Quasicrystals and icosians

Affine extensions of non-crystallographic Coxeter groups induced by projection

“…This connection between E 8 and H 4 can be exhibited in various ways, including Coxeter-Dynkin diagram foldings in the Coxeter group picture, 37 relating the root systems, 16,38,39 and in terms of the representation theory. 14,16,[37][38][39] For illustrative purposes, we focus on the folding picture first.…”

“…They include the groups H 2 , H 3 , and the largest non-crystallographic group H 4 ; the icosahedral group H 3 and its rotational subgroup I are of particular practical importance as H 3 is the largest discrete symmetry group of physical space. Thus, many 3-dimensional systems with high symmetry, such as viruses in biology, [5][6][7][8][9] fullerenes in chemistry, [10][11][12][13] and quasicrystals in physics, [14][15][16][17] can be modeled using Coxeter groups.…”

Affine Toda field theories related to Coxeter groups of noncrystallographic type

“…In [7] a similar identification has been made for the specific case of the embedding H 4 ֒→ E 8 , which relies on the property of an inflation map which mimics the action of ω entirely inside ∆. Here we avoid the introduction of such a quantity.…”

“…For our purposes it will be most important to achieve also the opposite, which can not be found in [7], namely to compute inner products in∆ from those in ∆. For this aim we introduce here the map 6) which acts on the simple roots in∆ as…”

“…It will be one of the results in this paper to show that there is an even more fundamental structure than E 8 underlying this particular model, the non-crystallographic Coxeter group H 4 . We draw here on the observation made first by Sherbak in 1988 [6], namely that H 4 can be embedded into E 8 , see also [7,8] for further developments of these mathematical structures. Loosely speaking, one may regard the E 8 -theory as two copies of H 4 -theories.…”