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broken line inside a regular polygon, spiral, equal angles, projections

Source: III All-Ukrainian Tournament of Young Mathematicians, Qualifying p11

May 20, 2021
geometryregular polygonUkrainian TYM

Problem Statement

A circle centered at point OO is separated by points A1,A2,...,AnA_1,A_2,...,A_n on nn equal parts (points are listed sequentially clockwise) and the rays OA1,OA2,...,OAnOA_1,OA_2,...,OA_n are constructed. The angle A2OA3A_2OA_3 is divided by rays into two equal angles at vertex OO, the angle A3OA4A_3OA_4 is divided into three equal angles, and so on, finally, the angle AnOA1A_nOA_1 divided into nn equal angles at vertex OO. A point belonging to the ray other than OA1OA_1, is connected by a segment with its orthogonal projection B0B_0 on the neighboring (clockwise) arrow) with ray OA1OA_1, pointB1 B_1 is connected by a segment with its orthogonal projection on the next (clockwise) ray, etc. As a result of such process it turns out the broken line B0B1B2B3...B_0B_1B_2B_3... infinitely "twists". Consider the question of giving the thus obtained broken numerical value of "length" L(n)L (n) and explore the value of L(n)L(n) depending on nn.
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