A topological coordinate system for the diamond cubic grid

Abstract

Topological coordinate systems are used to address all cells of abstract cell complexes. In this paper, a topological coordinate system for cells in the diamond cubic grid is presented and some of its properties are detailed. Four dependent coordinates are used to address the voxels (triakis truncated tetrahedra), their faces (hexagons and triangles), their edges and the points at their corners. Boundary and co-boundary relations, as well as adjacency relations between the cells, can easily be captured by the coordinate values. Thus, this coordinate system is apt for implementation in various applications, such as visualizations, morphological and topological operations and shape analysis.

On automatic determination of quasicrystal orientations by indexing of detected reflections

“…It is also applicable to periodic crystals with reflection indices specified in frames with n larger than three [17]. In particular, it can be applied to data specified in hexagonal four-index setting or in quadray coordinates [18,19].…”

The hyper‐Wiener Index of diamond nanowires

“…In addition to being the hardest natural substance, diamonds are a crucial component of jewellery and play a significant role in industry. There are also numerous theoretical studies about them, including, as we have already mentioned, computing different digital distance functions [32,33,36], providing topological and combinatorial coordinate systems [37][38][39] based on either the Cartesian or the symmetric coordinate frame, or creating a geometrical model-based software implementation of the morphology of growth of diamond crystals [40], to name a few. Special 2D structures, such as thin layers linked to graphite and diamond, including graphene, and 1D structures, such as different nanotubes and nanowires (based on carbon or silicon), are increasingly being employed in industry.…”