Fractals In Three-Dimensional Space. I-Fractals

In the paper "On the Increasing Role of Geometry", published in the electronic "Journal of Natural Science Research" in 2017, it was outspoken a hypothesis that now, at the time of innovative technologies, the importance of geometry is constantly increasing. The significance of geometry is also demonstrated by numerous Ph.D. and doctoral dissertations in the specialty No 05.01.01 - “Engineering Geometry and Computer Graphics”. It can be affirmed that all and everyone dissertations of technical and technological profile contain a geometric component to one degree or another. The "Geometry and Graphics" journal turned 8 (it was founded in June 2012). During this time, on its pages have been published numerous scientific papers, developing namely geometry and its branches: from simplest geometric constructions based on new properties of both lines and surfaces, to imaginary elements. Investigations were conducted in the following areas: “New Directions in Geometry”, “Fractal Geometry”, “Multidimensional Geometry”, “Geometric Constructions”, “Construction and Research of Surfaces”, “Imaginary Geometry”, “Practical Application of Geometry”, “Computer Graphics”, “Descriptive Geometry as Basis of other Branches of Geometry” ,”Geometry of Phase Spaces”. The journal publishes both recognized scientists and candidate for Ph.D. and doctor degrees. The considered array of papers clearly confirms the statement of the majority of authors, published in the journal, about geometry continuous development, which knocks out the ground for skeptics who decided that geometry is the science of the past centuries. As long as objects with shapes and surfaces surround us, geometry will be in demand. This, as they say, is unequivocal.

Representation of Engineering Geometry Development in “Geometry and Graphics” Journal

Сальков¹,

Kadykova²

2020

Spatial Geometric Cells — Quasipolyhedra

Efremov

Vereschagina

Kadykova

et al. 2021

Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.

To the Question of the Classification for Analytical Surfaces

Strashnov

Rynkovskaya

2022

Until this day, in scientific and technical literature there are descriptions of around 600 analytical surfaces, divided into 38 classes. In turn, it is possible to divide almost each class of surfaces into sub-classes, groups, or types, that is, to compose a classification for surfaces. It will make it easier their further study. For some classes of surfaces there is no sense to set up classification schemes, since these classes consist of a small number of surfaces, or the classification will contain a bare listing of surfaces. These surfaces can include minimal and one-sided surfaces, spiral and helical ones, surfaces of Ciceika and Edlinger, as well as undulating, wavy and corrugated surfaces, and several others. The "others" category includes surfaces which, although highlighted in separate classes, are few, or they are in the process of studying. The study of scientific literature has shown that a classification for revolution surfaces, as well as for transfer and Peterson ones has not yet been set up. This paper is devoted to these classes of analytical surfaces. As the study has showed, it is possible to draw interesting conclusions based on presented classifications. In this paper it has been shown how the appearance of scientific publications devoted to classification for surfaces promote the emergence of numerous works on application of relevant surfaces in architecture, construction, engineering, shipbuilding, light industry products, topography and cartography. But devotion to analytic surfaces’ geometric forms should not be an end in itself in actual practice. So, today architects are passionate about parametric architecture, which is based on knowledge of analytical surfaces’ geometry. However, some leading architects believe that the parametric architecture oversaturated the human life’s space, resulting in a style crisis. Despite architects and engineers’ conflicting opinions, geometers must go ahead of society's immediate demands, creating new forms, classifications, and proposals.