A group theoretical approach to structural transitions of icosahedral quasicrystals and point arrays

Abstract

In this paper we describe a group theoretical approach to the study of structural transitions of icosahedral quasicrystals and point arrays. We apply the concept of Schur rotations, originally proposed by Kramer, to the case of aperiodic structures with icosahedral symmetry; these rotations induce a rotation of the physical and orthogonal spaces invariant under the icosahedral group, and hence, via the cut-and-project method, a continuous transformation of the corresponding model sets. We prove that this approach allows for a characterisation of such transitions in a purely group theoretical framework, and provide explicit computations and specific examples. Moreover, we prove that this approach can be used in the case of finite point sets with icosahedral symmetry, which have a wide range of applications in carbon chemistry (fullerenes) and biology (viral capsids).

From affine A 4 to affine H 2: group-theoretical analysis of fivefold symmetric tilings

“…The Coxeter group D 6 embeds the icosahedral group H 3 as a subgroup (Kramer, 1993) and the exceptional Coxeter group E 8 embeds the non-crystallographic Coxeter group H 4 as a subgroup (Coxeter, 1973;Koca et al, 2001). It seems the affine extension of H 3 may play important roles in quasicrystals and in viral structures (Keef & Twarock, 2008;Indelicato et al, 2012;Dechant et al, 2013;Salthouse et al, 2015;Zappa et al, 2016). Structural transitions in quasicrystals induced by transitions in higher-dimensional lattices are discussed by Indelicato et al (2012).…”

Unveiling the Hidden Rules of Spherical Viruses Using Point Arrays

“…These gauge points indicate to the overall scaling of the point array. Another promising method of generating icosahedral point arrays has been offered by Zappa et al [11,12]. They propose generating point arrays from 6d projections of the hypercube.…”

Viral Phrenology

“…The structural protrusions were found by classifying the viruses using our modified fitting methods [2,3] for icosahedral point arrays [4,5]. Point arrays provide highly specific geometric constraints on the arrangement of viral proteins, and all spherical viruses studied so far conform to one or more of these arrays [3][4][5][6][7][8][9][10][11][12][13][14]. We have previously shown that all protruding features of spherical viruses are located on the gauge points of these arrays, though it was not yet known that these locations also indicated the genomic composition.…”