

Quaternion set (Julia)

Random fractals

Random fractals are very particular. They use a random and a chaotic motion. Brownian motion is the most striking example.

Moto browniano

Brownian motion

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