By Kardi Teknomo, PhD .
ComplexNumber


< Next | Previous | Index >

In this section, we will generate a fractal geometry using iterated map. The procedure is so simple that you can even make it in a spreadsheet.

  • Create basic shape and transform the coordinate of points to complex number Complex Number
  • Apply Complex Linear Transformation Complex Linear Transformation for any parameters Fractal Geometry from Complex Number and Fractal Geometry from Complex Number for any parameters Fractal Geometry from Complex Number and Fractal Geometry from Complex Number and Fractal Geometry from Complex Number
  • Transform transformation Complex Number back to Real and Imaginary for plotting back to Real and Imaginary for plotting
  • Use the previous results as input of transformation transformation Complex Number
  • Continue step 3 and 4 until say 100 steps
  • Plot the Complex Plane and you will see fractal figures with self-similarity as zoom in and zoom out.

Fractal Geometry Figures

The following fractal figures are made using a software. The original shape is on the top left (thick red). We use complex linear transformation with iterated map up to 1000 iteration and then we zoom in many times. You can see that as we zoom in, the shape remain the same and we will find out the same shape again and again. This characteristics is called self-similarity.

self-similarity Fractal Geometry from Complex Number Fractal Geometry from Complex Number

Fractal Geometry from Complex Number Fractal Geometry from Complex Number Fractal Geometry from Complex Number

Fractal Geometry from Complex Number Fractal Geometry from Complex Number Fractal Geometry from Complex Number

Below are basic shape and fractal generated using Microsoft Excel.

Fractal Geometry from Complex Number using Excel Fractal Geometry from Complex Number Fractal Geometry from Complex Number

If you change the basic shape and the parameters, you will obtain new fractal geometry

Fractal Geometry from Complex Number Fractal Geometry from Complex Number Fractal Geometry from Complex Number

See Also : Dynamical System , Complex Linear Transformation

< Next | Previous | Index >

Rate this tutorial or give your comments about this tutorial

This tutorial is copyrighted .