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Delightful Geometric Discovery

So, I wanted to make (regular-)hexagonal or square pie charts. Easy enough, right? Except, I wanted the area to represent the proportion, not the angle.

With a circle, those notions are equivalent. If you sweep out x% of 2π radians, you've covered x% of the area of the circle. Of course, you've also covered x% of the circumference.

Circle area and arc length

With regular polygons of fewer than infinitely many sides, the situation is not so clear. Sweeping out a small angle that doesn't include a corner covers less area than that same angle into a corner.

The discovery...Collapse )

In retrospect, I should have realized all of this when deriving the equations for integrating in polar coordinates. But, there, I thought that there was something special going on since every radius is perpendicular to the circle. Alas, this all delights me today.

The Lisp code that generated the images...Collapse )