Rhizomata 2025; 13(1): 27–52 Federico M. Petrucci* and Roberto Tadei A Proportional World: the Mathematical and Philosophical Outlook of the Proportion of the Platonic Solids https://doi.org/10.1515/rhiz-2025-0002 Abstract: Plato is the first philosopher to provide a robust account of the mathe- matical structure of the world; in this framework, the idea that the elements are proportionally arranged plays a crucial role. However, scholars agree nowadays that the proportional relation of the elements is not to be read at face value since no mathematical criterion has been found for which the proportion is appropriate. By contrast, this paper aims, on the one hand, to explain the mathematical reasons allowing Plato to state that the elements, each of which is to be associated with a perfect solid, are in proportion to one another (Ti. 31b4–32c5) and, on the other, to highlight the philosophical importance of ascribing to Plato the awareness of these reasons. More specifically, we show that Plato can arrange the solids in proportion ‘as far as it is possible’ by considering as a parameter the radius of the sphere into which each body is inscribed. This parameter establishes the proportion with a very low deviation, whose presence and tolerability can be explained from a tech- nical point of view and exploited from a philosophical point of view. Thus, our solution not only ensures the overall consistency of Plato’s account of elementary *Corresponding author: Federico Maria Petrucci, Department of Philosophy and Education Sciences, Università degli Studi di Torino, Via Sant’Ottavio 20, 10124 Torino (Italy), E-Mail: federicomaria.petrucci@unito.it Roberto Tadei, Department of Control and Computer Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino (Italy), E-Mail: roberto.tadei@polito.it Article Note: This paper was written within the framework of the PRIN 2022 project The ASAP Project: Ancient Science, Ancient Philosophy (2022BBPB8N). It stems from a serendipitous encounter between a mathematician with a passion for philosophy and a historian of ancient philosophy with a deep interest in ancient mathematics, and from the enriching interdisciplinary dialogue that followed a series of lectures on the Timaeus at the University of Turin. An earlier version of this paper was presented at the workshop Mathematics in Ancient Platonism, held in Gargnano sul Garda in June 2023, and we are grateful to the audience, especially to Michalis Sialaros and Phillip Horky, for their valuable feedback. We also owe a special debt of gratitude to Laura Marongiu and Giulia De Cesaris, who read an advanced version of the paper and provided very insightful suggestions. We are grateful to Furio Petrossi, who supported us with the realization of Figures 1 and 2, and to Martin Wad Thorsen, for his careful copy- editing. Finally, we extend our sincere thanks to the anonymous reviewer of Rhizomata and, above all, to István Bodnár, who has carefully engaged with our argument and has offered invaluable recommen- dations for its improvement.