A topological coordinate system for the diamond cubic grid

What is it about?

The diamond cubic grid describes the structure of the diamond (and some other materials). It can be also seen as a 3D tessellation of the space by triakis truncated tetrahedra. In various applications and algorithms, e.g., computing the surface, when using the diamond grid, it is very convenient if one directly access not only the triakis truncated tetrahedra themselves, but to their faces, edges and the points at their corners. Topological (combinatorial) coordinate systems offer that type of tools. Such a system is described in details with some possible applications.

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Why is it important?

Computing the surface (boundary tracking) of an object (crystal) can be done more easily if the used coordinate system allows to address the faces directly. Other geometric, image processing and visualization algorithm become more simple based on the provided coordinate system and the explained relations (e.g., boundary, co-boundary). The provided system, using 4 coordinates, reflects well the symmetry of the diamond cubic grid.

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