1.
Lattice (group)
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In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point

2.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space

3.
Translation group
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time

4.
Lattice (order)
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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of an ordered set in which every two elements have a unique supremum and a unique infimum. An example is given by the numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities, since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras and these lattice-like structures all admit order-theoretic as well as algebraic descriptions. If is an ordered set, and S ⊆ L is an arbitrary subset. A set may have many upper bounds, or none at all, an upper bound u of S is said to be its least upper bound, or join, or supremum, if u ≤ x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one, dually, l ∈ L is said to be a lower bound of S if l ≤ s for each s ∈ S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, a set may have many lower bounds, or none at all, but can have at most one greatest lower bound. A partially ordered set is called a join-semilattice and a meet-semilattice if each two-element subset ⊆ L has a join and a meet, denoted by a ∨ b, is called a lattice if it is both a join- and a meet-semilattice. This definition makes ∨ and ∧ binary operations, both operations are monotone with respect to the order, a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2. It follows by an argument that every non-empty finite subset of a lattice has a least upper bound. With additional assumptions, further conclusions may be possible, see Completeness for more discussion of this subject, a bounded lattice is a lattice that additionally has a greatest element 1 and a least element 0, which satisfy 0 ≤ x ≤1 for every x in L. The greatest and least element is called the maximum and minimum, or the top and bottom element. A partially ordered set is a lattice if and only if every finite set of elements has a join. Taking B to be the empty set, ⋁ = ∨ = ∨0 = ⋁ A and ⋀ = ∧ = ∧1 = ⋀ A which is consistent with the fact that A ∪ ∅ = A. A lattice element y is said to another element x, if y > x. Here, y > x means x ≤ y and x ≠ y

5.
Bragg plane
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In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, K, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography. We know then that for constructive interference we have, | d | cos θ + | d | cos θ ′ = d ⋅ = m λ where m ∈ Z and we can set one of the scattering centres as the origin of an array. This reciprocal space plane is the Bragg plane, x-ray crystallography Reciprocal lattice Bravais lattice Powder diffraction Kikuchi line Brillouin zone

6.
Bravais lattice
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This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox

7.
Diamond cubic
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The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. Diamond cubic is in the Fd3m space group, which follows the face-centered cubic Bravais lattice, the diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by 1/4 of the width of the unit cell in each dimension. Many compound semiconductors such as gallium arsenide, β-silicon carbide, and indium antimonide adopt the analogous zincblende structure, zincblendes space group is F43m, but many of its structural properties are quite similar to the diamond structure. The atomic packing factor of the cubic structure is π√3/16 ≈0.34. Zincblende structures have higher packing factors than 0.34 depending on the sizes of their two component atoms. The first-, second-, third-, fourth- and fifth-nearest-neighbor distances in units of the lattice constant are √3/4, √2/2, √11/4,1 and √19/4. Mathematically, the points of the cubic structure can be given coordinates as a subset of a three-dimensional integer lattice by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates satisfying the equations x = y = z, and x + y + z =0 or 1. There are eight points that satisfy these conditions, All of the points in the structure may be obtained by adding multiples of four to the x, y. Adjacent points in this structure are at distance √3 apart in the integer lattice and this structure may be scaled to a cubical unit cell that is some number a of units across by multiplying all coordinates by a/4. Alternatively, each point of the cubic structure may be given by four-dimensional integer coordinates whose sum is either zero or one. Two points are adjacent in the structure if and only if their four-dimensional coordinates differ by one in a single coordinate. The total difference in values between any two points gives the number of edges in the shortest path between them in the diamond structure. These four-dimensional coordinates may be transformed into three-dimensional coordinates by the formula →, because the diamond structure forms a distance-preserving subset of the four-dimensional integer lattice, it is a partial cube. Yet another coordinatization of the diamond cubic involves the removal of some of the edges from a grid graph. The diamond cubic is sometimes called the lattice but it is not, mathematically. However, it is still a highly symmetric structure, any incident pair of a vertex, moreover the diamond crystal as a network in space has a strong isotropic property. Another crystal with this property is the Laves graph, the compressive strength and hardness of diamond and various other materials, such as boron nitride, is attributed to the diamond cubic structure

