>>12825699Not sure if this is what you mean, but imagine that you keep bending the faces of a regular 2^n polygon into a regular 2^(n+1) polygon. Set the initial perimeter = 1. The perimeter never changes.
At n=1, you have two superimposed line segments.
The lengths of the 2 faces are 1/2.
The angles of the 2 vertices, in radians, are ? - ? = 0.
Circumradius = 1/4 csc ?/2
Apothem = 1/4 cot ?/2
At n=2, you have a square.
The lengths of the 4 faces are 1/4.
The angles of the 4 vertices are ? - ?/2 = ?/2.
Circumradius = 1/8 csc ?/4
Apothem = 1/8 cot ?/4
At n=k, you have a 2^k-gon.
The lengths of the 2^k faces are 1/2^k.
The angles of the 2^k vertices are ? - ?/2^(k-1) = ? * sum j=1->k-1 (1/2^j).
Circumradius = 1/2^(k+1) csc ?/2^k
Apothem = 1/2^(k+1) cot ?/2^k
As n->?, you have a 2^?-gon = infinite sides.
The length of each side -> 1/2^? = 0
The angle of each vertex -> ? * sum j=1->? (1/2^j) = ? * 1 = ?
Circumradius = lim k->? 1/2^(k+1) csc ?/2^k = 1/(2?)
Apothem = lim k->? 1/2^(k+1) cot ?/2^k = 1/(2?)
1/(2?) * 2 * ? = 1
Circumradius and apothem both converge to the radius of a circle of circumference 1.
