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

Sai Teja Somu; Vidyanshu Mishra

Home/Archive/arXiv

arXiv · 2109.11013

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

Sai Teja Somu·Vidyanshu Mishra

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Sai Teja Somu, Vidyanshu Mishra. 2021-09-22. On a Generalization of Tupper's Formula for $m$ Colours and $n$ Dimensions. https://arxiv.org/abs/2109.11013

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