Platonic solids generate their four-dimensional analogues

Dechant, Pierre-Philippe

Abstract

This paper shows 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 threedimensional 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 four-dimensional polytopes 16-cell, 24-cell, the F4 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 A1+I2(n) which induces I2(n)+I2(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold’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 three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

Citation

Dechant, P. (2013). Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A: Foundations and Advances, 69(6), 592-602. https://doi.org/10.1107/s0108767313021442

• Journal Article Type: Article
• Publication Date: Nov 1, 2013
• Deposit Date: Jan 20, 2014
• Publicly Available Date: Jan 26, 2014
• Journal: Acta Crystallographica Section A: Foundations and Advances
• Electronic ISSN: 2053-2733
• Publisher: International Union of Crystallography
• Peer Reviewed: Peer Reviewed
• Volume: 69
• Issue: 6
• Pages: 592-602
• DOI: https://doi.org/10.1107/s0108767313021442
• Keywords: Polytopes, Platonic solids, Four-dimensional geometry, Clifford algebras, Spinors, Coxeter groups, Root systems, Quaternions, Representations, Symmetries, Trinities, McKay correspondence.
• Public URL: https://durham-repository.worktribe.com/output/1442046