On a generalization of Tupper's formula for m colors and n dimensions

Abstract

Tupper's formula 12<⌊mod(⌊y17⌋2−17⌊x⌋−mod(⌊y⌋,17),2)⌋ has an interesting property: that for any monochrome image that can be represented using a 106×17 array of two-dimensional pixels, there exists a natural number k such that the graph of the inequality over the range 0≤x<106 and k≤y<k+17 is that image. In this paper, we give a generalization for m colors and n dimensions. In this paper, by “natural numbers” we mean non-negative integers. We give m formulae employing n free variables with the property that, for any n-dimensional object of m colors C1,…,Cm that can be represented by hypervoxels (hypervoxels being the multidimensional analogue of pixels) in an n-dimensional array of dimensions A1×⋯×An, there exists a natural number k such that, when the first formula is graphed using color C1, the second formula is graphed using color C2, …, and the mth formula is graphed using color Cm over 0≤x1<A1, 0≤x2<A2,…,0≤xn−1<An−1, and k≤xn<k+An, the union of all colored graphs is that n-dimensional object.