A Study on Directionality in the Ulam Square with the Use of the Hough Transform

摘要

A version of the Hough transform in which the direction of the line is represented by a pair of co-prime numbers has been used to investigate the directional properties of the Ulam spiral. The method reveals the detailed information on the intensities of the lines which can be found in the square and on the numbers of primes contained in these lines. This makes it possible to make quantitative assessments related to the lines. The analysis, among others, confirms the known observation that one of the diagonal directions is more populated with lines than the other one. The results are compared to those made for a square containing randomly located points with a density close to that for the Ulam square of a corresponding size. Besides its randomness, such square also has a directional structure resulting from the square shape of the pixel lattice. This structure does not depend significantly on the size of the square. The analysis reveals that the directional structure of the Ulam square is both quantitatively and qualitatively different from that of a random square. Larger density of lines in the Ulam square along one of the diagonal directions in comparison to the other one is confirmed.