this post was submitted on 27 May 2024
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The color wheel, for instance
A colour wheel is not even partially ordered, I don't think. There is a relation between some colours on the wheel but it's not an ordering.
You can impose a partial ordering on it. HSL uses a hue angle. If we assume full saturation and lightness and pick an arbitrary direction to be positive, there's your partial ordering. But not total ordering, there is no most clockwise color.
I feel like this doesn't qualify as an ordering relationship, because of the circular nature of the wheel: for any two elements a and b (a ≠ b) on the wheel it's both true that a is further clockwise than b and b is further clockwise than a (just keep rotating). This violates the antisymmetry property that an ordering relation should have.
You can fix it by establishing some point on the wheel as "least clockwise" (essentially unfolding it into just a straight line) but that immediately establishes a total ordering.
Measure acute only. Idk how to deal with 180° though