8.
Divisor summatory function
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In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the behaviour of the Riemann zeta function. The various studies of the behaviour of the function are sometimes called divisor problems. The divisor summatory function is defined as D = ∑ n ≤ x d = ∑ j, k j k ≤ x 1 where d = σ0 = ∑ j, k j k = n 1 is the divisor function. The divisor function counts the number of ways that the integer n can be written as a product of two integers. More generally, one defines D k = ∑ n ≤ x d k = ∑ m n ≤ x d k −1 where dk counts the number of ways that n can be written as a product of k numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a surface in k dimensions. Thus, for k=2, D = D2 counts the number of points on a lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis. Roughly, this shape may be envisioned as a hyperbolic simplex, finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations. The leading behaviour of the series is not difficult to obtain, peter Gustav Lejeune Dirichlet demonstrated that D = x log x + x + Δ where γ is the Euler–Mascheroni constant, and the non-leading term is Δ = O. Here, O denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the smallest value of θ for which Δ = O holds true, as of today, this problem remains unsolved. Many of the same work for this problem and for Gausss circle problem. Section F1 of Unsolved Problems in Number Theory surveys what is known, in 1904, G. Voronoi proved that the error term can be improved to O. In 1916, G. H. Hardy showed that inf θ ≥1 /4. In particular, he demonstrated that for some constant K, there exist values of x for which Δ > K x 1 /4, in 1922, J. van der Corput improved Dirichlets bound to inf θ ≤33 /100. In 1928, J. van der Corput proved that inf θ ≤27 /82, in 1950, Chih Tsung-tao and independently in 1953 H. E. Richert proved that inf θ ≤15 /46. In 1969, Grigori Kolesnik demonstrated that inf θ ≤12 /37, in 1973, Grigori Kolesnik demonstrated that inf θ ≤346 /1067. In 1982, Grigori Kolesnik demonstrated that inf θ ≤35 /108, in 1988, H. Iwaniec and C. J. Mozzochi proved that inf θ ≤7 /22

9.
Double lattice
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The orbit of any point under the action of a double lattice is a union of two Bravais lattices, related to each other by a point reflection. A double lattice in two dimensions is a p2 wallpaper group, in three dimensions, a double lattice is a space group of the type 1, as denoted by international notation. A packing that can be described as the orbit of a body under the action of a lattice is called a double lattice packing. In many cases the highest known packing density for a body is achieved by a double lattice, examples include the regular pentagon, heptagon, and nonagon and the equilateral triangular bipyramid. Włodzimierz Kuperberg and Greg Kuperberg showed that all convex planar bodies can pack at a density of at least √3/2 by use a double lattice

10.
Ehrhart polynomial
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In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a generalization of Picks theorem in the Euclidean plane. These polynomials are named after Eugène Ehrhart who studied them in the 1960s, informally, if P is a polytope, and tP is the polytope formed by expanding P by a factor of t in each dimension, then L is the number of integer lattice points in tP. More formally, consider a lattice L in Euclidean space ℝn, for any positive integer t, let tP be the t-fold dilation of P, and let L = # be the number of lattice points contained in the polytope tP. The Ehrhart polynomial of the interior of a convex polytope P can be computed as, L = d L. This result is known as Ehrhart–Macdonald reciprocity, let P be a d-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities, P =, then the t-fold dilation of P is a cube with side length t, containing d integer points. That is, the Ehrhart polynomial of the hypercube is L = d, additionally, if we evaluate L at negative integers, then L = d d = d L, as we would expect from Ehrhart–Macdonald reciprocity. Many other figurate numbers can be expressed as Ehrhart polynomials, let P be a rational polytope. In other words, suppose P =, where A ∈ ℝk × d and b ∈ ℤk, in this case, L is a quasi-polynomial in t. Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is, let P be a polygon with vertices, and. The number of points in tP will be counted by the quasi-polynomial L =7 t 24 +5 t 2 +7 + t 8. When P is a convex polytope, a0 =1. We can define a function for the Ehrhart polynomial of an integral n-dimensional polytope P as Ehr P = ∑ t ≥0 L z t. This series can be expressed as a rational function, additionally, Stanleys non-negativity theorem states that under the given hypotheses, h*i will be non-negative integers, for 0 ≤ j ≤ n. Another result by Stanley shows that if P is a polytope contained in Q. The h*-vector is in general not unimodal, but it is whenever it is symmetric, similar to the case of polytopes with integer vertices, one defines the Ehrhart series for a rational polytope. The case n = d =2 and t =1 of these statements yields Picks theorem, formulas for the other coefficients are much harder to get, Todd classes of toric varieties, the Riemann–Roch theorem as well as Fourier analysis have been used for this purpose

