Creating Self-Similar Fractals with Hutchinson Operators

translation

rotation and scaling

1

2

3

iterations

0

1

2

3

4

5

6

7

points

triangles

starting set

fixed points

constant points

A map

M

is a contraction mapping if for all points

x

,

y

,

|M(x)-M(y)|≤r|x-y|

, where

0≤r<1

. A similitude is a contraction mapping that is a composition of dilations, rotations, translations, and reflections. A two-dimensional Hutchinson operator maps a plane figure to the union of its images under a finite collection of similitudes. The orbit of a plane figure under such an operator can form a self-similar fractal. In this Demonstration you can vary three similitudes (without reflection) to see what self-similar fractals are possible.

As long as the initial subset of the plane is compact, iterations of the Hutchinson operator converge to the same fractal, yet the convergence is faster for some subsets than others; in particular, the set

F

of the three fixed points of the three similitudes gives fast convergence.

To better see what a Hutchinson operator does, use "constant points" to start with the same three points rather than

F

.