Original scientific paper

Regular Polytopes, Root Lattices, and Quasicrystals

R. Bruce King ; Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA

Abstract

The icosahedral quasicrystals of five-fold symmetry in two, three, and four dimensions are related to the corresponding regular polytopes exhibiting five-fold symmetry, namely the regular pentagon (H₂ reflection group), the regular icosahedron [3,5] (H₃ reflection group), and the regular four-dimensional polytope [3,3,5] (H₄ reflection group). These quasicrystals exhibiting five-fold symmetry can be generated by projections from indecomposable root lattices with twice the number of dimensions, namely A₄→H₂, D₆→H₃, E₈→H₄. Because of the subgroup relationships H₂ ⊂ H₃ ⊂ H₄, study of the projection E₈→H₄ provides information on all of the possible icosahedral quasicrystals. Similar projections from other indecomposable root lattices can generate quasicrystals of other symmetries. Four-dimensional root lattices are sufficient for projections to two-dimensional quasicrystals of eight-fold and twelve-fold symmetries. However, root lattices of at least six-dimensions (e.g., the A₆ lattice) are required to generate twodimensional quasicrystals of seven-fold symmetry. This might account for the absence of seven-fold symmetry in experimentally observed quasicrystals.

Keywords

polytopes; root lattices; quasicrystals; icosahedron

Hrčak ID:

102656

URI

https://hrcak.srce.hr/102656

Publication date:

31.5.2004.

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