A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems

The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular dodecahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cam-eras’ projection and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the ob-server ensuring synthesis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc.

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.

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.

Model for the architecture of caveolae based on a flexible, net-like assembly of Cavin1 and Caveolin discs

Caveolae are invaginated plasma membrane domains involved in mechanosensing, signaling, endocytosis, and membrane homeostasis. Oligomers of membrane-embedded caveolins and peripherally attached cavins form the caveolar coat whose structure has remained elusive. Here, purified Cavin1 60S complexes were analyzed structurally in solution and after liposome reconstitution by electron cryotomography. Cavin1 adopted a flexible, net-like protein mesh able to form polyhedral lattices on phosphatidylserine-containing vesicles. Mutating the two coiled-coil domains in Cavin1 revealed that they mediate distinct assembly steps during 60S complex formation. The organization of the cavin coat corresponded to a polyhedral nano-net held together by coiled-coil segments. Positive residues around the C-terminal coiled-coil domain were required for membrane binding. Purified caveolin 8S oligomers assumed disc-shaped arrangements of sizes that are consistent with the discs occupying the faces in the caveolar polyhedra. Polygonal caveolar membrane profiles were revealed in tomograms of native caveolae inside cells. We propose a model with a regular dodecahedron as structural basis for the caveolae architecture.

A Spherical Motor Driven by Electro-Magnets Based on Polyhedrons

This paper describes the development of a spherical motor, hereinafter called “14-12 spherical motor.” This spherical motor utilizes two polyhedrons – a truncated regular octahedron and a regular dodecahedron – for the arrangement of permanent magnets on the rotor and electro-magnets on the stator. The 14-12 spherical motor has two types of rotation axes and six rotation axes in all. Five-phase alternating current was applied to the electro-magnets to rotate the rotor. This study also developed a simulation model for the 14-12 spherical motor to numerically simulate the dynamic behavior of the motor. Basic performance was measured and simulated to evaluate (1) the relation between rotation speed and maximum output rotation torque and (2) cogging torque. Waveforms of the five-phase alternating current were improved using the simulation model in order to increase output rotation torque for the rotation axis with the smaller torque.

Symplectic Toric Geometry and the Regular Dodecahedron

The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.