The article suggests methods that allow creating the most complicated type of polygonal masonry found in Peru. This masonry type consists of large stone blocks weighing from several hundred kilograms to several tons fitted close to each other almost without a gap between complicated curved surfaces over a large area. The work provides a description of techniques, which apparently were used by builders who arrived from Europe. The techniques under discussion are based on the use of a reduced clay model, 3D-pantograph, topography translator and replicas. The use of the topography translator, reduced clay model and pantograph provides not only the unique appearance and high quality of masonry of large blocks, but also allows to increase the productivity of the builders significantly. As machines coping-scaling three-dimensional objects are known since the beginning of the 18th century, the stone structures under consideration should be approximately dated by this time. The remaining simpler types of polygonal masonry, when the stones are small or the fitted surfaces are almost flat, or the stones contact each other over a small area, or there are significant gaps between the stones, are quite consistent with the well-known methods of stone processing at that time or earlier, and, therefore, they do not require any additional explanations. The Fortress Sacsayhuaman is considered as an example of early star fortresses that has survived to our time. The polygonal structures in Peru, the polygonal Face Towers and polygonal bas-reliefs in Cambodia, symmetrical statues of pharaohs in Egypt are based on the same construction technologies, working methods, tools and technical contrivances. Therefore, with a high probability one can state that all these monuments were created by the same group of architects, sculptors, builders, and could not have appeared before the 17th century.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the $p$-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the "broom topology'' type.

Since the conventional split-merge algorithm is sensitive to the object scale variance and splitting starting point, a piecewise split-merge polygon approximation method is proposed to extract the object contour features. Specifically, the contour corner is used as the starting point for the contour piecewise approximation to reduce the sensitivity of the contour segment on the starting point; then, the split-merge algorithm is used to implement the polygon approximation for each contour segments. Both the distance ratio and the arc length ratio instead of the distance error are used as the iterative stop condition to improve the robustness to the object scale variance. Both the angle and length as two features describe the shape of the contour polygon, and affect each other along the contour order relationship. Since they have a strong coupling relationship. To improve the description correction of the contour, these two features are combined to construct a Coupled Hidden Markov Model to detect the object by calculating the probability of the contour feature. The proposed algorithm is validated on ETHZ Shape Classes and INRIA Horses standard datasets. Compared with other contour-based object detection algorithms, the proposed algorithm reduces the complexity of contour description, improves the robustness of contour features to scale variance, and has a higher object detection rate.