1984/09/01 by Peter Krämer, R. Neri · 6 citations
Physics and Astronomy · Materials Science · Mathematics · #Advanced Mathematical Theories and Applications #Quasicrystal Structures and Properties #Mathematics and Applications
paper · doi:10.1107/s0108767384001203
openalex publication_date 1984/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
A periodic lattice in \bb En is associated with an n-grid and its dual, and with a point symmetry group G. Given a subgroup H of G, a subspace \bb Em, m < n, of \bb En, invariant under H, is chosen and a projection of the n-grid from \bb En to \bb Em is defined. The translational and point symmetries of the projected n-grid are analyzed. A projection of the cubic n-grid from \bb En to \bb En-1 based on H = S(n) yields a periodic n-grid. A projection of the cubic 12-grid from \bb E12 to \bb E3 based on H = A(5) yields a non-periodic 12-grid. This 12- grid is characterized by three real numbers and from its projection has a well defined orientation. The dual to this 12-grid yields a generalization of the non-periodic Penrose patterns from two to three dimensions.