The research builds on the work of Erwin Schr\u00f6dinger, the physicist famous for his “Schr\u00f6dinger’s cat”<\/a> thought experiment, who, among other biological phenomena<\/a>, also studied color perception.<\/p>\n Combining results of color-perception studies within a geometric framework, the authors of the new study found shortcomings in Schr\u00f6dinger’s mathematical definitions of hue, saturation, and lightness. Beyond merely building on his work, they resolved these ambiguities and helped complete his work more than a century later.<\/p>\n “What we conclude is that these color qualities don’t emerge from additional external constructs such as cultural or learned experiences but reflect the intrinsic properties of the color metric itself,” says<\/a> lead author and data scientist Roxana Bujack.<\/p>\n “This metric geometrically encodes the perceived color distance \u2013 that is, how different two colors appear to an observer,” Bujack adds<\/a>.<\/p>\n Humans have trichromatic color vision<\/a>, which relies on three types of color-sensing cone cells in the retina. The sensitivity of each type of photoreceptor cell peaks at a different wavelength, and we use the combinations of signal strengths produced by these cells to perceive the color spectrum.<\/p>\n This process grants us three dimensions of color spaces, or organizations of color. These perceptual spaces<\/a> are like mental realms where we process our sensory perceptions into representations of the world around us.<\/p>\n In the 19th century, mathematician Bernhard Riemann introduced the idea that our perceptual spaces for color are curved rather than straight, a notion rooted in his eponymous branch of differential geometry<\/a>.<\/p>\n While a straight line is famously the shortest distance between two points in Euclidean space, Riemannian geometry often focuses on curved surfaces where the locally shortest path between two points, a geodesic, isn’t straight.<\/p>\n The physicist Hermann von Helmholtz suggested it’s possible to geometrically define individual color attributes based only on closest similarity in the Riemannian metric \u2013 a mathematical tool for studying certain manifolds, or higher-dimension analogs of surfaces.<\/p>\n In the 1920s, Schr\u00f6dinger used the Riemannian model of color perception to define the perceptual attributes of hue, lightness, and saturation. His definitions were based on a color’s location relative to the neutral axis, or the gradient of grays between black and white.<\/p>\n Subscribe to ScienceAlert’s free fact-checked newsletter<\/p>\n These definitions were largely accepted for the ensuing century, providing a framework for our understanding of color attributes. Yet as the authors of the new study worked on algorithms for scientific visualizations, they found problems with Schr\u00f6dinger’s work.<\/p>\n “With a little criticism, Schr\u00f6dinger’s geometric formulation of the color attributes has, in spirit, survived until today even though it, too, is in conflict with some phenomena observed in experiments,” they write<\/a>.<\/p>\n Schr\u00f6dinger never formally defined the neutral axis, they note, despite basing his definitions of color attributes on colors’ positions in relation to it.<\/p>\n Sensing an opportunity to advance the mathematics of color perception, the researchers sought to complete Schr\u00f6dinger’s work more than a century later.<\/p>\n They succeeded by defining the neutral axis based on the geometry of the color metric, which required working outside of the Riemannian model, they explain.<\/p>\n The researchers also made other important corrections. Schr\u00f6dinger’s view couldn’t explain the Bezold-Br\u00fccke effect, for example, a phenomenon in which varying light intensity induces a perceived change in hue.<\/p>\n Bujack and her colleagues corrected for this by replacing the straight-line definition of stimulus quality between a color and black with the shortest geodesic path in perceptual color space.<\/p>\n They also accounted for diminishing returns in color perception, which refers to our tendency to perceive large color differences as less than the sum of small color differences.<\/p>\n In a related 2022 paper<\/a>, many of the same researchers argued that this effect “cannot exist in a Riemannian geometry,” citing the need for improved methods for modeling color differences.<\/p>\n Related: Scientists Say They Found a New Color Humans Have Never Seen Before<\/a><\/p>\n With the new study, they outline a novel framework for modeling color in non-Riemannian space.<\/p>\n “Collectively, our solutions provide the first comprehensive realization of Helmholtz’s vision: formal geometric definitions of hue, saturation, and lightness derived entirely from the metric of perceptual similarity, without reliance on external constructs,” the researchers write<\/a>.<\/p>\n![\"Subscribe](\"https:\/\/www.newsbeep.com\/us\/wp-content\/uploads\/2025\/11\/41e99e2aab48d206d35bd401330a9ca6.jpeg\"\/)
<\/a><\/p>\n![\"chart](\"https:\/\/www.newsbeep.com\/us\/wp-content\/uploads\/2026\/03\/bd9142b477d2a0ca76fe2750734fb4c9.jpeg\"\/)
Results from the color perception experiments the team conducted: If the colors of the second and fourth columns match, then the closest perceived color to the neutral axis coincides with the color at the end of the shortest path. (LANL<\/a>)<\/p>\n