We considered Turing patterns on a spherical surface from the viewpoint of polyhedron geometry. We restrict our consideration to a set of parameters that produces a pattern of spots. We obtained numerical solutions for the Turing system on a spherical surface and approximated the solutions to convex polyhedrons. The polyhedron structure was dependent on both the radius of the sphere R and the initial condition. The number n of faces of the polyhedron increased with an increase in R. For small values of R, highly ordered structures were observed. With an increase of the value of R, a variety of structures were observed for each n, and the symmetry property of the spots, which determined the regularity of the polyhedron structure, gradually disappeared. We classified the numerical results according to their symmetrical properties of the approximated polyhedrons. The results revealed that the obtained Turing patterns lost symmetrical properties and varied the structures within same number of spots.

In the family of the area preserving Hénon maps (the Hénon maps), the mapping function is quadratic. Replacing the quadratic function with a piecewise linear function, we obtain the area preserving Lozi map (the Lozi map). For the Hénon map, the elliptic periodic orbits appearing through rotation bifurcation of the elliptic fixed point have one orbital point on the particular axis, i.e., the dominant axis. Thus, the dominant axis theorem holds for the Hénon map. For the Lozi map, the dominant axis theorem does not hold. We make clear the reasons from the study of bifurcations. For the Lozi map, a new theorem instead of the dominant axis theorem is obtained.

We report a practical method to assign parastichy numbers to spiral patterns formed by sunflower seeds and pineapple ramenta using a discrete Fourier transform. We designed various simulation models of sunflower seeds and pineapple ramenta and simulated their point patterns. The parastichy numbers can be directly and accurately assigned using the discrete Fourier transform method to analyze point patterns even when the parastichy numbers contain a divergence angle that results in two or more generalized Fibonacci numbers. The presented method can be applied to extract the structural features of any spiral pattern.

Quadratic curves are typical examples of geometric forms. This paper looks at selected quadratic curves and describes their transformation into cubic curves for use as axes for generating geometric patterns. Using periodic functions with t as an intermediary variable, we defined these curves as universal cubic curves expressed as x = f (t), y = g (t) and z = h (t). Using universal cubic curves selected as the axes and selected quadratic curves as motifs, we then generated original geometric patterns by applying affine transformations. This paper describes our investigation of mathematical modeling-based generation of 3D geometric patterns with cubic curves as the axes and quadratic curves as motifs, and of the transformation of the patterns into two dimensions to generate geometric pattern variations. Through pattern generation, this paper aims to provide a basic methodology that can be used in fields such as art and design.