Self-graphing equations
Samuel Allen Alexander
TL;DR
The paper investigates whether an xy-equation can graph itself on the plane, highlighting issues of typography-driven meaning and triviality. It resolves these by introducing a glyphed framework for equations and a self-constraint condition, then proves existence of self-graphing equations via computability theory, notably the Recursion Theorem and the Smn theorem. A concrete context demonstrates the approach, using an infinite-product construction to encode existential quantification and guarantee a self-graphing equation under the self-constraint. This work provides a rigorous, general foundation linking symbolic representations to geometric graphs, clarifying the role of typography and offering a robust method to obtain self-graphing equations.
Abstract
Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper's self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on typography) and it is trivial (for reasons we will explain). We fix these flaws by formalizing the problem, and we give a very general solution using techniques from computability theory.