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

TL;DR

This paper generalizes Tupper's formula to represent any n-dimensional, m-colour object within a hypervoxel array using m formulae with n free variables, extending the original 2D monochrome pixel representation.

Contribution

It introduces m formulae with n free variables that can represent any n-dimensional, m-colour object within a hypervoxel array, generalizing Tupper's original 2D monochrome formula.

Findings

Provides m formulae for n-dimensional objects

Ensures representation of any m-colour hypervoxel object

Extends Tupper's formula to higher dimensions and colours

Abstract

Tupper's formula $\frac{1}{2}<\bigg\lfloor \bmod \bigg(\lfloor \frac{y}{17}\rfloor 2^{-17\lfloor x \rfloor -\bmod (\lfloor y \rfloor,17)},2\bigg)\bigg\rfloor$ 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 $0\leq x <106$ and $k\leq y<k+17$, 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 $0\leq x_1<A_1$,$0\leq x_2<A_2,\cdots, 0\leq x_{n-1}<A_{n-1},k\leq x_n <k+A_n$, the union of all graphs is that $n$-dimensional object.