arXiv subjects

Alexander Poddiakov

Publications and source records attributed to Alexander Poddiakov.

3 recordsLinked to original sources

arXiv2023

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

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.

math.GM↗

arXiv2022

Intransitively winning chess players positions

Positions of chess players in intransitive (rock-paper-scissors) relations are considered. Namely, position A of White is preferable (it should be chosen if choice is possible) to position B of Black, position B of Black is preferable to position C of White, position C of White is preferable to position D of Black, but position D of Black is preferable to position A of White. Intransitivity of winningness of positions of chess players is considered to be a consequence of complexity of the chess environment -- in contrast with simpler games with transitive positions only. The space of relations between winningness of positions of chess players is non-Euclidean. The Zermelo-von Neumann theorem is complemented by statements about possibility vs. impossibility of building pure winning strategies based on the assumption of transitivity of positions of chess players. Questions about the possibility of intransitive positions of players in other positional games are raised.

math.HO↗

arXiv2018

Intransitive Machines

The intransitive cycle of superiority is characterized by such binary relations between A, B, and C that A is superior to B, B is superior to C, and C is superior to A (i.e., A>B>C>A - in contrast with transitive relations A>B>C). The first part of the article presents a brief review of studies of intransitive cycles in various disciplines (mathematics, biology, sociology, logical games, decision theory, etc.), and their reflections in educational materials. The second part of the article introduces the issue of intransitivity in elementary physics. We present principles of building mechanical intransitive devices in correspondence with the structure of the Condorcet paradox, and describe five intransitive devices: intransitive gears; levers; pulleys, wheels, and axles; wedges; inclined planes. Each of the mechanisms are constructed as compositions of simple machines and show paradoxical intransitivity of relations such as "to rotate faster than", "to lift", "to be stronger than" in some geometrical constructions. The article is an invitation to develop teaching materials and problems advancing the understanding of transitivity and intransitivity in various areas, including physics education.

math.HO↗