Some Properties of Points Arranged at Random on a Möbius Surface

Extract

If a series of points be pricked at random through a strip of paper, and the strip be held horizontally, certain of the points will appear as peaks, namely those which happen to be higher than either of their immediate neighbours. These we call maximal points. Similarly there are others, minimal points, which are lower than either of their immediate neighbours. Between each maximal point and an adjacent minimal point is a “run”, either a “run up” or a “run down”, the points in a run forming a steadily ascending (or descending) series. These runs will naturally be of various lengths, the length of a run being defined as the number of points in the run, including the first and the last. The shortest runs will thus be of length 2, others may be of length 3, 4, or some greater number, up to m, where m is the total number of points on the strip. Clearly where the points form a run of m there is only one run throughout the whole strip, the greatest possible number of runs is m-1, in which case each run is of length 2.