Platonic solids generate their four-dimensional analogues

Abstract: In this paper, we show how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic Solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the 4D polytopes 16-cell, 24-cell, the F_4 root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_1+I_2(n) which induces I_2(n)+I_2(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. We discuss these results in the wider mathematical context of Arnol'd's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real 3D systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as 4D geometries arising for instance in Grand Unified Theories and String and M-Theory.