Impossible by Degrees: Cohomology & Bistable Visual Paradox

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.

💡 Research Summary

The paper develops a unified cohomological framework for visual paradoxes built from bistable components—elements that admit exactly two local states (e.g., forward/backward depth of a Necker cube, clockwise/counter‑clockwise spin of a gear, convex/concave interpretation of a rhombic‑tiling corner). By restricting to such binary elements the authors can work entirely with coefficients in ℤ₂, turning the usual obstruction theory into simple parity counting.

The hierarchy proceeds from H⁰ through H², each degree capturing a qualitatively different kind of “paradox”. H⁰ detects ambiguity: multiple globally consistent assignments exist, i.e., the system admits more than one global reading. H¹ detects conflict (relative H¹) when boundary conditions force an inconsistency, and impossibility (absolute H¹) when the underlying constraint structure itself admits no global section. In algebraic terms a state x ∈ C⁰(Λ;ℤ₂) must satisfy the coboundary equation δx = λ, where λ ∈ C¹(Λ;ℤ₂) encodes the pairwise constraints (0 = agreement, 1 = opposition). The class