Impossible by Degrees: Cohomology & Bistable Visual Paradox

Abstract

The Penrose triangle, staircase, and related ``impossible objects'' have long been understood as related to first cohomology $H^1$: the obstruction to extending locally consistent interpretations around a loop. This paper develops a cohomological hierarchy for a class of visual paradoxes. Restricting to systems built from \emph{bistable} elements -- components admitting exactly two local states, such as the Necker cube's forward/backward orientations, a gear's clockwise/counterclockwise spin, or a rhombic tiling corner's convex/concave interpretation -- allows the use of $\mathbb{Z}_2$ coefficients throughout, reducing obstruction theory to parity arithmetic. This reveals a hierarchy of paradox classes from $H^0$ through $H^2$, refined at each degree by the relative/absolute distinction, ranging from ambiguity through impossibility to inaccessibility. A discrete Stokes theorem emerges as the central tool: at each degree, the connecting homomorphism of relative cohomology promotes boundary data to interior obstruction, providing the uniform mechanism by which paradoxes ascend the hierarchy. Three paradigmatic systems -- Necker cube fields, gear meshes, and rhombic tilings -- are studied in detail. Throughout, we pair cohomology with imagery and animation. To illuminate the underlying structure, we introduce the \emph{Method of Monodromic Apertures}, an animation technique that reveals monodromy through a configuration space of local sections.