SEARCH · Search NASA

Results for “space-filling polyhedra”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

3 records

All sources NASA NTRS DOE OSTI

DOE OSTI2024

Space-tiled colloidal crystals from DNA-forced shape-complementary polyhedra pairing

Generating space-filling arrangements of most discrete polyhedra nanostructures of the same shape is not possible. However, if the appropriate individual building blocks are selected (e.g., cubes), or multiple shapes of the appropriate dimensions are matched (e.g., octahedra and tetrahedra) and their pairing interactions are subsequently forced, space-filled architectures may be possible. With flexible molecular ligands (polyethylene glycol–modified DNA), the shape of a polyhedral nanoparticle can be deliberately altered and used to realize geometries that favor space tessellation. In this work, 10 new colloidal crystals were synthesized from DNA-modified nanocrystal building blocks that differed in shapes and sizes, designed to form space-filling architectures with micron-scale dimensions. The insights and capabilities provided by this new strategy substantially expand the scope of colloidal crystals possible and provide an expanded tool kit for researchers interested in designing metamaterials.

Science & Technology - Other Topics↗

DOE OSTI2022

The Euler characteristic as a basis for teaching topology concepts to crystallographers

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V – E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χ m ) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χ m has an elegant topological interpretation through the concept of orbifolds. Alternatively, χ m can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Full-potential KKR within the removed-sphere method: A practical and accurate solution to the Poisson equation

Here an efficient and accurate generalization of the removed-sphere method (RSM) to solve the Poisson equation for total charge density in a solid with space-filling convex Voronoi polyhedra (VPs) and any symmetry is presented. The generalized RSM avoids the use of multipoles and VP shape functions for cellular integrals, which have associated ill-convergent large, double-internal L sums in spherical-harmonic expansions, so that fast convergence in single-L sums is reached. Our RSM adopts full Ewald formulation to work for all configurations or when symmetry breaking occurs, such as for atomic displacements or elastic constant calculations. The structure-dependent coefficients A L that define RSM can be calculated once for a fixed structure and speed up the whole self-consistent-field procedure. The accuracy and rapid convergence properties are confirmed using two analytic models, including the Coulomb potential and energy. We then implement the full-potential RSM using the Green's function Korringa-Kohn-Rostoker (KKR) method for real applications and compare the results with other first-principle methods and experimental data, showing that they are equally as accurate.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