Tupper's self-referential formula

Article

July 3, 2022

Tupper's self-referencing formula is a two-variable inequality. When the set of points in the plane that satisfy this inequality are plotted, part of the plane represents the formula itself. Created by Jeff Tupper in 2001, this is an example of self-reference.

Features

Definition

The formula is an inequality defined by: 1 2 < ⌊ m oh ( ⌊ there 17 ⌋ 2 − 17 ⌊ x ⌋ − m oh ( ⌊ there ⌋ , 17 ) , 2 ) ⌋ {\displaystyle {1 \over 2}<\left\lfloor \mathrm {mod} \left(\left\lfloor {y \over 17}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm { mod} (\lfloor y\rfloor ,17)},2\right)\right\rfloor } where ⌊ ⋅ ⌋ {\displaystyle \lfloor \cdot \rfloor } is the integer function and mod is the modulo operator,,,.

Plot

Let k be the number of 543 digits equal to: 960939379918958884971672962127852754715004339660129306651505519271702802395266 424689642842174350718121267153782770623355993237280874144307891325963941337723 487857735749823926629715517173716995165232890538221612403238855866184013235585 136048828693337902491454229288667081096184496091705183454067827731551705405381 627380967602565625016981482083418783163849115590225610003652351370343874461848 378737238198224849863465033159410054974700593138339226497249461751545728366702 369745461014655997933798537483143786841806593422227898388722980000748404719

Operation

Because of the integer part and the modulo 2, the right part of the inequality can only take as value 0 or 1. The solutions of the inequality are therefore those of the equality: ⌊ m oh ( ⌊ there 17 ⌋ 2 − 17 ⌊ x ⌋ − m oh ( ⌊ there ⌋ , 17 ) , 2 ) ⌋ 1 {\displaystyle \left\lfloor \mathrm {mod} \left(\left\lfloor {y \over 17}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm {mod} (\lf