Fractal Wallpaper

Background

We will make stages of self-similar fractal patterns created with IFS.

To do this we will restrict ourselves to patterns that do not require reflections, because these would require mirror images of the pictures.

Several patterns not using reflections are shown here.

The boxes with the T show the direction of the rotations; combined with the translations these give the rules for the IFS generating the pattern.

A box with no rotation is shown as .

In these examples the fractal pattern are generated by a deterministic IFS with contraction factors and translations arranged so that at each iteration all the scaled copies are contained in a square of side length one.

Here we take the opposite approach. Starting with a picture one inch square for stage 0,

Stage 1 consists of 3 one inch pictures that fit in a square of side length 2,

Stage 2 consists of 9 one inch pictures that fit into a square of side length 4,

Stage n consists of 3n one inch pictures that fit in a square of side length 2n.

The first three stages of Nial's cat becoming the fractal pattern 4

Here we find how many pictures we need to make the pattern for display on the wall.

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