Regular polygons

Abstract

The construction of regular polygons with a compass and straightedge is a well-known task and this problem has interested mathematicians for a long time. In particular, for a long time they could not answer the question of whether is it possible to construct a regular 17-gon with a compass and straightedge. C. F. Gauss solved this problem in 1796. He proved later that it is possible to construct with a compass and straightedge the regular polygons with $n=2^m n_1\cdots n_l$ sides, where $n_1,\cdots, n_l$ are different prime numbers of the form $\; n_k=2^{2^{\nu_k}}+1$. P. Wantzel proved in 1837 that only these regular polygons can be constructed. Essential is here the construction of the regular polygons with $n_k=2^{2^{\nu_k}}+1$ sides. The currently known prime numbers of the form $n=2^{2^{\nu}}+1$ are $3, 5, 17, 257$ and $65537$. In the paper we present a new approach for solving this task. Among other things we analyze in detail the case of $n=65537$. J. G. Hermes announced in 1894 that he had a full description of the construction of the 65537-gon. This was the result of 10 years of work, but his text was too extensive and was never published. We show exactly and without gaps how the regular 65537-gon can be constructed.