I was actually thinking, during the previous thread involving Complex numbers ...

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`It may not have any practical use, but if one wanted to define an ordering for`

`complex numbers that was deterministic and relatively unbiased, a way to do this`

`would be based on what I'll call for now the "spiral distance".`

`Conceptually, you take an infinite length spiral line that starts at and is`

`centered on the origin, where for each turn the current spot on the spiral`

`increases an infinitesimal radius from the origin, or a distance approaching`

`zero, in the calculus sense. Complex numbers closer to the origin on the spiral`

`will be ordered earlier than those further from the spiral.`

`Actually calculating this is a simple comparison of the radius and angle`

`components of the two complex numbers in the polar coordinate system. If the`

`radius value is different, then the one with the smaller radius is ordered`

`before the one with the larger; if the two radius values are the same, then the`

`one with the smaller angle is ordered first; if both are the same, then the two`

`complex numbers are equal.`

`The math is just as simple as a naive comparison that just compares the real`

`component and then imaginary component in a cartesian coordinate system, but the`

`result is much more reasonable I think.`

`This whole principle of "distance from origin" method of ordering does also, I`

`suspect, scale to any number of dimensions; the one-dimensional version is`

`simply comparing first the absolute value of the two numbers, and then saying`

`that either the positive or negative version orders first.`

-- Darren Duncan