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ITERATION INTO THE COMPLEX PLANE:
SYMMETRIES AND TRANSFORMATIONS

JAVIER BARRALLO & SANTIAGO SÁNCHEZ




Name: Barrallo J. Computer Science PhD. (1992), (b. Bilbao, Spain, 1964)

Address: E.T.S. Arquitectura. Plaza Onati, 2. 20018 San Sebastian. Spain. 

E-mail: mapbacaj@sa.ehu.es

Fields of interest: Dynamical Systems, Mathematical Education, Structural Analysis of Heritage Buildings.

Awards: Hobby Press Award, 1992.

Publications and/orExhibitions: 

Fractal Geometry (book 1992), The Frontier between Art & Science (Exhibition 1997 - 2001), Mathematics & Design Conference (1998), ISAMA Conference (1999), The Basque Country Gothic Churches (2000)
 
 

Abstract: Three techniques to generate more intricate fractal symmetry patterns are described in this paper. The first one is based on the development of new fractal formulae, the second is based on mappings of classic formulae according to the 17 crystallographic groups and the third one is the use of kaleidoscopes and tessellations.
 
 

1. INTRODUCTION

Sets obtained under iteration into the complex plane usually show simple symmetric patterns. These sets often have fractal structure, especially when using quadratic functions.

The basic symmetries shown by some elementary functions like the Mandelbrot, Julia, Koch or Sierpinski sets can be modified in order to show more intricate symmetries and visual effects. We will discuss three different techniques:

          a) The creation of new families of formulae that exhibit more sophisticated symmetries. For example a quadratic equation with simple X or Y-axis symmetry can be transformed into a new one showing more symmetric properties.

          b) The use of plane mappings that copy a certain region of the complex plane into others, according to the scheme of any of the 17 crystallographic groups of symmetry.

         c) The design of transformations that produce kaleidoscopes, self-similarity patterns, Truchet tessellations, L-Systems and Penrose and Escher like tiling.
 

2. FORMULA CREATION

Some formulae can be altered to manifest symmetries that usually are not present on them. For example the Mandelbrot and the Sierpinski sets can be transformed as in the following example to present a new and more complex appearance. 
 
 

FOUR MANDELBROTS

init:
   z = 1 + 0i
   c = pixel
loop:
   z = c * (z^2 + 1 / z^2)
bailout:
   |z| < 1000
 
 

SIERPINSKI NET

init:
   z = 0.2 * abs (pixel^3)
loop:
   IF (imag(z) > 0.5)
   z = 2 * real(z) + (2 * imag(z) - 1) * I
   ELSEIF (real(z) > 0.5)
   z = 2 * real(z) - 1 + 2 * imag(z) * I
   ELSE
   z = 2 * real(z) + 2 * imag(z) * I
   ENDIF
Bailout:
   |z| <= 100

 
 

3. PLANE MAPPINGS

Any region of the plane can be mapped into another regions with the same or different shape producing symmetric patterns. The 17 crystallographic groups of symmetry can be used as a basic catalogue to generate these transformations.

The structures obtained iterating the complex variable values into a dynamical system are mapped all along the plane creating an amazing family of graphics with several applications to Art and Design.

 
 

4. TRANSFORMATIONS

After a formula is introduced into a dynamical system, iterated enough times and mapped along the complex plane, the resulting values can be modified by external transformations, not related with the formulae.

These transformations can be designed to produce kaleidoscopes, self-similar patterns, L-Systems, Truchet tessellations and Penrose and Escher like tiling.

Other transformations can be designed to create symmetry effects. For example the following code recreates the reflection of an object over the water surface.
 
 

LAKE TRANSFORMATION

y = imag(pixel)
if (y < imag(screen_centre))
   dy = imag(screen_centre) y
   y = 2 * imag(screen_centre) y
   y = y + alpha * dy * sin(beta * dy^0.2)
   pixel = real(pixel) + y I
 

All these techniques will be widely shown with several new images that have been specially created for the ISIS conference.


 
 

All images created by Javier Barrallo 2001

 

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