  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.
