bmonk wrote:rdococ wrote:bmonk wrote:It's nice, though. And fits my thoughts on the subject. The formula for the angles of a polygon is not real obvious or simple to my mind, but going by the uninternal angles makes it very simple: the sum is 720° or τ, no matter how many angles are involved.

You forgot already that [tex]\tau[/tex] = 360 degrees, because [tex]\pi[/tex] is wrong.

True. But the point is that the sum of any polygon's external angles is 2[tex]\tau[/tex]. If you want the sum of the internal angles, then you need to calculate a bit more.

The sum of any polygon's external angles is just tau, not 2τ, and just 360°, not 720°. Are you thinking of something else?

In addition to external, internal, and "uninternal" angles, there are also "unexternal" angles, which always sum to (n-1)τ. I don't know when they would be important, but I just thought of them randomly.

The formula for the measure of one external angle of a regular n-gon is τ/n. The measure of an interior angle is equally simple with either, (1/2-1/n)τ or (1-2/n)π. The formula for the measure of each "uninternal" angle would be (1/2+1/n)τ or (1+2/n)π, and the formula for each "unexternal" angle would be (1-1/n)τ.

I also found it interesting that for polygons that cross over themselves the external angle sum is always a multiple of tau. I also think the fact that the internal angle is only found by taking a straight angle and subtracting the exterior angle, as well as the fact that when going around the polygon's perimeter, you pivot at the same angle as the exterior angle, not the interior angle, show that perhaps exterior angles are more fundamental than interior angles, although they are both important, of course.

I also have a comment on the Pi Manifesto's argument when it mentions that the sum of internal angles of a triangle is pi. Aren't squares more important than triangles, being the shape we use to measure area, having opposite sides parallel, and being the only polygon whose interior and exterior angles are equal? The sum of both the interior and exterior angles of a square is tau. Isn't that a stronger argument for tau than the sum of the interior angles of a triangle being pi (but the sum of the exterior angles still being tau) is for pi?