Obstructions to Reality: Torsors & Visual Paradox

TL;DR

This paper introduces a mathematical framework using cohomology and torsors to rigorously analyze and classify visual paradoxes like the Penrose staircase, revealing their fundamental obstructions to global consistency.

Contribution

It develops a novel cohomological approach to characterize visual paradoxes as non-trivial network torsors, including the first nonabelian paradox and paradoxes driven by boundary conditions.

Findings

Classified the first nonabelian visual paradox using Klein bottle torsors.

Analyzed paradoxes driven by boundary conditions with non-constant sheaves.

Established a categorical framework linking diverse visual paradoxes.

Abstract

Visual paradoxes like the Penrose staircase present a fundamental tension: locally coherent geometric relationships that cannot be realized globally. Inspired by Penrose's observations connecting such paradoxes to cohomology, we develop a mathematical framework that precisely characterizes this phenomenon through network torsors and sheaf cohomology. Network torsors capture the essential nature of visual paradoxes by formalizing relative geometric attributes (height changes, orientation flips) without requiring absolute measures. We demonstrate that a significant class of visual paradoxes can be rigorously characterized as non-trivial network torsors, with their obstruction to global consistency quantified by elements of $H^1$. This framework enables analysis of classical paradoxes and construction of novel examples on various topological spaces. Key contributions include: (1) the first nonabelian visual paradox, classified by an infinite dihedral torsor on a Klein bottle; (2) paradoxes driven by boundary conditions rather than loops, analyzable via non-constant structure sheaves; and (3) a categorical framework for comparing paradoxes that reveals unexpected connections between visually distinct figures. Our approach unifies diverse visual paradoxes under a single mathematical principle: the obstruction to globalizing locally consistent geometric relationships.