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You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as its point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middlemost.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as its point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middlemost.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3 = 6$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure, must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as it's point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middle-est.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as its point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middlemost.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure, must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as it's point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middle-est.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adoring the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure, must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as it's point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middle-est.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.