Laves graph
In geometry and crystallography, the Laves graph is an infinite cubic symmetric graph. It can be embedded into three-dimensional space, with integer coordinates, to form a structure with chiral symmetry in which the three edges at each vertex form 120° angles to each other. It can also be defined more abstractly as a covering graph of the complete graph on four vertices.
H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K4 crystal,(10,3)-a network,diamond twin,triamond, and the srs net.
As Coxeter (1955) describes, the vertices of the Laves graph can be defined by selecting one out of every eight points in the three-dimensional integer lattice, and forming their nearest neighbor graph. Specifically, one chooses the points
and all the other points that can be formed by adding multiples of four to these coordinates. The edges of the Laves graph connect pairs of points whose Euclidean distance from each other is √2 (these pairs differ by one unit in two coordinates, and are the same in the third coordinate). The other non-adjacent pairs of vertices are farther apart, at a distance of at least √6 from each other. The edges of the resulting geometric graph are diagonals of a subset of the faces of the regular skew polyhedron with six square faces per vertex, so the Laves graph is embedded in this skew polyhedron.
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