Two subdivision methods based on the regular octahedron for single- and double-layer spherical geodesic domes

The paper presents the topological–geometric analysis of a selected number of space frames configurations for geodesic domes which are generated from the regular octahedron. Two subdivision methods for spherical triangles, proposed by Fuliński, were used to create two families of structures. The first family consists of six single-layer and six double-layer geodesic domes shaped on the basis of the first method of subdivision, while the second family contains six single-layer and six double-layer geodesic domes shaped on the basis of the second method of subdivision. The calculated results of the geometric parameters of the analyzed structures were used to create original formulas that allow for more advanced structures to be achieved, that is, with a larger number of nodes and struts. The geometric results were also used to create nomograms showing the range of struts of the same length for double-layer geodesic domes. In both single-layer and double-layer domes, the number of groups of struts of equal lengths and the number of faces with different areas are smaller for structures created according to the first method of subdivision. The comparison of the resulting element quantities of two methods shows that the largest differences appear between the domes with a larger number of struts (up to 67%). Here, the analysis might help the designer reach a final decision on the better choice of topology, in particular, when this aspect is combined with other design goals, such as efficiency, economy, utility, and elegance in the design of the structure and the cover of large areas.

Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Volume Preserving Maps Between p-Balls

We construct a volume preserving map U p from the p-ball B p ( r ) = x ∈ R 3 , ∥ x ∥ p ≤ r to the regular octahedron B 1 ( r ′ ) , for arbitrary p > 0 . Then we calculate the inverse U p − 1 and we also deduce explicit expressions for U ∞ and U ∞ − 1 . This allows us to construct volume preserving maps between arbitrary balls B p ( r ) and B p ′ ( r ˜ ) , and also to map uniform and refinable grids between them. Finally we list some possible applications of our maps.

Linear and Nonlinear Random Walks in 1d, 2d and 3d Space.

The regular octahedron [1] refers to the number of five Platonic figures. It can be composed of eight equal equilateral triangles or twelve identical segments. "The octahedron is dual to the cube" [2]. The regular octahedron can also be composed of many identical small cubes just as in Ancient Egypt were pyramids of stone blocks. The construction of an octahedron using small cubes can be obtained by considering a random walk in three-dimensional (3D) space. In [3] we considered a visual model of a 3D random linear and nonlinear walk in an octahedron. In [3,4] we reviewed and systematized the visual models of 1D, 2D and 3D random linear and nonlinear walks too.

Octahedron-based spatial bar structures - the form of large areas covers

The large areas covers may be designed as the spatial dome constructions where the basis of their shaping are regular polyhedra. The paper presents eight new designed spatial bar structures as geodetic domes with a span of 50 m. The basis of their shaping is the regular octahedron. This polyhedron has not been recognized in detail as the basis for geodesic domes designing. Using second method of the division of the initial equilateral triangle proposed by professor Fuliński, bar domes generated from 2904-hedron, 3456-hedron, 4056-hedron, 4704-hedron, 5400-hedron, 6144-hedron, 6936-hedron and 7776-hedron were obtained. The designed eight bar structures were subjected to thorough geometric and static analysis showing the behaviour of the geodesic bar domes generated according to the presented in the paper method of the division of original face of regular octahedron. Own formulas were developed to determine the number of nodes and bars. The designed eight bar systems in the form of geodesic domes, which the basis of shaping is regular octahedron can be used as the covers of large areas without the necessity of the internal supports usage.

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.

Novel red-emitting phosphors A2HfF6:Mn4+(A = Rb+, Cs+) for solid-state lighting

New red-emitting phosphors A2HfF6(A = Rb+, Cs+) doped with Mn4+located at the center of a regular octahedron show intense red emission under blue light excitation and find potential application in LED chips.