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nomadreid

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- On a variation of the lexicographical order, but in polar coordinates (identifying angles with the same position in the plane), this would naively seem to be both a linear order (not amenable to a field structure!) and to cover the plane... but this would be too easy, so I am overlooking something basic....

On a plane with a selected origin point and a selected zero rotation direction, identify each point p with (r

p=*q if they are identical,

p <* q if

[1] r

[2] r

(that is, if the plane consists of concentric circles around the origin, then those on the inner circles are less than those on the outer circles, and inside a circle, the points at a smaller non-negative angle (mod 2π) are smaller than the points with a larger one).

Either what is wrong with this definition (and could it be easily fixed), why isn't it a linear order (yes, I know that it can't be made into a field structure), and/or why isn't it a space-filling curve?

_{p},θ_{p}), where r_{p}is the distance to the origin and θ_{p}is the angle in [0, 2π). Define an order ≤* between points p and q as bp=*q if they are identical,

p <* q if

[1] r

_{p}< r_{q}, or[2] r

_{p}= r_{q}& θ_{p}= θ_{q}(that is, if the plane consists of concentric circles around the origin, then those on the inner circles are less than those on the outer circles, and inside a circle, the points at a smaller non-negative angle (mod 2π) are smaller than the points with a larger one).

Either what is wrong with this definition (and could it be easily fixed), why isn't it a linear order (yes, I know that it can't be made into a field structure), and/or why isn't it a space-filling curve?