Extended Data Fig. 1: Similarity as a function of distance in color space.

a, Data from all distances in the fixed distance triad task (Fig. 1c). On each trial, there was a target color, here always at 0°, and participants’ task was to choose which of two other colors was closer to the target color in color space. The two choice colors always differed by 30°. The x-axis shows the closer color of the two choice colors. That is, the 150° label on the x-axis reflects performance on a condition where the two choices were 150° and 180° away from the target color. As shown with a subset of this data in Fig. 1c, increasing distance from the target results in a decreased ability to tell which of two colors is closer to the target in color space. This shows the non-linearity of color space with respect to judgments of color similarity. Note that this function does not depict the actual psychophysical similarity function: Roughly speaking, the d’ estimate in this graph is the estimate of instantaneous slope (over a 30 deg. range) in the similarity function in Fig. 1f. b, Despite being conceived of as a color wheel in many memory experiments, in reality, participants internal representation of color–and thus the confusability between colors–ought to be a function of their linear distance in an approximately 3D color space, not their angular distance along the circumference of an artificially imposed wheel. Since the colors are equal luminance, we can conceive of this on a 2D plane. Thus, on this plane the confusability of a color “180 degrees away” on the wheel is only slightly lower than one “150 degrees away” on the wheel, since in 2D color space it is only slightly further away. This simple non-linearity from ignoring the global structure of the color ‘wheel’ partially explains the long tails observed in typical color report experiments, although it does not explain the full degree of this non-linearity, which is additionally attributable to psychophysical similarity being a non-linear function even of distance across 2D color space. c, The similarity function remains non-linear even in 2D color space. Distances here are scaled relative to the color wheel rather than in absolute CIELa*b* values., for example, an item 180 degrees opposite on the color wheel is “120” in real distance since if the distance along the circumference is 180, 120 is the distance across the color wheel. d, Plotted on a log axis, the similarity falls off approximately linearly, indicating that similarity falls of roughly exponentially with the exception of colors nearby the target. The non-exponential fall-off near the 0 point reflects perceptual noise/lack of perceptual discriminability between nearby colors. As shown in Fig. 1, when you convolve measured perceptual noise with an exponential function, this provides a very good fit to the similarity function, consistent with a wide-variety of evidence about the structure of similarity and generalization¹⁹.