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Regular Polytopes, Root Lattices, and Quasicrystals
R. Bruce King
; Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA
Puni tekst: engleski, pdf (261 KB)
King, R.B. (2004). Regular Polytopes, Root Lattices, and Quasicrystals. Croatica Chemica Acta, 77(1-2), 133-140. Preuzeto s https://hrcak.srce.hr/102656
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.
polytopes; root lattices; quasicrystals; icosahedron
Hrčak ID: 102656
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