Impossible Hues: Color Space Geometry and Representation Learning

I once argued against Newton’s reduction of color to wavelength. My Farbenlehre insisted: color emerges from the living interplay of light and darkness, not from mechanical decomposition. Modern color science vindicates neither of us fully—yet reveals something more profound. Color exists as geometry.

The Architecture of Perception

The XYZ color space maps all perceivable hues onto three-dimensional coordinates determined not by physical wavelength but by psychological reality—the response patterns of three cone types in the human retina. Here is the crucial observation: perception creates coordinate systems. Just as I recognized that color depends on relational context rather than isolated stimuli, XYZ demonstrates that our experience organizes itself geometrically, with distances measuring perceptual similarity and boundaries marking the limits of what our biological encoders can construct.

Consider the phenomenon of impossible colors—“reddish green” or “yellowish blue,” mathematically valid coordinates in XYZ space that no physical light can produce. The laboratory-created “olo” translates infrared into visible experience through techniques that bypass normal visual processing. These impossible hues occupy regions of representation space that exist geometrically but remain unreachable through natural stimulus. They reveal perception’s constructed nature: what we experience is not reality itself but a geometric manifold shaped by encoder constraints.

Geometries of Meaning Beyond the Eye

Neural networks learn similar representation spaces. A geographic coordinate passes through layers of transformation—mapped onto learned plane heights, folded by ReLU functions, scaled and shifted—until Belgium and Netherlands occupy cleanly separable regions in a final geometric space. The network doesn’t classify raw inputs; it transforms them through increasingly abstract geometries where complex patterns become simple.

The hippocampus performs analogous work: place cells encode physical location through population firing patterns, creating an allocentric spatial map independent of sensory modality or viewing angle. Two-dimensional navigation becomes a high-dimensional neural representation—each place cell contributes one coordinate axis to a geometric space where position itself becomes the encoded variable.

Both systems demonstrate a deeper principle: dimensionality as liberation rather than curse. Just as impossible colors require escaping three-cone constraints, neural optimization in high-dimensional spaces benefits from abundance. With 1.2 billion parameters, gradient descent finds escape routes from local minima through countless dimensions—the same dimensionality that makes exhaustive search impossible makes continuous optimization tractable.

The Space Between Possible and Real

Here is what color space and representation learning share: they illuminate the gap between mathematical possibility and phenomenological actuality. XYZ admits impossible colors geometrically valid but perceptually unreachable. Neural representations contain adversarial examples—valid coordinates in learned space that no natural input produces. Place cells can fire for locations never physically visited, encoding potential rather than only actual experience.

Perception, whether human or artificial, constructs geometric manifolds where relationships matter more than absolute values. My insight about color’s relational nature finds echo in learned embeddings capturing semantic similarity through geometric proximity. The boundaries aren’t between light and dark, but between what geometric structures our encoders can construct and what physical reality does provide.

We don’t perceive the world. We perceive the geometry our constraints allow us to build from it.