Counting in Cycles

Abstract

Counting in terms of cycles allows modeling many processes of Nature. We make use of a slight numerical incongruence within the numbering system to find a translational mechanism which connects sequential ↔ commutative properties of assemblies. The algorithms allow picturing the logical syntax Nature uses when reading the DNA. Ordering a collection on two different properties of its members will impose two differing sequences on the members. The coordinates of a point on a plane of which the axes are the two sorting orders sidestep the logical contradiction arising from the different linear assignments. During a reorder, elements aggregate into cycles. Using an etalon collection of simple logical symbols (which are pairs of natural numbers), which we reorder, we see typical movement patterns along the path of the string of elements that are members of the same cycle. We split the {value, position} descriptions of a natural number and observe the places the unit occupies at specific instances of time among its peers, while being a member of a cycle, under different orders prevailing. Which places elements occupy among their peers, under specific order conditions, is determined by numeric properties of the natural numbers that make up the element. The model presented is therefore tautologic, and can be compared to a hybrid of sudokus with the ultimate form of Rubik‘s cube. It is necessary for the reader to do some self-education on the matter of cycles as a prerequisite to understanding the working principle of the model. The contraption being a new development, it can’t have references in literature. One cannot lose a bet on the idea that sorting and ordering a collection of elementary logical elements will turn up typical patterns and that these archetypes of patterns will be of interest to Theoretical Physics, Chemistry, Biology, Information Theory, and some other fields, too.