The kissing number of the regular pentagon
The kissing number (or Newton number) of a convex disc \(D\) is the maximum number of congruent copies of \(D\) that can be arranged so that each touches \(D\) but no two of them overlap. The kissing number of a regular \(n\)-gon has been determined for all \(n\neq 5\) by \textit{K. Böröczky} [Period. Math. Hung. 1, 113-119 (1971; Zbl 0231.52008)] and later again by \textit{L. Zhao} [Discrete Math. 188, No. 1-3, 293-296 (1998; Zbl 0958.05022)]. Here the authors consider the case \(n=5\) and want to prove that the kissing number of the regular pentagon is equal to \(6\). Unfortunately, their proof does not seem to be valid, since it is based on the following assertion: ``Let \(AOB\) be the triangle spanned by the centers of three mutually touching congruent regular pentagons. Then the angle at \(O\) is at least \(54\) degrees.'' This is not true, since it is not difficult to find three such pentagons with the corresponding angle measuring even less than \(53\) degrees. The main point is that the authors erroneously believe that the minimum angle is attained if either the three pentagons have a common vertex or the two pentagons centered at \(A\) and \(B\) have a common side. On the other hand, the kissing number of the regular pentagon is actually equal to \(6\), as has already been proved previously by the reviewer [Period. Math. Hung. 4, 315-328 (1973; Zbl 0283.52002)].
scientific article; zbMATH DE number 1792435