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martin.ankerl.com

Describes how to use the golden ratio to generate a uniform distribution of hues with similar lightness values.

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I came in expecting the usual revelation about “Oh, RGB < HSL/HSV for this sort of work”, but the golden ratio bit was quite nifty. Good submission @etc !

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Thanks!

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This is excellent. I knew that we see golden-ratio rotation in how some plants position their leaves, leading to each new leaf only minimaly overlapping all lower-lying leaves; but it had never occurred to me to use golden-ratio rotation on an HSL colour’s hue, leading to each new colour only minimally overlapping with all prior colours.

This solves a particular problem for me: the plotting tools I use uniformly-spaced colour scales. This means that if I make a plot with two groups, and then a plot with three groups, the new plot does not use the same two colours plus another one. A golden-ratio scale will solve this beautifully.

Looking for the mechanism behind golden-ratio rotation in plants I found a fascinating article giving a bird’s-eye view of the subject, The mathematical lives of plant. Some bits that struck me particularly forcefully:

• The primordia (points where offshoots grow from) form in a small region at the tip of a stem. Hofmeister proposed that the precise spot in which they form within that region is the spot that is furthest from older primordia. The primordia then move outward and downward along the stem as the tip continues to grow.
• a simple mathematical model […] showed that the forces Hofmeister described—outward, downward, and away from other primordia—produced golden angle spirals.
• The flowers could also produce their primordia at angles of approximately 99.5º. In that case, the numbers of spirals in each direction would not be Fibonacci numbers, but the closely related Lucas numbers, which begin with 1, 3, 4, 7 . . . ,
• Plants could produce [primordia] at angles that vary but repeat. For instance, Hotton found that the angle could be 131, then 88, then 88 again, then 131, then 89, then 87, then 131, then 315, and then go back to 131 and start over. “What’s interesting about this is that the pattern that actually forms would be hardly distinguishable from the one where the angle was the same,” Hotton says. “You could actually see opposing pairs of spirals. You could count them and see that there were five in one direction and eight in the other. But the angles wouldn’t be the same every time; it would be following this periodic sequence.”

Addendum: deterministic sequences that fill a space more-or-less uniformly are apparently called low-discrepancy sequences.

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While HSV is definitely better, and harmonics are good, the next “level” here is to use CIELCH, which is a colorspace that maps more closely to our perception of color. It’s another polar space, like HSV (LCH = luminance, chromaticity, hue), so the same basic logic works, but the math for moving about the space is different.

//Used to build software for a paint company, they were big on CIE because they cared about the perceptual properties of each color.

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If you want a uniform distribution, why not just use a uniform distribution instead of faking it with golden ratio subdivisions?