Regular Polyhedra Research Articles

A Precision Evaluation Index System for Remote Sensing Data Sampling Based on Hexagonal Discrete Grids

With the rapid development of earth observation, satellite navigation, mobile communication, and other technologies, the order of magnitude of the spatial data we acquire and accumulate is increasing, and higher requirements are put forward for the application and storage of spatial data. As a new form of data management, the global discrete grid can be used for the efficient storage and application of large-scale global spatial data, which is a digital multiresolution georeference model that helps to establish a new model of data association and fusion. It is expected to make up for the shortcomings in the organization, processing, and application of current spatial data. There are different types of grid systems according to the grid division form, including global discrete grids with equal latitude and longitude, global discrete grids with variable latitude and longitude, and global discrete grids based on regular polyhedrons. However, there is no accuracy evaluation index system for remote sensing images expressed on the global discrete grid to solve this problem. This paper is dedicated to finding a suitable way to express remote sensing data on discrete grids, as well as establishing a suitable accuracy evaluation system for modeling remote sensing data based on hexagonal grids to evaluate modeling accuracy. The results show that this accuracy evaluation method can evaluate and analyze remote sensing data based on hexagonal grids from multiple levels, and the comprehensive similarity coefficient of the images before and after conversion is greater than 98%, which further proves the availability of the hexagonal-grid-based remote sensing data of remote sensing images. This evaluation method is generally applicable to all raster remote sensing images based on hexagonal grids, and it can be used to evaluate the availability of hexagonal grid images.

Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra

Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra

Planetary Systems and the Hidden Symmetries of the Kepler Problem

The question of whether the solar distances of the planetary system follow a regular sequence was raised by Kepler more than 400 years ago. He could not prove his expectation, inasmuch as the planetary orbits are not transformed into each other by the regular polyhedra. In 1989, Barut proposed another relation, which was inspired by the hidden symmetry of the Kepler problem. It was found to be approximately valid for our Solar System. Here, we investigate if exoplanet systems follow this rule. We find that the symmetry-governed sequence is valid in several systems. It is very unlikely that the observed regularity is by chance; therefore, our findings give support to Kepler’s guess, although with a different transformation rule.

Visualizing surface formation of semi-regular polyhedra of Archimedes

The most common method of forming semi-control polyhedra consists in cutting off angles and ribs of regular polyhedra by planes. The aim of the work - to consider the automated formation of a number of surfaces of semi-regular Archimedean polyhedra based on the dodecahedron. These include the truncated dodecahedron, the icosododecahedron, the romboicosododecahedron and the truncated icosododecahedron. The formation of surfaces is carried out by the kinematic method in AutoCAD using programs compiled in the AutoLISP language. Methods. The methodology for the formation of these polyhedra provides for truncation of the angles and edges of the dodecahedron. This requires the calculation of a number of geometric parameters of these polyhedra and dodecahedron, such as the value of the truncation of the dodecahedron edges, the size of the edges of truncated polyhedra, the centers of faces, dihedral angles, etc. In order to generate these surfaces, a frame is constructed because the frame lines are used as guides to form surfaces in a kinematic way. The electronic model of each polyhedron is constructed as a set of compartments of surfaces of all its faces, and each compartment is assigned to a certain layer of the drawing. The frame and electronic model of the polyhedra under study are formed by means of user programs composed in the functional language AutoLISP. The process of forming surfaces of selected polyhedra in the AutoCAD environment is provided by special programs that are also compiled in the AutoLISP language. Results. Software was created to demonstrate the process of formation of a number of Archimedes polyhedra on the monitor screen.

Random generation of convex aggregates for DEM study of particle shape effect

In this study, the random generation of 3D convex aggregates is proposed for the numerical study of the particle shape effect in DEM simulation. First, the Bezier curve fitting technique is employed to construct a curve skeleton inside a randomly generated polyhedron. Then, a solid surface is generated based on the curve skeleton using the Gregory surface interpolation algorithm. Next, convex aggregates are created inside both regular and irregular polyhedrons to demonstrate the capability of the proposed algorithm. Finally, DEM simulation of convex aggregates with different roundness features is performed to study the effect of roundness on the shear behavior of granular aggregates.

