Chemistry 2011.org
Chemistry2011.org
All About Chemistry... 2011 and beyond

Related Stories

Platonic solids generate their four-dimensional analogues

Alicia Boole Stott, the third daughter of mathematician George Boole, is probably best known for establishing the term "polytope" for a convex solid in four dimensions. Alicia was also a long time collaborator of HSM Coxeter, one of the greatest geometers of the 20th Century.

Platonic solids are regular bodies in three dimensions, such as the cube and icosahedron, and have been known for millennia. They feature prominently in the natural world wherever geometry and symmetry are important, for instance in lattices and quasi-crystals, as well as fullerenes and viruses.

Platonic solids have counterparts in four dimensions, and the Swiss mathematician Ludwig Schlaefli and Alicia Boole Stott showed that there are six of them, five of which have very strange symmetries. Stott had a unique intuition into the geometry of four dimensions, which she visualised via three-dimensional cross-sections.

A paper by Dechant published in Acta Crystallographica Section A: Foundations and Advances 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. The rotations of the Platonic solids can be interpreted naturally as four-dimensional objects, known as spinors.

These in turn generate symmetry (Coxeter) groups in four dimensions, and yield the analogues of the Platonic solids in four dimensions. In particular, this spinorial construction works for any three-dimensional symmetry (Coxeter) group, such that these cases explain all "exceptional objects" in four dimensions, i.e. those four-dimensional phenomena that do not have counterparts in arbitrary higher dimensions. This also has connections to other "exceptional phenomena" via Arnold's trinities and the McKay correspondence. This spinorial understanding of four-dimensional geometry explains for the first time the strange symmetries that these four dimensional objects have.

This connection between the geometry of four dimension and that of three dimensions via rotations/spinors has not been noticed for centuries despite many of the great mathematicians working on Platonic solids and their symmetries, and is very different from Alicia Boole Stott's way of visualizing three-dimensional sections. It sheds new light on both sides, from real three-dimensional systems with polyhedral symmetries such as (quasi) crystals, viruses and fullerenes to four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

Story Source:

The above story is based on materials provided by International Union of Crystallography. Note: Materials may be edited for content and length.

Share this story with your friends!

Social Networking

Please recommend us on Facebook, Twitter and more:

Other social media tools

Global Partners
Feedback

Tell us what you think of Chemistry 2011 -- we welcome both positive and negative comments. Have any problems using the site? Questions?

About us

Chemistry2011 is an informational resource for students, educators and the self-taught in the field of chemistry. We offer resources such as course materials, chemistry department listings, activities, events, projects and more along with current news releases.

Events & Activities

Are you interested in listing an event or sharing an activity or idea? Perhaps you are coordinating an event and are in need of additional resources? Within our site you will find a variety of activities and projects your peers have previously submitted or which have been freely shared through creative commons licenses. Here are some highlights: Featured Idea 1, Featured Idea 2.

About you

Ready to get involved? The first step is to sign up by following the link: Join Here. Also don’t forget to fill out your profile including any professional designations.

Global Partners