The Journal

Circles in the stream

Update 4/5/2026, deleted a lot of unnecessary math.

Also, the name comes from a record that I've never listened to, and it was the first thing that came in mind to me when thinking of circles. Whatever.

The important things are that

1. The area of a regular polygon with N sides and length L is A=L2⋅N4⋅tan⁡(π/N). For a triangle A3=34⋅L2, for a square A4=L2 and for an hexagon A6=3⋅32⋅L2. Big revelation for the square, and the hexagon is 6 times the triangle.

2. The area of a regular polygon with N sides and a distance R from any corner to the center is A=R2⋅N⋅sin⁡(π/N)⋅cos⁡(π/N). The interesting thing here is that a regular polygon with infinite sides it looks pretty similar to a circle. And yes, when N tends to +Infinity N sin(pi/N) cos(pi/N) tends to Pi. Wait, what?limN→+∞⁡N⋅sin⁡(π/N)⋅cos⁡(π/N)=π

3. Also I had mathematical proof that ∑i=1Ni3=(i⋅(i+1)2)2 ...but seriously, who cares.

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