Dynamics of Gas Bubbles Inside a Ball-like Area at the Nodes of a Uniform Cubic Mesh Under Sudden Liquid Pressure Rise

The influence of hydrodynamic interaction between spherical gas bubbles inside a ball-like area at the nodes of a uniform cubic mesh on their dynamics under sudden liquid pressure rise has been investigated. Main attention is payed to the dynamics of a bubble located in the center of such a cluster of cubic structure. Prior to the pressure rise, the liquid and the bubbles are at rest, the liquid pressure is 1 bar, the radii of the bubbles are 0.25 mm, the size of the cubic mesh cells is 5 mm. The increase in the liquid pressure is 3 bar. For the sake of comparison, ball-like clusters with spherical and stochastic structures are also considered, with similar position of the central bubble and with initial minimum distance between the centers of bubbles not exceeding 5 mm. In the clusters of spherical structure, the non-central bubbles are located on concentric spheres with radii of 5 and 10 mm at the nodes of equally-oriented regular polyhedra (icosahedra and dodecahedra). In the clusters of stochastic structure, the non-central bubbles are distributed stochastically in a sphere with radius of 10 mm. A mathematical model is applied, in which the dynamics of bubbles is governed by the ordinary differential equations in the radii of bubbles and the position-vectors of their centers. It is shown that in the case of a cluster with cubic structure, the interaction between the bubbles leads to strengthening the collapse rate of the central bubble and to non-monotonic decay of oscillations of their pressure. Similar features are characteristic of the clusters with spherical and stochastic structures.

Geometric realizations of abstract regular polyhedra with automorphism group H3.

A geometric realization of an abstract polyhedron {\cal P} is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Γ. The method entails finding a real orthogonal representation of Γ of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H3.

Безымянные запахи: теория обоняния в «Тимее» Платона

Timaeus, theory of perception, sense of smell and odours, Plato, regular polyhedra. Abstract. The article is devoted to the problem of the sense of smell in Plato’s Timaeus. Any study of Plato’s doctrines requires first of all an examination of the rich tradition of the exegesis of his texts. In this article, author discusses the works of such famous scholars of Plato’s philosophy as Francis Cornford, Gregory Vlastos, Denis O’Brien and others. Despite the fact that the problem of the sense of smell is not among the central themes in Plato's cosmology, it turns out to be connected with many of them: the problem of the correspondence elements to forms of regular polyhedra, the problem of division into species of elements, the question of the structure of surfaces of regular polyhedra, etc. The author tried to present the question of the sense of smell in the Timaeus as a consistent theory, which can be placed in the context of the theories of his predecessors, such as Alcmaeon, Empedocles and Democritus, and his followers, first of all Aristotle. The author accepts the view that odours are mixtures of water and air (Vlastos). Reconstructing the theory of the sense of smell as a part of the whole theory of perception, the author comes to the conclusion that odours have no species and names because they don’t belong to the one idea of the element. Unlike colors, sounds and juices (which retain their identity even being mixed with other elements: color – with fire, taste – with water, sound – with air) odours have a half-formed, mixed, nature. In the light of this reconstruction, Plato’s theory of odours appears to pave the way to a more developed Aristotle’s theory of smell.

正多面体的七种不同类型的着色公式

正多面体的七种不同类型的着色公式

Golden section based method of 3D modeling of icosahedron and dodecahedron in construction design

This paper provides an introduction of a three-dimensional modeling of the two most complex regular polyhedrons: icosahedron and dodecahedron. It is suggested to apply a method of the icosahedron and dodecahedron building by using rectangles with the sides ratio like in the golden section, and taking the icosahedron’s golden rectangles as a basis. The author gives information on the first mentioning of this building method by the Italian mathematician L. Pacioli in his Divine Proportion book. Based on this method, the building of the icosahedron and dodecahedron inscribed in the regular hexahedron and circumscribed about it is performed. This may be important for the design of facilities, based on the shapes of these geometrical figures, with inscribed regular hexahedrons for the purpose of a better organization of residential space of storeys. The examples of using the geometrical shapes of regular polygons (RP) in architecture are given. The building is performed using the AutoCAD package, but it also can be repeated in other known CAD software.

Mobile assembly of two Bennett linkages and its application to transformation between cuboctahedron and octahedron

The Bennett linkage has been proposed as a building block of deployable structures as it can generate desirable spatial motions with the least number of links. However, these deployable structures are based on the tiling of Bennett linkages. No spatial assembly of Bennett linkages has been developed to form any transformation between regular and semi-regular polyhedrons. In this paper, we propose a new type of mobile assembly consisting of a pair of Bennett linkages connected by four spherical joints. We show that this particular assembly can be used to construct transformable polyhedrons. Using an alternative form of such assembly, a polyhedral transformation between cuboctahedron and octahedron with one degree of freedom is realised. The axes of the revolute joints within the Bennett linkages are determined by the geometrical relationship between the deployed geometry and the folded one. The transformation between two polyhedral shapes has no bifurcation which is proven through kinematic analysis and demonstrated by a physical model. This transformable polyhedron has great potential for the aerospace applications where transportability and protection of payload are critical design features.

