Demystifying Tupper's formula

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2023-05-23 03:30:03

A book I was recently reading mentioned a mathematical curiosity I haven't seen before - Tupper's self-referential formula. There are some resources about it online, but this post is my attempt to explain how it works - along with an interactive implementation you can try in the browser.

For this purpose, it's more useful to think of Tupper's formula not as a function but as a relation, in the mathematical sense. In Tupper's paper this is a relation on , meaning that it's a set of pairs in \mathbb{R} \times \mathbb{R} that satisfy the inequality.

For our task we'll use discrete indices for x and y, so the relation is on \mathbb{N}. We'll plot the relation by using a dark pixel (or square) for a x,y coordinate where the inequality holds and a light pixel for a coordinate where it doesn't hold.

The "mind-blowing" fact about Tupper's formula is that when plotted for a certain range of x and y, it produces this:

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