Some psychological and pedagogical principles of mathematical education

The article deals with the construction of regular inscribed polygons, which are of great scientific and practical importance. As a result of solving such problems, new formations arise, new systems of connections are formed, new properties, qualities of the mind (flexibility, depth, awareness), which mark a progressive shift in mental development. This is why the effect of math training is directed at this side of the psyche. To increase its developing effect, it is necessary to take into account the specifics of thinking, the ratio of age and individual mental characteristics of schoolchildren. Let us now consider the problem of humanizing higher education. Humanitarianization presupposes, first of all, the introduction of a young person to the humanitarian culture of mankind. In other words, humanitarization is usually seen as an additional and necessary component of professional education. The author draws attention to the issues of humanitarization of mathematical education in Uzbekistan for further improving the system of teaching mathematical science at all levels of education, support the effective work of teachers, expand the scale and increase the practical significance of research, and strengthen ties with the international community.

Limit theorems for generalized perimeters of random inscribed polygons. II

Lao and Mayer (2008) recently developed the theory of U-max statistics, where instead of the usual sums over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Examples include the greatest distance between random points in a ball, the maximum diameter of a random polygon, the largest scalar product in a sample of points, etc. Their limit distributions are related to distribution of extreme values. This is the second article devoted to the study of the generalized perimeter of a polygon and the limit behavior of the U-max statistics associated with the generalized perimeter. Here we consider the case when the parameter y, arising in the definition of the generalized perimeter, is greater than 1. The problems that arise in the applied method in this case are described. The results of theorems on limit behavior in the case of a triangle are refined.

Limit theorems for generalized perimeters of random inscribed polygons. I

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

Random Polygons and Estimations of π

In this paper, we study the approximation of π through the semiperimeter or area of a random n-sided polygon inscribed in a unit circle in ℝ2. We show that, with probability 1, the approximation error goes to 0 as n → ∞, and is roughly sextupled when compared with the classical Archimedean approach of using a regular n-sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.

Ellipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law

Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear. Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis. Results: The average form deviation of vertexes from the fitted ellipse was 0±3.1 % (8,291 vertices in 1375 cells were examined). Area of the polygonal cell was 0.9±0.1 times of area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells can be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Then, an improved relation of Lewis’s law for a n-edged cell was derived \[cell\ area=0.5nab\sin(\frac{2\pi}{n})(1-\frac{3}{n^2})\] where, a and b are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. This study also improved the relation of Aboav-Weaire’s law \[number\ of\ neighboring\ cells=6+\frac{6-n}{n}(\frac{a}{b}+\frac{3}{n^2})\] Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.

Archetypal Mathematics

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month, photographs of ancient arches lead to a discussion of inscribed polygons.