Laves graph
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.
It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain.
H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932.
[1][2] It has also been called the K4 crystal,[3] (10,3)-a network,[4] diamond twin,[5] triamond,[6][7] and the srs net.
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.
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.
The edges of the Laves graph connect pairs of points whose Euclidean distance from each other is the square root of two,
The edges meet at 120° angles at each vertex, in a flat plane.
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.
[1] 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-
, the construction produces the (abstract) Laves graph, but does not give it the same geometric layout.
Then, fix the set of vertices of the covering graph to be the ordered pairs
The resulting graph is independent of the chosen spanning tree, and the same construction can also be interpreted more abstractly using homology.
However, its vertices are in different positions than the more-symmetric, conventional geometric construction.
[12] Another subgraph of the simple cubic net isomorphic to the Laves graph is obtained by removing half of the edges in a certain way.
The resulting structure, called semi-simple cubic lattice, also has lower symmetry than the Laves graph itself.
However, the overall structure is chiral: no sequence of translations and rotations can make it coincide with its mirror image.
[10][1][9] The numbers of vertices at distance 0, 1, 2, ... from any vertex (forming the coordination sequence of the Laves graph) are:[14] If the surrounding space is partitioned into the regions nearest each vertex—the cells of the Voronoi diagram of this structure—these form heptadecahedra with 17 faces each.
Experimenting with the structures formed by these polyhedra led physicist Alan Schoen to discover the gyroid minimal surface,[15] which is topologically equivalent to the surface obtained by thickening the edges of the Laves graph to cylinders and taking the boundary of their union.
[16] The Laves graph is the unique shortest triply-periodic network, in the following sense.
Triply-periodic means repeating infinitely in all three dimensions of space, so a triply-periodic network is a connected geometric graph with a three-dimensional lattice of translational symmetries.
A fundamental domain is any shape that can tile space with its translated copies under these symmetries.
Any lattice has infinitely many choices of fundamental domain, of varying shapes, but they all have the same volume
One can also measure the length of the edges of the network within a single copy of the fundamental domain; call this number
[17] A sculpture titled Bamboozle, by Jacobus Verhoeff and his son Tom Verhoeff, is in the form of a fragment of the Laves graph, with its vertices represented by multicolored interlocking acrylic triangles.
[19] The Laves graph may also give a crystal structure for boron, one which computations predict should be stable.
[20] Other chemicals that may form this structure include SrSi2 (from which the "srs net" name derives)[8] and elemental nitrogen,[9][20] as well as certain metal–organic frameworks[21] and cyclic hydrocarbons.
[23][24] The structure of the Laves graph, and of gyroid surfaces derived from it, has also been observed experimentally in soap-water systems, and in the chitin networks of butterfly wing scales.
