Three sides to every tale - Timaeus 54a-54b

Abstract

Plato’s Timaeus (54a-55c) presents a curious method for constructing the faces of re- gular polyhedra — the square and the equilateral triangle — from two types of ‘fairest’ right-angled triangles. This paper explores geometric properties inherent in Plato’s chosen constructions. Distinct from a historical analysis of Plato’s original intent, this exploration reveals mathematically elegant fe- atures of the constructions that would likely have appealed to ancient Greek geometric and aesthetic sensibilities. We examine how these properties, such as the preservation of symmetry through specific multi-triangle compositions (four isosceles triangles for a square, and six scalene triangles for an equila- teral triangle), lead to unique geometric configurations, including the natural emergence of face centers that are crucial for understanding polyhedral duality.