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About algorithmic worldsIntroduction
The piling operator
As we have seen above, a pictorial algorithm working "pixel by pixel" takes the coordinates of a pixel, and returns a color. Mathematically, it is nothing but a function from the plane into color space. The computer produces a digital image by evaluating this function at the positions of the pixels. The image actually computed is only a part, limited in size and in resolution, of an infinite image extending over the whole plane. For most algorithms, however, only a tiny part (of measure zero, in mathematical terms) of the plane displays aesthetically interesting structures.
One can therefore wonder if it would be possible to create a pictorial algorithm with the following properties:
Note that the latter requirement is the (informal) definition of a fractal. Most fractals do not satisfy the first requirement, however.
The picture produced by such an algorithm would be very similar in nature to the virtual satellite maps which appeared over the last few years on the net (typically, Google Maps). Such a satellite photograph displays structures on a very wide scale range (from a few meter to a few thousand kilometers), and these structures are present everywhere (at least on the continents).
By analogy, I would like to name the products of algorithms satisfying the two conditions above algorithmic worlds. With Ultra Fractal, it is possible to freely travel in these worlds (just like with Google Maps or with the zoomable images on this
website, but without any restriction on zooming or panning). An image
produced by such an algorithm is only a local photograph limited in
resolution of the underlying and infinitely more complex algorithmic