On the Properties of the Symbols of One Class of Hypoelliptic Equations

We consider regular hypoelliptic operators and study some properties of completely regular polyhedra. Basing on the obtained properties, we find an upper bound for the functional dimension of the solution spaces of hypoelliptic equations.

Золотое сечение и золотые прямоугольники при построении икосаэдра, додекаэдра и тел Архимеда, основанных на них

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Визуализация правильных многогранников в процессе их образования

In this work the automated formation of surfaces correct to convex polyhedrons of Platon and two regular not convex star-shaped polyhedrons of Kepler-Poinsot by the kinematic method. Researches on realization of a goal were carried out in the environment of AutoCAD with use of the programs developed in the functional Autolisp programming language which is built in AutoCAD. The AutoLisp language and the AutoCAD environment are chosen for achievement of a goal as they allow showing bodies in the movement. The technique of formation of electronic models of the polyhedrons necessary for performance of visualization of polyhedrons is stated. The model is a set of compartments of a surface, issued in the form of the block. The user function in the AutoLisp language which identifier is team in the environment of AutoCAD is developed for each model. Each compartment was placed in the drawing layer which is taken away for it. When developing the user functions were taken into account to a possibility of the AutoCAD environment – the available teams for formation of surfaces. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made by the defrosting method of the block containing surface compartments. In the course of "defrosting" of layers with compartments on the screen of the monitor process of formation of a surface is shown – drawings of compartments of a surface appear one by one. The last drawing is an image of a surface. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made. The fragment of the program by training of one side of a tetrahedron is given Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. The possibility of formation of surfaces of regular polyhedrons is shown by a kinematic method: the movement rectilinear forming on the directing lines as which edges of polyhedrons are used.

Вероятностное и динамическое уравнения коллоида

We obtained two equations that characterize the structure of the colloid: the equation of the Schrodinger type that specifies the redistribution of heat and potential energy in the colloid and material equation – the diffusion equation with the operator of Liesegang associated directly with a substance that allows you to find the discontinuities of the structures caused by the vibrations of electrically charged particles. This procedure based on the assumption of the instability of the colloidal state, caused by the movement of charged particles. The reality is not collected in parts from the particles of matter in the course of evolution from the past to the future, and is all at once from the past to the future for a given pattern, that is, for specific PATTERNS, as defined by quantum theory. Without going deep into the theory of Kulakov, we will accept its fundamental provisions as a certain given. The forms of this structural data were obtained experimentally and mathematically confirmed. Let there be a certain angle of the skeleton, where due to the unevenness and partial randomness of the structure of the core grids forms "defects" – that is, electrical or magnetic moments of a particular order. Then small mobile clusters are attracted to it by electrostatic or electromagnetic forces, which are then adsorbed and somehow arranged on the "defects" in accordance with their dipole moments. This circumstance can be determined by "magic numbers", that is, as the number of clusters "stuck" to the defect of the core structure, with the formation of chemical bonds in the future. We can assume that the spanning structure of Coxeter can form small clusters form regular polyhedrons, and may occur or other structure having more complicated form.

Multilayer Formation in Self-Shaping Emulsion Droplets.

In several recent studies, we showed that micrometer-sized oil-in-water emulsion droplets from alkanes, alkenes, alcohols, triglycerides, or mixtures of these components can spontaneously "self-shape" upon cooling into various regular shapes, such as regular polyhedrons, platelets, rods, and fibers ( Denkov , N. , Nature 2015 , 528 , 392 ; Cholakova , D. , Adv. Colloid Interface Sci. 2016 , 235 , 90 ). These drop-shape transformations were explained by assuming that intermediate plastic rotator phase, composed of ordered multilayers of oily molecules, is formed beneath the drop surface around the oil-freezing temperature. An alternative explanation was proposed ( Guttman , S. , Proc. Natl. Acad. Sci. USA 2016 113 , 493 ; Guttman , S. , Langmuir 2017 , 33 , 1305 ), which is based on the assumption that the oil-water interfacial tension decreases to very low values upon emulsion cooling. Here, we present new results, obtained by differential scanning calorimetry (DSC), which quantify the enthalpy effects accompanying the drop-shape transformations. Using optical microscopy, we related the peaks in the DSC thermograms to the specific changes in the drop shape. Furthermore, from the enthalpies measured by DSC, we determined the fraction of the intermediate phase involved in the processes of drop deformation. The obtained results support the explanation that the drop-shape transformations are intimately related to the formation of ordered multilayers of alkane molecules with thickness varying between several and dozens of layers of alkane molecules, depending on the specific system. The new results provide the basis for a rational approach to the mechanistic explanation and to the fine control of this fascinating and industrially relevant phenomenon.

