اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPacking theory derived from phyllotaxis and products of linear forms
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textit{Parastichies} are spiral patterns observed in plants and numerical patterns generated using golden angle method. We generalize this method for botanical pattern formation, by using Markoff theory and the theory of product of linear forms, to obtain a theory for (local) packing of any Riemannian manifolds of general dimensions $n$ with a locally diagonalizable metric, including the Euclidean spaces. Our method is based on the property of some special lattices that the density of the lattice packing maintains a large value for any scale transformations in the directions of the standard Euclidean axes, and utilizes maps that fulfill a system of partial differential equations. Using this method, we prove that it is possible to generate almost uniformly distributed point sets on any real analytic Riemann surfaces. The packing density is bounded below by approximately 0.7. A packing with logarithmic-spirals and a 3D analogue of the Vogel spiral are obtained as a result. We also provide a method to construct $(n+1)$-dimensional Riemannian manifolds with diagonal and constant-determinant metrics from $n$-dimensional manifolds with such a metric, which generally works for $n = 1, 2$. The obtained manifolds have the self-similarity of biological growth characterized by increasing size without changing shape.
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