Enumerate all images (2-colourings of the n*n grid Z_n^2), for instance by lexicographically ordering them for each value of n, and then ordering the sets in increasing order of n. So, we get something like: [0] [1] [0 0] [0 0] [0 0] [0 1] ... [1 1] [1 0] [1 1] [1 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] ... We can take the nth image of this sequence, and scale it to occupy a square of side length (½)^n. These can easily be placed on the diagonal of a unit square. Colour everything else arbitrarily. The resulting 2- colouring of the unit square thus contains scaled-down versions of arbitrarily close approximations to all 2-colourings of the unit square, including itself. I propose this property be called 'omni-similarity'. It is stricter than the quasi-self-similarity found in Julia sets. I have heard that the Mandelbrot Set exhibits omni-similarity. Is this true? It certainly seems implausible. Sincerely, Adam P. Goucher