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Square grid graph Triangular grid graph. In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense.
2 lattice as three triangular tilings: + + The vertex arrangement of the triangular tiling is called an A 2 lattice. [2] It is the 2-dimensional case of a simplectic honeycomb. The A * 2 lattice (also called A 3 2) can be constructed by the union of all three A 2 lattices, and equivalent to the A 2 lattice. + + = dual of =
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. [1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices. In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb ...
Equivalence in the sense of generating the same lattice is represented by the modular group: : + represents choosing a different third point in the same grid, : / represents choosing a different side of the triangle as reference side 0–1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in ...
This page was last edited on 20 April 2006, at 16:55 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may ...
A 2D lattice gives an irregular honeycomb tessellation, with equal hexagons with point symmetry; in the case of a regular triangular lattice it is regular; in the case of a rectangular lattice the hexagons reduce to rectangles in rows and columns; a square lattice gives the regular tessellation of squares; note that the rectangles and the ...
The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice.
Therefore, each triangle has area , as needed for the proof. [5] A different proof that these triangles have area is based on the use of Minkowski's theorem on lattice points in symmetric convex sets. [10] Subdivision of a grid polygon into special triangles