Applications of Geometry of Numbers to Phyllotaxis and Crystallography

The golden angle method, originally known from phyllotaxis in botany, has been used to generate dense point packings on surfaces of revolution. In my recent work, I have extended this method to general surfaces and higher-dimensional manifolds by employing the theories of products of linear forms in number theory, diagonalizable metrics in differential geometry, and local solutions of quasilinear hyperbolic equations. This extension suggests that any biological forms can exhibit phyllotactic patterns locally regardless of their morphology, while the overall pattern is influenced by their global properties in the embedded space. On the algebraic side, it is interesting that the same ideas used for phyllotaxis can also be applied to pseudorandom number generation over F2 = {0, 1}.
This work is motivated by my previous research in crystallography. Time permitting, I will also introduce some of the research, which contributes to the analytical foundations of crystallography and is also an application of the geometry of numbers.