Self-Similar Structures of Nontransitive Dice Sets: Examples of Nested "Rock-Paper-Scissors" Relations Based on Numbers from The Lo Shu Magic Square

Abstract

Nontransitive dice are dice beating one another in a cyclic way: die A wins die B, B wins C, and C wins A (like in a rock-paper-scissors game). In this article, it has been shown that a structure of mutual wins of 3 nontransitive dice (with numbers equal to numbers from The Lo Shu magic square) can be repeated at least twice in different scales: - in the nontransitive relations between three sets, each of which consists of three nontransitive dice (total 9 dice); and - in the nontransitive relations between three sets, each of which consists of three nontransitive subsets, each of which consists of three nontransitive dice (total 27 dice). In other words, structures of nontransitive superiority relations can be self-similar. Aims of the future study can be: - to show opportunities for building self-similar structures of nontransitive relations of arbitrary depths of nestedness; and - to design a recursive algorithm of filling the structures with the appropriate numbers. Perhaps a geometrical presentation of these numbers forms a fractal structure.