Discrete Self-Similar Fractals

Discrete self-similar fractals and a subset of them that we call 'nice' discrete self-similar fractals are defined below. Note that the Fibered Fractal Tiler allows you to simulate the 'growth' of any discrete self-similar fractal, but you can only generate fibered tilesets and their seeds for the 'nice' fractals.

Definition of discrete self-similar fractals

The following is a formal definition for discrete self-similar fractals. Following that is a higher level, more intuitive description. Note that these definitions and descriptions only deal with the full shape forming in the first quadrant of the plane (i.e. all points having positive x and y coordinates). However, it is trivial to rotate the shapes to be in any of the four quadrants.

A discrete self-similar fractal can be thought of as a shape that is produced in an infinite number of stages (since it is an infinite shape). The 0th stage consists of only the point (0,0). The first stage (which is the stage that you must specify in order to generate a fibered tileset or to simulate the fractal's growth) consists of the points (0,0) plus some additional, arbitrary points in the first quadrant. Every subsequent stage (and there are an infinite number of these) consists of one entire copy of the previous stage for every point that was in the first stage. Each of these copies are placed so that, if the image was zoomed out so that the previous stage appeared as the single point (0,0), they fill the locations equivalent to each of the points in the first stage.

Definition of 'Nice' Discrete Self-Similar Fractals

Nice discrete self-similar fractals are a subset of discrete self-similar fractals. The restrictions in terms of the definition of the first stage of the fractal are listed below: