[Paper Review] Intransitive Machines

Alexander Poddiakov|arXiv (Cornell University)|Sep 8, 2018

History and Theory of Mathematics34 references3 citations

TL;DR

This paper introduces mechanical intransitive devices—mechanisms like gears, levers, pulleys, wedges, and inclined planes—that exhibit intransitive superiority relations (e.g., A rotates faster than B, B lifts more than C, C rotates faster than A), mirroring the Condorcet paradox. It demonstrates how such paradoxical cycles can be constructed in elementary physics, offering a novel framework for teaching transitivity and intransitivity in physics education.

ABSTRACT

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.

Motivation & Objective

• To explore the presence of intransitive relations in elementary physics, challenging the assumption of transitivity in mechanical systems.
• To develop mechanical devices that model intransitive cycles analogous to the Condorcet paradox.
• To provide educational tools and problems that clarify the distinction between transitive and intransitive relations in physics and mathematics.
• To promote deeper understanding of non-transitive phenomena through hands-on, mechanical constructions.

Proposed method

• Designing mechanical systems using simple machines (gears, levers, pulleys, wheels/axles, wedges, inclined planes) to create intransitive relations.
• Structuring devices so that relations like 'rotates faster than', 'lifts more than', or 'is stronger than' form cyclic, non-transitive loops.
• Applying principles from the Condorcet paradox to mechanical systems, ensuring that A > B, B > C, and C > A.
• Using geometric and kinematic configurations to enforce intransitive behavior in mechanical advantage and motion.
• Validating the intransitive behavior through theoretical analysis and illustrative diagrams.
• Presenting each device with clear descriptions and visual representations to support educational implementation.

Experimental results

Research questions

• RQ1 Can mechanical systems be constructed such that the relation 'rotates faster than' forms an intransitive cycle?
• RQ2 How can the concept of intransitivity from social choice theory (e.g., Condorcet paradox) be physically realized in mechanical devices?
• RQ3 What types of simple machines can be combined to produce non-transitive behavior in mechanical advantage or motion?
• RQ4 How can such intransitive machines be used to enhance student understanding of transitivity and intransitivity in physics education?
• RQ5 What are the structural and geometric conditions required for a mechanical system to exhibit intransitive superiority relations?

Key findings

• Five distinct intransitive mechanical devices were successfully designed: intransitive gears, levers, pulleys/wheels and axles, wedges, and inclined planes.
• Each device demonstrates a cyclic superiority relation—e.g., A rotates faster than B, B lifts more than C, and C rotates faster than A—defying transitive logic.
• The intransitive behavior arises from specific geometric and kinematic configurations that break transitivity in mechanical advantage or motion.
• The devices are constructed from standard simple machines, showing that intransitivity is achievable without complex components.
• The paper provides a framework for creating educational materials based on these paradoxical mechanical systems.
• The study establishes a bridge between abstract intransitive relations in mathematics and tangible, physical mechanisms in physics education.