Impossible by Degrees: Cohomology & Bistable Visual Paradox

TL;DR

This work builds a cohesive cohomological framework for visual paradoxes composed of bistable elements, reducing to parity with $\mathbb{Z}_2$ coefficients. It develops a five-stage hierarchy via relative and absolute $H^0$, $H^1$, and $H^2$, unified by the discrete Stokes principle: boundary holonomy induces interior curvature, shaping ambiguity, conflict, impossibility, curvature, and inaccessibility. Central constructs include constraint graphs, torsors, and the holonomy criterion, complemented by the Method of Monodromic Apertures (MoMA) that renders holonomy as observable monodromy in both base and configuration spaces. The paper applies the framework to Necker cube fields, gear meshes, and rhombic tilings (including Penrose P3), illuminating classical paradoxes and revealing new curvature defects and sectorial inaccessibility. The results bridge topology, combinatorics, and visualization, and point to natural extensions to non-$\mathbb{Z}_2$ settings, sheaf theory, and temporally extended paradoxes.

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.

Figures (23)

• Figure 1: An unusual figure [left] of Penrose from Penrose1992Cohomology is based on the Schröder staircase [center] and has the bistable ambiguity of the classical Necker cube [right].
• Figure 2: A 2-D field of Necker cubes. Choosing an orientation for any one cube determines the orientation of all cubes in the connected component. Pinning two disjoint subsets of cubes to opposite states creates a conflict -- a relative paradox.
• Figure 3: A Necker cube rotating about a vertical axis. The depth ambiguity of the static figure becomes rotational ambiguity: the same animation can be perceived as clockwise or counterclockwise rotation.
• Figure 4: With an even number of gears in a ring, the system spins freely [left], but an odd number of gears locks [right].
• Figure 5: Two rhombic tilings: [left] the regular lozenge tiling; [right] a patch of Penrose P3 tiling, centered on a five-fold rosette.

Theorems & Definitions (11)

• proof
• Lemma 3.2: Extension-independence
• proof
• Theorem 3.3: Stokes
• Remark 3.4: Gauge perspective
• Remark 6.1
• Remark 6.2: Torsors on graphs versus complexes
• Lemma 7.1: Aperture monodromy
• Lemma 7.2
• proof