Erwin Schrödinger proposed in the 1920s that color perception has a geometry. The space of perceivable colors is not flat — perceptual distances between colors don't scale linearly with physical differences in wavelength. A small shift in hue near blue looks larger than the same shift near yellow. Schrödinger argued that the correct mathematical framework is Riemannian geometry: color space is a curved manifold, and the perceived difference between two colors is the geodesic distance along it.
The framework was elegant and partly successful, but it never fully worked. The Bezold-Brücke effect — the observation that changing brightness shifts perceived hue — couldn't be accommodated. The relationship between lightness, hue, and saturation resisted unified mathematical treatment. For a century, the problem was assumed to be one of parameter fitting: the right metric tensor hadn't been found yet.
Bujack and colleagues at Los Alamos National Laboratory (Computer Graphics Forum, 2026) show that the problem was not the parameters but the framework. Color space is not Riemannian. The correct geometry is non-Riemannian — a metric space where the axioms of Riemannian geometry do not all hold. Specifically, the neutral axis — the line from black through gray to white — cannot be assumed. It must be derived from the geometry itself.
For a century, every color model placed the neutral axis by definition: gray is the achromatic point, the axis of zero saturation, and it runs from black to white as an input to the model. Bujack's framework derives it as an output. The neutral axis emerges from the metric structure the way a geodesic emerges from curvature — it is the set of points equidistant from all hues. When you derive it rather than assume it, hue, saturation, and lightness all emerge from the mathematics simultaneously. The Bezold-Brücke effect becomes a curvature effect: brightness distorts perceived hue because the metric is non-Riemannian and the neutral axis shifts under changes in luminance.
The general principle: when a mathematical framework almost works but has persistent, unfixable discrepancies, the problem may not be in the parameters. The problem may be in the framework's axioms. Schrödinger's geometry was correct in spirit — color perception has curvature — but wrong in specifics because he assumed a kind of curvature that was too constrained. The fix was not to find better numbers within Riemannian geometry but to abandon Riemannian geometry for something less familiar.
The hundred-year stall was caused by a definitional assumption that felt too obvious to question. Gray is gray. The neutral axis is neutral. Who would derive what everyone already knows? But “everyone already knows” is exactly the condition under which an assumption becomes invisible. The missing piece was not a discovery about color. It was a discovery about which part of the model was assumed rather than earned.