1 ; 1 ; in Month : December (2019) Article No : gjaap-v1-1001
A. V. Yurkin

##### Abstract
Short Communication The regular octahedron refers to the number of five Platonic figures. It can be composed of eight equal equilateral triangles or twelve identical segments. "The octahedron is dual to the cube". The regular octahedron can also be composed of many identical small cubes just as in Ancient Egypt were pyramids of stone blocks. The construction of an octahedron using small cubes can be obtained by considering a random walk in three-dimensional (3D) space. In we considered a visual model of a 3D random linear and nonlinear walk in an octahedron. In we reviewed and systematized the visual models of 1D, 2D and 3D random linear and nonlinear walks too. In this paper we explore some new features, patterns and fractions of numbers in visual 3D models of random linear and nonlinear walks in an octahedron composed of small cubes. Our studies of the deterministic models and visual constructions of linear (without any acceleration in and nonlinear (with the simplest uniformly acceleration random walk and arithmetic figures given in this paper show various geometric properties and nonlinear effects of 1D, 2D and 3D spaces. In 1D space with a linear random walk a linear arithmetic triangle (Pascal's triangle) is densely filled with numbers. In 1D space with a nonlinear random walk a nonlinear arithmetic triangle is loosely (contains gaps) filled with numbers. In 2D space with linear and nonlinear random walk linear and nonlinear arithmetic squares are densely filled with numbers (without gaps) in both cases. In 3D space with a linear random walk Figure 1: The third linear arithmetic octahedron (the third iteration 3). 1-fraction on the surface of octahedron and 2-fraction inside it. the linear arithmetic octahedron is almost densely filled with numbers but the neighboring areas inside the octahedron remain are empty (contains gaps) until the next iteration. cervical cancer. Diagnosis was made only after In 3D space with a nonlinear random walk the nonlinear arithmetic octahedron is not completely filled with numbers (contains gaps) as in the case of a nonlinear 1D random walk; some neighboring regions inside the nonlinear octahedron remain empty (contains gaps) until the next iteration and some remain empty during several or many iterations. But gaps and “islands of numbers” or separate structures of numbers consistently appear and disappear after several or many iterations in a nonlinear 3D case.

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