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Abstract
Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of threedimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D4, F4 and H4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the noncrystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the twodimensional noncrystallographic Coxeter groups I2(n) and the threedimensional groups A3, B3, as well as the icosahedral group H3. IPPP/12/49, DCPT/12/98
Original language  English 

Title of host publication  Conference proceedings 
Subtitle of host publication  Applied Geometric Algebras in Computer Science and Engineering 2012 (AGACSE 2012), July 24, La Rochelle, France 
Publisher  Springer 
Pages  89108 
Number of pages  20 
Volume  24 
Edition  1 
DOIs  
Publication status  Published  Mar 2014 
Publication series
Name  Advances in Applied Clifford Algebras 

Publisher  Birkhauser Verlag Basel 
ISSN (Print)  01887009 
Bibliographical note
(c) Springer Basel 2013. This is an author produced version of a paper published in Advances in Applied Clifford Algebras. Uploaded in accordance with the publisher's selfarchiving policy.Activities
 1 Workshop

Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017
PierrePhilippe Dechant (Keynote/plenary speaker)
6 Aug 2017 → 12 Aug 2017Activity: Talk or presentation › Workshop