In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance 2 {\displaystyle {\sqrt {2}}} . It can also be defined, divorced from its geometry, as an abstract undirected graph, 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. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space. Several crystalline chemicals have known or predicted structures in the form of the Laves graph. Thickening the edges of the Laves graph to cylinders produces a related minimal surface, the gyroid, which appears physically in certain soap film structures and in the wings of butterflies.
Constructions
From the integer grid
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
( 0 , 0 , 0 ) , ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) , ( 2 , 2 , 2 ) , ( 3 , 0 , 1 ) , ( 0 , 1 , 3 ) , ( 1 , 3 , 0 ) , {\displaystyle {\begin{aligned}(0,0,0),\quad (1,2,3),\quad (2,3,1),\quad (3,1,2),\\(2,2,2),\quad (3,0,1),\quad (0,1,3),\quad (1,3,0),\\\end{aligned}}}
and all the other points 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 the square root of two, 2 {\displaystyle {\sqrt {2}}} , as the points of each pair differ by one unit in two coordinates, and are the same in the third coordinate. The edges meet at 120° angles at each vertex, in a flat plane. All pairs of vertices that are non-adjacent are farther apart, at a distance of at least 6 {\displaystyle {\sqrt {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. It is possible to choose a larger set of one out of every four points of the integer lattice, so that the graph of distance- 2 {\displaystyle {\sqrt {2}}} pairs of this larger set forms two mirror-image copies of the Laves graph, disconnected from each other, with all other pairs of points farther than 2 {\displaystyle {\sqrt {2}}} apart.
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