Rationals, periodicity and chaos: A Pythagorean view and a conjecture into socio-spatial dynamics

Deep in the fascinating world of numbers there still might lurk useful insights into the processes of the socio-spatial world. A rich section of the world of numbers is of course Number Theory and its pantheon of findings, a part of which is revisited here. It is suggested in this note that a smooth sequence of seemingly random periodic cycles hides the absence of chaotic dynamics in the sequence. Put differently, a seemingly chaotic sequence of periodic cycles, no matter the bandwidth, implies absence of chaotic motion at any point in the sequence; and conversely, the presence of chaotic motion at any specific point in the sequence implies smooth sequence of periodic cycles at any point in the sequence prior to the onset of quasi periodic or chaotic motions. To make this conjecture, the paper draws material from the well known property of rational numbers in Number Theory, namely that the division of unity by any integer will always produce a sequence of decimals in some form of periodicity. The conjecture is taken in a liberally interpreted “Pythagorean type” context, whereby a general principle is suggested to be present in all natural or social systems dynamics. Thus, the paper's subtitle.