Hanging Around with Platonic Solids

The Platonic solids, also known as the five regular polyhedra, are the five solids whose faces are congruent regular polygons of the same type. Polyhedra is plural for polyhedron, derived from the Greek poly + hedros, meaning “multi-faces.” The five Platonic solids include the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Photographs 1a-d show several regular polyhedra

Analytical Form-Finding for Highly Symmetric and Super-Stable Configurations of Rhombic Truncated Regular Polyhedral Tensegrities

Tensegrities have exhibited great importance and numerous applications in many mechanical, aerospace, and biological systems, for which symmetric configurations are preferred as the tensegrity prototypes. Besides the well-known prismatic tensegrities, another ingenious group of tensegrities with high symmetry is the truncated regular polyhedral (TRP) tensegrities, including Z-based and rhombic types. Although Z-based TRP tensegrities have been widely studied in the form-finding and application issues, rhombic TRP tensegrities have been much less reported due to the lack of explicit solutions that can produce their symmetric configurations. Our former work presented a unified solution for the rhombic TRP tensegrities by involving the force-density method which yet cannot control structural geometric sizes and may produce irregular shapes. Here, using the structural equilibrium matrix-based form-finding method, we establish some analytical equations, in terms of structural geometric parameters and force-densities in elements, to directly construct the self-equilibrated, symmetric configurations of rhombic TRP tensegrities, i.e., tetrahedral, cubic/octahedral, and dodecahedral/icosahedral configurations. Moreover, it is proved, both theoretically and numerically, that all of our obtained rhombic TRP tensegrities are super-stable and thus can be stable for any level of the force-densities without causing element material failure, which is beneficial to their actual construction. This study helps to readily design rhombic tensegrities with high symmetry and develop novel biomechanical models, mechanical metamaterials, and advanced mechanical devices.

TWIN GEODESIC DOMES WITH ONE RADIUS

The paper proposes a method of forming geometric networks of one arch radius using regular spherical polyhedrons. A variant of network placing on a spherical icosahedron and, accordingly, on a sphere is proposed. The placement on the sphere of arches of one radius differs from the placement of meridians and is an effective solution for network with a smallest size of arch segments, with nodes of two intersecting arches formed on circles of the same radius and on regular spherical polyhedrons. The problem is solved by constructing an arches of the same radius using paired circles with two standard arch sizes. Several variants of the twin geodetic domes with the given stability loss are suggested.

The fraction of overlapping interphase around 2D and 3D polydisperse non-spherical particles: Theoretical and numerical models

The fraction of interphase is an important microstructure parameter in the prediction of macroscopic properties of particulate composites. Currently, some researchers have presented theoretical and numerical investigations on the interphase fraction for spherical particle systems, and even quantify the influence of interphase fraction on the overall elastic and transport properties of particulate composites. However, the overlapping interphase fraction in polydisperse non-spherical particle systems is still an open issue. In this work, a generic theoretical model is formulated to derive the overlapping interphase fraction for polydisperse 2D non-circular and 3D non-spherical particle systems by means of the statistical geometry of composites. In this model, the morphology of interphase coated onto the surface of non-spherical particles can be characterized by circularity and sphericity of particles that are important parameters to describe the shape of particles. On the other hand, numerical simulations for the one-point probability function of interphase are presented to verify the proposed theoretical framework. In the numerical simulations, a novel algorithm is developed by reducing the problem of identifying the precise location between an arbitrary spatial point and interphase to the basic issue of finding the distance from the point to the surface of particles. If the distance is less than the interphase thickness, the point definitely locates inside interphase. In addition, the algorithm can be further used to detect the overlap between adjacent ellipsoidal particles (2D ellipses). Moreover, a variety of particle shapes (such as regular polygons and ellipses with the same circularity, and regular polyhedrons and ellipsoids with the same sphericity) are taken into account to generate particle packing structures. Then, the fraction of overlapping interphase in each packing structure is statistically obtained. Results show that statistical values are consistent with their theoretical values under the same conditions. This validates the reliability of the theoretical framework. Finally, the effects of particle size distributions and interphase thicknesses on the interphase fraction are investigated. It can be found that the interphase fraction increases with the increase of particle volume fraction, particle fineness and interphase thickness.