Open AccessColored lattices
Author(s) -
David Harker
Publication year - 1978
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.75.11.5264
Subject(s) - colored , combinatorics , row , permutation group , mathematics , permutation (music) , abelian group , cyclic permutation , symmetric group , computer science , physics , materials science , database , acoustics , composite material
Combinations of translations and color permutations are derived that leave a periodic array of colored points—a colored lattice—apparently unchanged. It is found that there are three types of colored lattices: (1 ) those in which all rows and nets have more than one color, (2 ) those in which there are rows with only one color, and (3 ) those in which there are both rows and nets with only one color. The color permutation groups of colored lattices are all Abelian. The direct product of three independent cyclic subgroups is required by type1 , but only two are required by type2 ; in type3 the color permutation group consists of then powers of a cyclic permutation of alln colors present—i.e., the group consists of a single cycle.