In the twenty-first century, in the era of 3D printers and advanced technologies for the production of complex parts, it becomes possible to create objects that have a fractal structure. This gives relevance to the attempts of applying the knowledge of fractal geometry. This paper presents effectiveness confirmation and feasibility studies of using a fractal algorithm for making stronger structures. The algorithm is based on the Sierpiński fractal. The essence of the algorithm is to increase the buckling strength of the rods by fixing their midpoints. This principle makes it possible to create complex flat and three-dimensional structures, as well as to solve some engineering problems. Analytical strength calculations were run in Wolfram Mathematica, whilst Solid Works, a CAD and CAE program, were used to compute the strength.

It has long been known that there are fractals, which construction resolve into cutting out of elements from lines, curves or geometric shapes according to a certain law. If the fractal is completely self-similar, its dimensionality is reduced relative to the original object and usually becomes fractional. The whole fractal is often decomposing into a set of separate elements, organized in the space of corresponding dimension. German mathematician Georg Cantor was among the first to propose such fractal set in the late 19th century. Later in the early 20th century polish mathematician Vaclav Sierpinski described the Sierpinski carpet – one of the variants for the Cantor set generalization onto a two-dimensional space. At a later date the Austrian Karl Menger created a three-dimensional analogue of the Sierpinski fractal. Similar sets differ in a number of parameters from other fractals, and therefore must be considered separately. In this paper it has been proposed to call these fractals as i-fractals (from the Latin interfican – cut). The emphasis is on the three-dimensional i-fractals, created based on the Cantor and Sierpinski principles and other fractal dependencies. Mathematics of spatial fractal sets is very difficult to understand, therefore, were used computer models developed in the three-dimensional modeling software SolidWorks and COMPASS, the obtained data were processing using mathematical programs. Using fractal principles it is possible to create a large number of i-fractals’ three dimensional models therefore important research objectives include such objects’ classification development. In addition, were analyzed i-fractals’ geometry features, and proposed general principles for their creation.

One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.

Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.