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A cubic lattice is a lattice whose points lie at positions in the Cartesian three-space, where , , and are integers.The term is also used to refer to a regular arrangement of spheres whose unit cell form a cube. There are three types of cubic lattices, corresponding to three types of cubic close packing: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). The illustrations above show the Brillouin zones for these types of lattice.

The theta series of a lattice is the generating function for the number of vectors with norm in the lattice.Theta series for a number of lattices are implemented in the Wolfram Language as LatticeData[lattice, "ThetaSeriesFunction"].The following table summarized lattice with closed-form theta series. Here, is a Jacobi theta function.latticetheta series generating functionBarnes-Wall latticebody-centered cubic latticeCoxeter-Todd latticeface-centered cubic latticehexagonal close packing latticehexagonal latticeLeech latticesimple cubic latticesquare latticetetrahedral packing latticeThe following tables gives the first few terms of the series for these lattices.latticeOEIStheta seriesBarnes-Wall latticeA008409body-centered cubic latticeA004013Coxeter-Todd latticeA004010face-centered cubic latticeA004015hexagonal close packing latticehexagonal latticeLeech latticeA008408simple..

Clean tile is a game investigated by Buffon (1777) in which players bet on the number of different tiles a thrown coin will partially cover on a floor that is regularly tiled. Buffon investigated the probabilities on a triangular grid, square grid, hexagonal grid, and grid composed of rhombi. Assume that the side length of the tile is greater than the coin diameter . Then, on a square grid, it is possible for a coin to land so that it partially covers 1, 2, 3, or 4 tiles. On a triangular grid, it can land on 1, 2, 3, 4, or 6 tiles. On a hexagonal grid, it can land on 1, 2, or 3 tiles.Special cases of this game give the Buffon-Laplace needle problem (for a square grid) and Buffon's needle problem (for infinite equally spaced parallel lines).As shown in the figure above, on a square grid with tile edge length , the probability that a coin of diameter will lie entirely on a single tile (indicated by yellow disks in the figure) is given by(1)since the shortening of the side..

For every positive integer , there exists a circle which contains exactly lattice points in its interior. H. Steinhaus proved that for every positive integer , there exists a circle of area which contains exactly lattice points in its interior.Schinzel's theorem shows that for every positive integer , there exists a circle in the plane having exactly lattice points on its circumference. The theorem also explicitly identifies such "Schinzel circles" as(1)Note, however, that these solutions do not necessarily have the smallest possible radius. For example, while the Schinzel circle centered at (1/3, 0) and with radius 625/3 has nine lattice points on its circumference, so does the circle centered at (1/3, 0) with radius 65/3.Let be the smallest integer radius of a circle centered at the origin (0, 0) with lattice points. In order to find the number of lattice points of the circle, it is only necessary to find the number in the..

A point lattice is a regularly spaced array of points.In the plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, etc. Unless otherwise specified, point lattices may be taken to refer to points in a square array, i.e., points with coordinates , where , , ... are integers. Such an array is often called a grid or a mesh.Point lattices are frequently simply called "lattices," which unfortunately conflicts with the same term applied to ordered sets treated in lattice theory. Every "point lattice" is a lattice under the ordering inherited from the plane, although a point lattice may not be a sublattice of the plane, since the infimum operation in the plane need not agree with the infimum operation in the point lattice. On the other hand, many lattices are not point lattices.Properties of lattice are implemented in the Wolfram Language as LatticeData[lattice, prop].Formally,..

Buffon's needle problem asks to find the probability that a needle of length will land on a line, given a floor with equally spaced parallel lines a distance apart. The problem was first posed by the French naturalist Buffon in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104).Define the size parameter by(1)For a short needle (i.e., one shorter than the distance between two lines, so that ), the probability that the needle falls on a line is(2)(3)(4)(5)For , this therefore becomes(6)(OEIS A060294).For a long needle (i.e., one longer than the distance between two lines so that ), the probability that it intersects at least one line is the slightly more complicated expression(7)where (Uspensky 1937, pp. 252 and 258; Kunkel).Writing(8)then gives the plot illustrated above. The above can be derived by noting that(9)where(10)(11)are the probability functions for the..

For every positive integer , there exists a square in the plane with exactly lattice points in its interior. This was extended by Schinzel and Kulikowski to all plane figures of a given shape. The generalization of the square in two-dimensional to the cube in three dimensions was also proved by Browkin.

A tree is planted at each lattice point in a circular orchard which has center at the origin and radius . If the radius of trees exceeds units, one is unable to see out of the orchard in any direction. However, if the radii of the trees are , one can see out at certain angles.

A regularly spaced array of points in a square array, i.e., points with coordinates , where , , ... are integers. Such an array is often called a grid or mesh, and is a special case of a point lattice.The fraction of lattice points visible from the origin, as derived in Castellanos (1988, pp. 155-156), is(1)(2)(3)Therefore, this is also the probability that two randomly picked integers will berelatively prime to one another.The number of the lattice points which can be picked with no four concyclic is (Guy 1994, p. 241).Any parallelogram on the lattice in which two oppositesides each have length 1 has unit area (Hilbert and Cohn-Vossen 1999, pp. 33-34).A special set of polygons defined on the regular lattice are the golygons. A necessary and sufficient condition that a linear transformation transforms a lattice to itself is that it be unimodular. M. Ajtai has shown that there is no efficient algorithm for finding any fraction..

