For those situations where there's a need to automatically generate a unique color for each item in a set, I've figured out a formula that yields a good distribution based on the oklch color space. The attached document has some pretty infographics. You can also see the source code of the document in Typst. Typst is already the future of programming.

@Squirrel interesting read. For me as a colour blind person, the 50 colours are very hard to distinguish. So be aware when using them. How do other colour blinds see this?

I'm not aware of being colour blind (but everyone has to find out sometime I guess?) but I also find the colours quite similar. E.g. I perceive the runs I've marked in the screenshot below as being extremely similar, to the point that it looks like a repeating pattern to me. Marking them out with arrows shows how the colour does change, to the point that the colour at the end of my perceived "run" is somewhat different from the starting colour - maybe laying them out in this ordered fashion isn't showing them off in an ideal situation?

@Erik Stel Even with excellent color vision, 50 colors are hard to distinguish. To me the first 14 colors look especially distinct. The point is not to construct 50 clearly distinct colors, because that's an impossible task, but to make an infinite series, which happens to repeat itself after 50 colors. I justify the repetition because after 50 colors, you can't distinguish the colors anyway, even if you use a more complex algorithm that has a period of more than 50.

Right, that makes sense - thanks for clarifying!

That’s a fun investigation. In case you didn’t know, coming up with schemes of colors for different items (“categorical variables”) is a well-established & difficult problem in graphic design & data visualization. See, for instance, the eleven palettes that D3 includes: https://d3js.org/d3-scale-chromatic/categorical.

@Joshua Horowitz I wasn't aware of prior research in categorical colors. I only studied prior research in continuous color scales. It appears that the problem is considered difficult because people didn't use the oklch color space, which has only been around since 2020. It was interesting to see the color schemes in D3. Unfortunately, ten out of eleven are black box blobs with no description of how the colors were picked or what they are supposed to be good for. The eleventh one has a blog post describing how it was made, which was by having a graphical designer subjectively pick the colors using an in-house color picker. Plotting all the D3 color schemes, it appears as if the colors were picked by throwing darts at a color picker. The only deliberate property I see is that some of them try to avoid clustering in a/b. All eleven vary widely in lightness and chroma, which is good for making the colors look more distinct, but bad if you require the colors to be consistent. My color formula seems to be the only one designed with the property that you can take an arbitrary slice of it, and all colors in the slice will be almost uniformely distributed, and add more colors at any time and maintain near uniform distribution without having to rebalance the whole set of colors. I've attached my plots. I've added my own color scheme on the last page for comparison. Source code is available.

Color space might be slightly different but https://blog.brownplt.org/2018/06/11/philogenic-colors.html landed on very similar approach of "find the jump around the circle that's least-close to any fraction".