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
\sqrt2
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.
As 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 pointsand 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,
\sqrt{2}
\sqrt{6}
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-
\sqrt{2}
\sqrt{2}
K4
K4
K4
K4
A maximal abelian covering graph can be constructed from any finite graph
G
K4
G
d
Zd
(v,w)
v
G
w
Zd
uv
v
G
(v,w)
(u,w\plusmn\epsilon)
\epsilon
uv
uv
Using the same construction, the hexagonal tiling of the plane is the maximal abelian covering graph of the three-edge dipole graph, and the diamond cubic is the maximal abelian covering graph of the four-edge dipole. The
d
d
The unit distance graph on the three-dimensional integer lattice has a vertex for each lattice point; each vertex has exactly six neighbors. It is possible to remove some of the points from the lattice, so that each remaining point has exactly three remaining neighbors, and so that the induced subgraph of these points has no cycles shorter than ten edges. There are four ways to do this, one of which is isomorphic as an abstract graph to the Laves graph. However, its vertices are in different positions than the more-symmetric, conventional geometric construction.
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.
The Laves graph is a cubic graph, meaning that there are exactly three edges at each vertex. Every pair of a vertex and adjacent edge can be transformed into every other such pair by a symmetry of the graph, so it is a symmetric graph. More strongly, for every two vertices
u
v
u
v
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The girth of this structure is 10—the shortest cycles in the graph have 10 vertices—and 15 of these cycles pass through each vertex. The numbers of vertices at distance 0, 1, 2, ... from any vertex (forming the coordination sequence of the Laves graph) are:
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. They are plesiohedra, polyhedra that tile space isohedrally. Experimenting with the structures formed by these polyhedra led physicist Alan Schoen to discover the gyroid minimal surface, 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.
V
L
V
L
6\sqrt2
L3/V
27/\sqrt2
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. It was installed in 2013 at the Eindhoven University of Technology.
The Laves graph has been suggested as an allotrope of carbon, analogous to the more common graphene and graphite carbon structure which also have three bonds per atom at 120° angles. In graphene, adjacent atoms have the same bonding planes as each other, whereas in the Laves graph structure the bonding planes of adjacent atoms are twisted by an angle of approximately 70.5° around the line of the bond. However, this hypothetical carbon allotrope turns out to be unstable.
The Laves graph may also give a crystal structure for boron, one which computations predict should be stable. Other chemicals that may form this structure include SrSi2 (from which the "srs net" name derives) and elemental nitrogen, as well as certain metal–organic frameworks and cyclic hydrocarbons.
The electronic band structure for the tight-binding model of the Laves graph has been studied, showing the existence of Dirac and Weyl points in this structure.
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.