11.
Eisenstein integer
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The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers form a ring of algebraic integers in the algebraic number field Q — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the polynomial z 2 − z +. In particular, ω satisfies the equation ω2 + ω +1 =0, the product of two Eisenstein integers a + b ω and c + d ω is given explicitly by ⋅ = + ω. The norm of an Eisenstein integer is just the square of its modulus and is given by | a + b ω |2 = a 2 − a b + b 2, thus the norm of an Eisenstein integer is always an ordinary integer. Since 4 a 2 −4 a b +4 b 2 =2 +3 b 2, the group of units in the ring of Eisenstein integers is the cyclic group formed by the sixth roots of unity in the complex plane. Specifically, they are These are just the Eisenstein integers of norm one, if x and y are Eisenstein integers, we say that x divides y if there is some Eisenstein integer z such that y = zx. This extends the notion of divisibility for ordinary integers, ordinary primes congruent to 2 mod 3 cannot be factored in this way and they are primes in the Eisenstein integers as well. Every Eisenstein integer a + bω whose norm a2 − ab + b2 is a prime is an Eisenstein prime. In fact, every Eisenstein prime is of form, or is a product of a unit. The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by N = a 2 − a b + b 2. This can be derived as follows, N = | a + b ω |2 = = a 2 + a b + b 2 = a 2 − a b + b 2. The quotient of the complex plane C by the lattice containing all Eisenstein integers is a torus of real dimension 2. This is one of two tori with maximal symmetry among all such complex tori and this torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. Gaussian integer Kummer ring Systolic geometry Hermite constant Cubic reciprocity Loewners torus inequality Hurwitz quaternion Quadratic integer Eisenstein Integer--from MathWorld

12.
Euclid's orchard
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In mathematics, informally speaking, Euclids orchard is an array of one-dimensional trees of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclids orchard is the set of segments from to. The trees visible from the origin are those at lattice points, the name Euclids orchard is derived from the Euclidean algorithm. If the orchard is projected relative to the origin onto the plane x + y =1 the tops of the form a graph of Thomaes function. Opaque forest problem Euclids Orchard, Grade 9-11 activities and problem sheet, Texas Instruments Inc. Project Euler related problem

13.
Fokker periodicity block
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Fokker periodicity blocks are a concept in tuning theory used to mathematically relate musical intervals in just intonation to those in equal tuning. They are named after Adriaan Daniël Fokker and these are included as the primary subset of what Erv Wilson refers to as constant structures, where each interval occurs always subtended by the same number of steps. The basic idea of Fokkers periodicity blocks is to represent just ratios as points on a lattice and this zero-dimensional set of pitches is a periodicity block. In other words, all pitches and intervals can be considered as residues modulo octave and this simplification is commonly known as octave equivalence. Typically, n ranges from one to three, in the two-dimensional case, the lattice is a square lattice. In the 3-D case, the lattice is cubic, the lattice points correspond to the integers, with the point at position x being labeled with the pitch value 3x/2y for a number y chosen to make the resulting value lie in the range from 1 to 2. Thus, A =1, and surrounding it are the values, 128/81, 32/27, 16/9, 4/3,1, 3/2, 9/8, 27/16, 81/64. In the two-dimensional case, corresponding to 5-limit just tuning, the defining the lattice are a perfect fifth. This gives a square lattice in which the point at position is labeled with the value 3x5y2z, again, z is chosen to be the unique integer that makes the resulting value lie in the interval [1, 2). Once the lattice and its labeling is fixed, one chooses n nodes of the other than the origin whose values are close to either 1 or 2. The vectors from the origin to one of these special nodes are called unison vectors. These domains form the tiles in a tessellation of the original lattice, each tile is called a Fokker periodicity block. The area of each block is always a number equal to the number of nodes falling within each block. Example 1, Take the 2-dimensional lattice of perfect fifths and just major thirds, choose the commas 128/125 and 81/80. The result is a block of twelve, showing how twelve-tone equal temperament approximates the ratios of the 5-limit, the periodicity blocks form a secondary, oblique lattice, superimposed on the first one. e. Unison vectors and periodicity blocks in the three-dimensional harmonic lattice of notes, koninklijke Nederlandsche Akademie van Wetenschappen, B72. Paul Erlich, A Gentle Introduction to Fokker Periodicity Blocks, Part 1, Part 2, etc

