1997/12/31 by Adam Doliwa, A. Doliwa, P. M. Santini +3 · 9 citations
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Algebra over a field #Geometric transformation #Point (geometry) #Quadrilateral #Quasicrystal Structures and Properties #Transformation (genetics) #nlin.SI #solv-int
paper · pdf · doi:10.1063/1.533175
published as J. Math. Phys. 41 (2000) 944-990 · 50 pages, 15 figures; minor corrections, added references
arxiv created 1998/01/17 · openalex publication_date 2000/02/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂̄ formalism.