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Within a mathematical view, lattices deal with periodic point sets, sometimes considered as being connected by their short lattice root vectors, i.e. as such nets they do reflect the full symmetry of the according space group. And conversely, the full space group thus acts transitively on its vertex set. However, there clearly are further such nets too in a more general sense. Those can be considered in the same view as Archimedean polyhedra extend the Platonic ones, for there multiple different regular face polygons as facets become allowed, while here, for such non-lattices, there come in multiple different vertex configurations, where, although all being locally congruent, this congruence transformation is not included any longer within the full structure symmetry. Esp. in crystallography several such high-symmetrical non-lattice point structures are non-the-less likewise important. And this even more, if they fulfill the following properties:
For those 3 properties it has been proven by T. Sunada in 2008 that the only non-lattice structures fulfilling them would be either the often so called Diamond "lattice" or the chiral Laves' graph. Thence the latter also got the comparing name "Diamond Twin". Also the name "SRS Net" is used by crystallographers, as it gets being used by the Si atoms within the SrSi crystal structure.
The diamond net has 2 local vertex configurations, as displayed on the right. These are to be mutually attached by matching vertex colors (yellow / green). Note, that the configurations could be considered as the vertices and the body-center of a cube-inscribed tetrahedron each, just within their alternate orientations. And in fact, This diamond structure indeed can be understood from the Delone complex of the A3 lattice (octet), when inscribing the first of those configurations in all its accordingly oriented tets. According to the lattice repetition unit of A3, i.e. the encasing cube of each octahedron of octet the here being used encasing cube of those tets are smaller by a factor 2. Moreover, when considering those correctly oriented tets only, these smaller cubes within that repetition unit are situated at alternate positions only.
By means of the afore mentioned color symmetry of vertices the other configuration unit well could have been used likewise, then clearly using the other oriented tets in a slightly displaced octet. This displacement, in units of the large repetition unit cube, then clearly is 1/4 of its body diagonal.
As an aside, for sure, both these local configurations also could have been used simultanuosly, when getting inscribed into both tets of octet. However, then not only the color coding of vertices becomes arbitrary at the small encasing cubes' vertices, rather the formerly equivalence of vertex positions at the tets vertices and body-center does now break down. Then there would be 2 different vertex types instead, 4-valent vertices at the tets' centers and 8-valent at the octs' vertices. Altogether the Delone complex of this structure then is nothing but the rhombidodecahedral honeycomb. Or, in other words, that structure then happens to be an overlay of 2 differently positioned diamond nets.
The struts of this diamond net feature hexagonal holes. These holes have the shape of a skew hexagon with an always alternating zigzag path {(+,-)3}. From the partial path of 3 consecutive vertices this can be continued therefore in 2 different ways, bending thereafter in either direction. Thence, as the encasing tet of each vertex configuration has 6 edges and each such edge supports such a triple (vertex pair and body-center), there is a total of 12 such skew hexagonal holes per vertex.
The Voronoi cell of the diamond structure is the triakis truncated tetrahedron, a tut with 4 of the 3-fold vertex-pyramids of rad attached. Accordingly the edge ratio of the Voronoi complex is sqrt(3) : sqrt(8). Note that those to be attached pyramids also are nothing but the centri-pyramids underneath the faces of a tet. This in turn shows, that this Voronoi complex is closely related to cytatoh.
An integer-based description of those net coordinates is {(x,y,z) ∈ ℤ3 | x=y=z (mod 2) & x+y+z=0 or 1 (mod 4)}.
The diamond structure also can be constructed with polyhedral struts. Then eg. icosahedra could be used at vertex positions, while octahedra would replace their connections. This is what is known as skew polyhedral Well's diamond triangulated surface.
In contrast to the former, where the 2 elemetary units got based on cube-inscribed tets each, here they get based on the 4 cube-inscribed diagonal regular hexagons, or, more precisely, in their alternations in turn, i.e. in planar triangles, again together with the body-center. As a cube has 4 such inscribed hexagons, there are 4 such elemetary units here too. Those are depicted on the right, together with their matching vertex colors (red / yellow / green / blue). The 4 here being used planes have a pairwise angle of arccos(1/3) = 70.528779°.
An alternate description of the Laves' graph can be given by black-white subsymmetry of the mucube, when on the white squares the net by the square's diagonals is drawn. Alternatively this could have been done on the black ones too. As it turns out, these 2 however are not symmetrically isomorph, rather they just represent an enantiomorph pair.
Holes of this net are larger, despite of the left seemingly hexagonal projection view. They represent skew decagons here of the cyclic form {(o,o,+,o,-)2}. Because of the low symmetry of those decagons it here happens that there are 15 such holes incident to every vertex.
The Voronoi cell of the Laves' graph is the highly irregular heptadecahedron shown on the right.
The number of vertices at distance n from a fixed vertex is 1, 3, 6, 12, 35, 48, 69, 86, ... (OEIS A038620).
Like for the diamond net, a more voluptuous polyhedral strut description can be provided here too, just that it becomes here a bit more elaborate. Re-consider the above being shown cubical units. Construct to each there shown strut an orthogonal line segment, which then connects the centers of the 2 vertex-incident square faces. Together with their intersecting cubical edge, those clearly define sections of octahedra, which are centered around those outer vertex points – while the central one obviously deals as the center of the dual, cube-inscribed octahedron. Accordingly, the thus optained octahedral structure is just edge-connected and clearly is a substructure of octet. In fact, all the octahedral bits along those cube's edges are missing, where no vertex is being placed. If moreover a truely face-connected polyhedral structure is being searched for instead, then one simply could add in those tetrahedra as well, which are edge-connected to those first constructed orthogonal edges, i.e. matching into the cavities between 2 neighbouring so far existing octs. The thus obtained substructure of octet got displayed on the left.
Out of curiosity it might be mentioned here, that the vertex set of the above described (and at the left being shown) substructure of octet could be used to derive its own Voronoi complex. The overall symmetry of the such in early 2024 by T. Dorozinski constructed complex then surely is the same as of the original Laves' graph and thus esp. is chiral in that sense that it cannot be mapped onto its enantiomorph. In contrast however to the original's Voronoi cell, this one then would result in a much simpler hendecahedron only, which now has only a 2-fold rotational axis and no further symmetry. However, because being a Voronoi cell, it clearly is a pleisiohedron too or, equivalently, the thereby obtained complex is monotopal.
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