14.
Fundamental pair of periods
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In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the object with which elliptic functions. Although the concept of a lattice is quite simple, there is a considerable amount of specialized notation. This article attempts to review this notation, as well as to present some theorems that are specific to the two-dimensional case, the fundamental pair of periods is a pair of complex numbers ω1, ω2 ∈ C such that their ratio ω2/ω1 is not real. In other words, considered as vectors in R2, the two are not collinear, the lattice generated by ω1 and ω2 is Λ = This lattice is also sometimes denoted as Λ to make clear that it depends on ω1 and ω2. It is also denoted by Ω or Ω, or simply by 〈ω1. The two generators ω1 and ω2 are called the lattice basis, the parallelogram defined by the vertices 0, ω1 and ω2 is called the fundamental parallelogram. It is important to note that, while a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair, a number of properties, listed below, obtain. Two pairs of numbers and are called equivalent if they generate the same lattice. The fundamental parallelogram contains no further points in its interior or boundary. Conversely, any pair of points with this property constitute a fundamental pair. Note that this belongs to the matrix group S L. This equivalence of lattices can be thought of as underlying many of the properties of elliptic functions, the abelian group Z2 maps the complex plane into the fundamental parallelogram. That is, every point z ∈ C can be written as z = p + m ω1 + n ω2 for integers m, n, with a point p in the fundamental parallelogram. Since this mapping identifies opposite sides of the parallelogram as being the same, equivalently, one says that the quotient manifold C / Λ is a torus. Define τ = ω2/ω1 to be the half-period ratio, then the lattice basis can always be chosen so that τ lies in a special region, called the fundamental domain. Alternately, there exists an element of PSL that maps a lattice basis to another basis so that τ lies in the fundamental domain. The fundamental domain is given by the set D, which is composed of a set U plus a part of the boundary of U, U =, where H is the upper half-plane

15.
Gaussian integer
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with addition and multiplication of complex numbers, form an integral domain. This integral domain is a case of a commutative ring of quadratic integers. It does not have an ordering that respects arithmetic. Formally, the Gaussian integers are the set Z =, where i 2 = −1, note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice. The norm of a Gaussian integer is the square of its value as a complex number. It is the natural number defined as N = a 2 + b 2 = ¯ =, the norm is multiplicative, since the absolute value of complex numbers is multiplicative, i. e. one has N = N N. The latter can also be verified by a straightforward check, the units of Z are precisely those elements with norm 1, i. e. the set. The Gaussian integers form a principal ideal domain with units, for x ∈ Z, the four numbers ±x, ±ix are called the associates of x. As for every principal ideal domain, Z is also a unique factorization domain and it follows that a Gaussian integer is prime if and only if it is irreducible. The prime elements of Z are also known as Gaussian primes, an associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes, the positive integer Gaussian primes are the prime numbers that are congruent to 3 modulo 4. One should not refer to only these numbers as the Gaussian primes, which refers to all the Gaussian primes, many of which do not lie in Z. In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, for example,5 = · and 13 = ·. If p =2, we have 2 = = i2, the ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q consisting of the complex numbers whose real and imaginary part are both rational. It is easy to see graphically that every number is no farther than a distance of 22 from some Gaussian integer. Put another way, every number has a maximal distance of 22 N units to some multiple of z, where z is any Gaussian integer, this turns Z into a Euclidean domain. The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his monograph on quartic reciprocity

