The Color Circle

Choose any point on the map and put the marker somewhere in the middle. The readout gives three numbers and no spectrum, because no spectrum is determined. Infinitely many spectra sum to those three numbers — a single wavelength diluted with white, two wavelengths from either side, a lamp with a hundred peaks — and the eye reports each of them as this one color. Such pairs are called metamers, and they are why a screen works at all: a screen’s three lights produce spectra that no daylight scene ever produces, and the eye, keeping only the sums, cannot tell.

Turn on two wavelengths and drag the mix. The mixed color moves along the straight line between the two lights, because adding two triples and dividing out brightness gives a point on the segment between them. It cannot leave that segment. The mixture of any two colored lights lies on the line between them on this map.

Choose across white. The two lights are placed so that the line between them passes through daylight white, and the mix is set to the proportion that lands on it. Two saturated lights, a yellow and a blue, and the three sums are daylight’s, so the eye sees white. A complementary pair is two colors whose connecting line passes through white. Every wavelength from 380 to about 493, and every one from about 567 to 700, has a complement on the rim; the instrument finds it by extending the ray from the color through white to the far side.

The mix slider divides brightness, not power. A 700 nm light is so far out on the eye’s red sensitivity curve that a watt of it is almost invisible; to see it against a green, nearly all the power has to go to the red. The readout shows both proportions.

Pick one wavelength and slide to 520 nm, a green. Straight across white from it there is no wavelength at all. The ray reaches the dashed line, and the dashed line is not part of the spectrum: it is the straight segment from 700 nm to 380 nm that closes the curved edge, and every point on it is a mixture of the two ends, deep red with violet. Those mixtures are the purples. No single wavelength produces one, and a rainbow, which lays the wavelengths out in order, contains none.

Yet the color wheel is a wheel, and the purples sit on it between violet and red. The spectrum is a line with two ends, and the eye’s three sums place it on the plane as a curve that still has two ends; nothing folds it or doubles it. But because the sums add, every mixture of the two ends is also a color the eye can see, and those mixtures fill in the straight segment between the ends. With the segment filled in, the rim runs all the way round. The purples are what closes the circle.

On the pitch page the line of frequencies became a circle by folding: $2f$ landed on $f$. Halving a wavelength doubles the frequency, so an octave above 700 nm would be 350 nm — outside the eye’s range, though not by much, since violet at 380 is near. If the eye folded the way the ear does, deep red and violet would be the same hue and the circle would close by identification.

It does not. Red and violet are neighbors on the wheel because the purples bridge them, and the bridge is made of mixtures, not of a doubling. So the two circles of these pages are different constructions. One is a line wrapped onto itself, $x \bmod 1$, and each point of it is a class of frequencies. The other is a line laid down in a plane and closed by a straight segment, and the circle is the angle around a center. The ear’s circle has no inside; every point of the eye’s circle has one, with white at the middle.

The horseshoe is the 1931 CIE diagram, and the three numbers $X$, $Y$, $Z$ behind it are not the cones’ sums directly but a fixed linear combination of them, chosen so that $Y$ measures brightness alone and none of the three goes negative. Any other invertible combination would give a map with the same points in different places, and the angles around white would come out differently. The wheel as drawn here is that convention, and it is not perceptually uniform: equal steps of angle are not equal steps of visible difference.

What does not depend on the convention is the shape: a curve with two ends, closed by a straight segment, with a point inside from which every ray meets the rim exactly once. Three linear sums over a line will always produce that, because the line is one-dimensional, because sums add and so mixtures fill in straight segments, and so the region of possible colors is bounded partly by the spectrum and partly by a chord. The purples are not an artifact of 1931.