June 17th: Maybe it should be this…

    This post is centered on understanding (and then proving):

    Key question (that I had thought was too obvious to ask before): how do we define angles?

    An angle is a ratio of the length of the subtending arc (of the circle at which the angle is centered at) to the length of the radius.

    My previous interpretation of the above diagram has been wrong. Updated interpretation:

    And so, approximately, the arc length would be:

    If we set the origin to be at O and a local coordinate to be x = r sin(\phi), y = r cos(\phi),

    d\theta = \frac{\text{arc length}}{\text{radius length}} \approx \frac{sin(\phi)dS}{r} = \frac{\frac{x}{r}dS}{\sqrt{x^2 + y^2}} = \frac{x dS}{x^2 + y^2}, which we know to be a harmonic function.

    In this way, we can generalize it to any dimension.

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