Modeling 13 Archimedean solids by and object-oriented language

The computational study of structures with chemical relevance is preceded by its modeling in such manner that no calculations can be submitted without the knowledge of their spatial atomic arrangement. In this regard, the use of an object-oriented language can be helpful both to generate the Cartesian coordinates (.xyz file format) and to obtain a ray-traced image. The modeling of chemical structures based on programming has some advantages with respect to other known strategies. The more important advantage is the generation of Cartesian coordinates that can be visualized easily by using free of charge software. Our approach facilitates the spatial vision of complex structures and make tangible the chemistry concepts delivered in the classroom. In this article an undergraduate project is described in which students generate the Cartesian coordinates of 13 Archimedean solids based on a geometrical/programming approach. Students were guided along the project and meetings were held to integrate their ideas in a few lines of programmed codes. They improved their decision-making process and their organization and collecting information capabilities, as much as their reasoning and spatial depth. The final products of this project are the coded algorithms and those made tangible the grade of learning/understanding derived of this activity.

Groups, Platonic solids and Bell inequalities

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.

Polyhedra, Tiles and Graphene Defects: The Case of Tetraoctite

: This mini review concerns hydrocarbons, (CH)n, and carbon allotropes, Cn, and their relationships with regular solids and regular surfaces, respectively. Platonic and Archimedean solids and surfaces related to carbon allotropes are described as an introduction. An overview is then provided on how Stone–Wales defects lead to a series of structures: pentaheptite, Haeckelite, net C, net W, planar C4, biphenylene, graphyne, graphdiyne, and tetraoctite. This last compound is discussed in detail together with its relation to the Mills–Nixon effect on cyclooctatetraene (COT).

POLYHEDRA AND QUASI- POLYHEDRA IN ARCHITECTURE OF CIVIL AND INDUSTRIAL ERECTIONS

Innovative spatial forms appear and develop at the joint of science, art, and architecture. Geometry is the most important, fundamental components of architectural forming. Now, having passed the stages of passion for the large-span shells, the sky-scrapers, typical inexpensive buildings, architectural bionics and ergonomics; pneumatic, membrane, wire rope and shrouds erections, the architects and designers payed attention at analytically non-given forms of erections and at the polyhedron. It is noticeably especially at the last 10-15 years. In a paper, the problems of application of the polyhedron and their modifications in architecture, building, and technics are analyzed. They consider prisms, pyramids, prismatoids, Platonic and several Archimedean solids, quasi-polyhedrons, and some figures constituted on their base. Polyhedral domes, umbrella shells, and hipped plate constructions are presented too. Large quantity of the illustrations devoted to the architecture of buildings and erections, to the landscape architecture and to the sculptural compositions is presented for the confirmation of increasing interest to these structures. 31 titles of the used original sources are given.

Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them

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.