Smallest maximal snakes of translates of convex domains
A maximal snake of size \(n\) is a set \((K_1,\dots,K_n)\) of non-overlapping translates of a convex domain \(K\) in the plane, such that \(K_i\) touches \(K_j\) if and only if \(|i -j|= 1\) and no translate of \(K\) can touch \(K_1\) or \(K_n\) without intersecting an additional \(K_i\). The authors prove that the size of the smallest maximal snake is 11 if \(K\) is a parallelogram and 10 otherwise, thereby generalizing the corresponding theorem on circular disks. For the proof, some properties of Minkowski planes are studied in detail, and a modification of the usual notion of angles in such planes is introduced. For these angles, a partial ordering and a partial addition are defined. The angles of equilateral triangles turn out to be of particular importance.
scientific article; zbMATH DE number 806197