On a Generalization of Tupper's Formula for $m$ Colours and $n$ Dimensions

Abstract

Tupper's formula $\frac{1}{2}<\lfloor \mathrm{mod}(\lfloor \frac{y}{17}\rfloor 2^{-17\lfloor x\rfloor -\mathrm{mod}(\lfloor y\rfloor ,17)},2)\rfloor$ has an interesting property that for any monochrome image that can be represented by pixels in a two dimensional array of dimensions $106\times 17$, there exists a natural number $k$ such that the graph of the equation in the range $0\le x<106$ and $k\le y<k+17$, is that image. In this paper, we give a generalization for $m$ colours and $n$ dimensions. We give $m$ formulae consisting of $n$ free variables, with the property that, for any $n$ dimensional object of $m$ colours $C_1,\cdots ,C_m$, that can be represented by hypervoxels(multidimensional analogue of pixel) in a $n$ dimensional array of dimensions $A_1\times \cdots \times A_n$, there exists a natural number $k$ such that, when the first formula is graphed using colour $C_1$, second formula is graphed using colour $C_2$,$\cdots$, $m$th formula is graphed using colour $C_m$ in the range $0\le x_1<A_1$,$0\le x_2<A_2,\cdots ,0\le x_{n-1}<A_{n-1},k\le x_n<k+A_n$, the union of all graphs is that $n$-dimensional object.

5 pages, submitted it to INTEGERS