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 . 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.[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.[8] 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.
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the Lavesgraph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three...
Society from 1956 to 1958. He is the namesake of Laves phases and the Laves tilings; the Lavesgraph, a highly-symmetrical three-dimensional crystal structure...
Australian Aboriginal languages. Laves phase, a particular class of intermetallic phases Lavesgraph, a periodic spatial graphLaves tiling, dual of an Archimedean...
ordered y-edge. Another (hypothetical) crystal with this property is the Lavesgraph (also called the K4 crystal, (10,3)-a, or the diamond twin). The compressive...
generated by the hexagonal close-packing The 17-sided Voronoi cells of the Lavesgraph Many other plesiohedra are known. Two different ones with the largest...
z\cos x=0} The gyroid structure is closely related to the K4 crystal (Laves' graph of girth ten). In nature, self-assembled gyroid structures are found...
edges or faces.) The infinite Lavesgraph has convex heptadecahedral Voronoi cells. Because of the symmetries of the graph, these heptadecahedra are plesiohedra...
pentagons, hexagons, and heptagons which can either be flat or tubular. The Lavesgraph or K4 crystal is a theoretically predicted three-dimensional crystalline...
as diaphragm Planar separator theorem, a theorem in graph theory Vertex separator, a notion in graph theory Geometric separator, a line that separates a...
automatically using graph drawing techniques. In some sources, the phrase "Hasse diagram" has a different meaning: the directed acyclic graph obtained from...
Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability...
joins points of equal value. It is a plane section of the three-dimensional graph of the function f ( x , y ) {\displaystyle f(x,y)} parallel to the ( x ...
monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases. There are five Platonic solids in three-dimensional...
lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted...
elections have provided the following results in Laval since the 1979 election. In the late 1970s and 1980s Laval replicated the global Quebec results, with...
multiple of four. The concept was proposed by Georg O. Brunner and Fritz Laves and later developed by Michael O'Keefe. The coordination sequences of many...
closure and transitive reduction are also used in the closely related area of graph theory. A relation R on a set X is transitive if, for all x, y, z in X,...
graph are derived from the axons interconnecting those areas. Thus connectomes are sometimes referred to as brain graphs, as they are indeed graphs in...
post. Jack Ketch 1663–1686 (London) Paskah Rose 1686 (Bleackley (1929) graphs his name as Pasha Rose; London) John Price 1714–1715 (London) William Marvell...
census. As can be seen by the graph, the majority of people, 44 people, are aged between 45 and 54. 22% percent of Magdalen Laver residents are 65 or over...
metrics. Voronoi diagrams of 20 points under two different metrics The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites)...
lattices". Approximate formula for site-bond percolation on a honeycomb lattice Laves lattices are the duals to the Archimedean lattices. Drawings from. See also...
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