Given a convex plane region with area and perimeter ,where is the number of enclosed lattice points (Nosarzewska 1948). This improves on Jarnick's inequality

A grid usually refers to two or more infinite sets of evenly-spaced parallel lines at particular angles to each other in a plane, or the intersections of such lines.The two most common types of grid are orthogonal grids, with two sets of lines perpendicular to each other (such as the square grid), and isometric grids, with three sets of lines at 60-degree angles to each other (such as the triangular grid). It should be noted that in most grids with three or more sets of lines, every intersection includes one element of each set.There are other types of planar grids, like hexagonal grids, which are formed by tessellating regular hexagons in the plane. These are often found in strategy and role-playing games because of the lack of single points of contact characteristic of isometric and orthogonal grids. The collection of cells created by a grid is often called a "board" when these cells are used as resting places for pieces in a game.Grids can..

Two lattice points and are mutually visible if the line segment joining them contains no further lattice points. This corresponds to the requirement that , where denotes the greatest common divisor. The plots above show the first few points visible from the origin.If a lattice point is selected at random in two dimensions, the probability that it is visible from the origin is . This is also the probability that two integers picked at random are relatively prime. If a lattice point is picked at random in dimensions, the probability that it is visible from the origin is , where is the Riemann zeta function.An invisible figure is a polygon all of whose vertices (with possibly degenerate edges when restricted on a grid) are invisible from the origin. There are invisible sets of every finite shape. The lower left-hand corner of the invisible squares on a square grid with having smallest -coordinate and side lengths 1 and 2 are (20, 14) and (54, 20), respectively...

A bounded plane convex region symmetric about a lattice point and with area must contain at least three lattice points in the interior. In dimensions, the theorem can be generalized to a region with area , which must contain at least three lattice points. The theorem can be derived from Blichfeldt's theorem.

Count the number of lattice points inside the boundary of a circle of radius with center at the origin. The exact solution is given by the sum(1)(2)(3)(Hilbert and Cohn-Vossen 1999, p. 39). The first few values for , 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (OEIS A000328).The series for is intimately connected with the sum of squares function (i.e., the number of representations of by two squares), since(4)(Hardy 1999, p. 67). is also closely connected with the Leibniz series since(5)where is a Lerch transcendent and is a digamma function, so taking the limit gives(6)(Hilbert and Cohn-Vossen 1999, p. 39).Gauss showed that(7)where(8)(Hardy 1999, p. 67).The first few values of are 5, 13/4, 29/9, 49/16, 81/25, 113/36, 149/49, 197/64, 253/81, 317/100, 377/121, 49/16, ... (OEIS A000328 and A093837). As can be seen in the plot above, the values of such that are , 3, 4, 6, 11, 16, 21, 36, 52, 53, 86, 101, ... (OEIS A093832).Writing..

Let be a set of simple polygonal obstacles in the plane, then the nodes of the visibility graph of are just the vertices of , and there is an edge (called a visibility edge) between vertices and if these vertices are mutually visible.

When a closed interval is partitioned by points , the lengths of the resulting intervals between the points are denoted , , ..., , and the value is called the mesh size of the partition.

A point lattice which can be constructed from an arbitrary parallelogram of unit area. For any such planar lattice, the minimum distance between any two points is a quantity characteristic of the lattice. This distance satisfies(Hilbert and Cohn-Vossen 1999, p. 36). For a lattice in three dimensions,(Hilbert and Cohn-Vossen 1999, p. 45).

A 24-dimensional Euclidean lattice. An automorphism of the Leech lattice modulo a center of two leads to the Conway group . Stabilization of the one- and two-dimensional sublattices leads to the Conway groups and , the Higman-Sims group HS and the McLaughlin group McL.The Leech lattice appears to be the densest hypersphere packing in 24 dimensions, and results in each hypersphere touching others. The number of vectors with norm in the Leech lattice is given by(1)where is the divisor function giving the sum of the 11th powers of the divisors of and is the tau function (Conway and Sloane 1993, p. 135). The first few values for , 2, ... are 0, 196560, 16773120, 398034000, ... (OEIS A008408). This is an immediate consequence of the theta function for Leech's lattice being a weight 12 modular form and having no vectors of norm two. has the theta series(2)(3)(4)(5)where is the Eisenstein series, which is the theta series of the lattice (OEIS A004009),..

Let denote an integral convex polytope of dimension in a lattice , and let denote the number of lattice points in dilated by a factor of the integer ,(1)for . Then is a polynomial function in of degree with rational coefficients(2)called the Ehrhart polynomial (Ehrhart 1967, Pommersheim 1993). Specific coefficients have important geometric interpretations. 1. is the content of . 2. is half the sum of the contents of the -dimensional faces of . 3. . Let denote the sum of the lattice lengths of the edges of , then the case corresponds to Pick's theorem,(3)Let denote the sum of the lattice volumes of the two-dimensional faces of , then the case gives(4)where a rather complicated expression is given by Pommersheim (1993), since can unfortunately not be interpreted in terms of the edges of . The Ehrhart polynomial of the tetrahedron with vertices at (0, 0, 0), (, 0, 0), (0, , 0), (0, 0, ) is(5)where is a Dedekind sum, , , (here, GCD is the greatest common divisor),..

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