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.

A Non-Regular Icosahedron Geometry Satellite Form, Mirror Invested Polyhedro Heliotrope, For Optical Tracking

Working on relationships of three circles in common ratio [4/π or square root of the golden number ] and drawing lines of related tangents, squares and triangles, viewed on the paper plan, a figure having the shape of a section [Hexagonal] similar to that of an Icosahedron or Dodecahedron. This gave me the idea of searching for an existing probable Polyhedron built upon this traced shape. In fact this Polyhedron was built[ 4x scale], whose geometry relates to the Icosahedron and the Dodecahedron. It is a non regular Icosahedron having 12 Isosceli triangles and 8 Equilateral triangles. Mirror triangles cut to size, invested the structure for the configuration of a “Polyhedroheliotrope”Satellite Optical Tracking application.

An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities a\dot (ab \dot c) \dot c = b; a\dot (a \dot bc) \dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(\frac{1}{2} (1 +\sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

Toric Geometry of the Regular Convex Polyhedra

We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.

Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra

The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.

Theoretical Study of Iron Heterogeneous Growth on the Surface of C60 Molecule

Direct molecule dynamics (MD) simulations have also been performed to study heterogeneous nucleation and growth of iron on C60 molecule. The grown mechanism of this crystallization process was explored. The results indicate that 92 iron atoms attach to C60 molecule surface can form new covalent bond, forming a closed regular icosahedron. More atoms grow in layer to form bigger regular closed clathrate base on the structure of former one. As increase of atoms number, there will appear some crystal faces.

ROCHE: Analysis of Eclipsing Binary Multi-Dataset Observables

Code ROCHE is devoted to modeling multi-dataset observations of close eclipsing binaries such as radial velocities, multi-wavelength light curves, and broadening functions. The code includes circular surface spots, eccentric orbits, asynchronous or/and differential rotation, and third light. The program makes use of synthetic spectra to compute observed UBVRIJHK magnitudes from the surface model and the parallax. The surface grid is derived from a regular icosahedron to secure more-or-less equal (triangular) surface elements with observed intensities computed from synthetic spectra for supplied passband transmission curves. The limb-darkening is automatically interpolated from the tables after each computing step. All proximity effects (tidal deformation, reflection effect, gravity darkening) are taken into account. Integration of synthetic curves is improved by adaptive phase step (important for wide eclipsing systems).The code is still under development. It is planned to extend its capabilities towards low mass ratios and widely different radii of components to facilitate modeling of extrasolar planet transits. Another planned extension of the code will be modeling of spatially-resolved eclipsing binaries using relative visual orbits and/or interferometric visibilities.