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Squared rays



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The square lengths of the xth ray can be expressed using the law of cosines: 2 - 2 cos(x * 2pi / n).

If we sum this over all the rays, the cosine terms cancel out, so we are left with 2n.


I had to think about this point: does it matter there are only n-1 rays?



That's an excellent question.


We include an imaginary 0th ray, the square length of which is 2 - 2 cos(0) = 0.

This is necessary to have all the cosine terms cancel each other out.

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Yeah, I had to think that point through for a moment.

I saw it as having a sum that I didn't know how to evaluate.

So I added and subtracted the term that would make the sum zero.

That step simultaneously evaluated the sum and added 2 to the mix.


Your approach is neater, reasoning that the answer is not changed if you add the squared length of the nth ray.


That's a nice insight.

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