Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups

Pierre-Philippe Dechant; Celine Boehm; Reidun Twarock

doi:10.1088/1751-8113/45/28/285202

Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups

Pierre-Philippe Dechant, Celine Boehm, Reidun Twarock

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Abstract

Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H-2, H-3 and H-4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H-3 generate (twist) translations along two-, three-and five-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H-2 and H-4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).

Original language	English
Article number	285202
Number of pages	14
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	45
Issue number	28
DOIs	https://doi.org/10.1088/1751-8113/45/28/285202
Publication status	Published - 20 Jul 2012

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© 2012 IOP Publishing Ltd. This is an author produced version of a paper published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's self-archiving policy.

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10.1088/1751-8113/45/28/285202

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© 2012 IOP Publishing Ltd. This is an author produced version of a paper published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's self-archiving policy.
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Cite this

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abstract = "Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H-2, H-3 and H-4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H-3 generate (twist) translations along two-, three-and five-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H-2 and H-4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).",

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AB - Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H-2, H-3 and H-4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H-3 generate (twist) translations along two-, three-and five-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H-2 and H-4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).

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Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups

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