16.
Hexagonal lattice
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The hexagonal lattice or triangular lattice is one of the five 2D lattice types. Three nearby points form an equilateral triangle, in images, four orientations of such a triangle are by far the most common. Two orientations of an image of the lattice are by far the most common and they can conveniently be referred to as hexagonal lattice with horizontal rows, with triangles pointing up and down, and hexagonal lattice with vertical rows, with triangles pointing left and right. They differ by an angle of 30°, the hexagonal lattice with horizontal rows is a special case of a centered rectangular grid, with rectangles which are √3 times as high as wide. Of course for the orientation the rectangles are √3 times as wide as high. Its symmetry category is wallpaper group p6m, a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. For an image of a structure, again two orientations are by far the most common. They differ by an angle of 90°, or equivalently 30°, the term honeycomb lattice could mean a corresponding hexagonal lattice, or a structure which is not a lattice in the group sense, but e. g. one in the sense of a lattice model. A set of points forming the vertices of a honeycomb shows the honeycomb structure and it can be seen as the union of two offset triangular lattices, shown here red and blue. A triangular lattice itself can be divided into 3 offset triangular lattices, shown above in red, green, a triangular lattice is also called an A2 lattice, A2, and the union of three triangular lattices is A*2. In addition to points, or instead of them, the sides of the hexagons may be shown. The set of points can be partitioned into three sets with these larger translation distances. Within each set of directions the directions differ by an angle of 60°, for a hexagonal lattice with horizontal rows one of the three directions is horizontal, and for a hexagonal lattice with vertical rows one of the three directions is vertical. Conversely, for a given lattice we can create a lattice that is √3 times as fine by adding the centers of the equilateral triangles. Since there are twice as many triangles as vertices, this triples the number of vertices, a pattern with 3- or 6-fold rotational symmetry has a lattice of 3-fold rotocenters that is this finer lattice relative to the lattice of translational symmetry. For reflection axes, there are two sets of directions, mentioned above. If there are reflection axes in the main directions, one of the three sets of rotocenters play a different role than the other two, these reflection axes pass through them. With p6 one set is special because of being 6-fold, square lattice hexagonal tiling close-packing centered hexagonal number Eisenstein integer Voronoi diagram Loewners torus Born, M

17.
Lattice reduction
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In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice, one measure of nearly orthogonal is the orthogonality defect. This compares the product of the lengths of the vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same, any particular basis of n vectors may be represented by a matrix B, whose columns are the basis vectors b i, i =1, …, n. If the number of vectors is less than the dimension of the underlying space, for a given lattice Λ, this volume is the same for any basis, and hence is referred to as the determinant of the lattice det or lattice constant d. If the lattice reduction problem is defined as finding the basis with the smallest possible defect, then the problem is NP complete. However, there exist polynomial time algorithms to find a basis with defect δ ≤ c where c is some constant depending only on the number of basis vectors and this is a good enough solution in many practical applications. For a basis consisting of just two vectors, there is a simple and efficient method of reduction closely analogous to the Euclidean algorithm for the greatest common divisor of two integers. As with the Euclidean algorithm, the method is iterative, at each step the larger of the two vectors is reduced by adding or subtracting a multiple of the smaller vector. Lattice reduction algorithms are used in a number of modern number theoretical applications, although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm can find a short basis in polynomial time with guaranteed worst-case performance. LLL is widely used in the cryptanalysis of public key cryptosystems, the LLL algorithm for computing a nearly-orthogonal basis was used to show that integer programming in any fixed dimension can be done in polynomial time. The following algorithms reduce lattice bases and they can be compared in terms of runtime and approximation to an optimal solution, always relative to the dimension of the given lattice. If there are implementations of these algorithms this should also be noted here

18.
Niemeier lattice
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In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Hans-Volker Niemeier. Venkov gave a proof of the classification. Witt has a mentioning that he found more than 10 such lattices. One example of a Niemeier lattice is the Leech lattice, Niemeier lattices are usually labeled by the Dynkin diagram of their root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number, there are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams. The complete list of Niemeier lattices is given in the following table and it is the square root of the discriminant of the root lattice. G0×G1×G2 is the order of the group of the lattice G∞×G1×G2 is the order of the automorphism group of the corresponding deep hole. If L is an odd unimodular lattice of dimension 8n and M its sublattice of even vectors, then M is contained in exactly 3 unimodular lattices, one of which is L and the other two of which are even. There may be several lines between the pair of vertices, and there may be lines from a vertex to itself. Kneser proved that this graph is always connected, the number on the right is the Coxeter number of the Niemeier lattice. In 32 dimensions the neighborhood graph has more than a billion vertices, some of the Niemeier lattices are related to sporadic simple groups. The Leech lattice is acted on by a cover of the Conway group. The Niemeier lattices, other than the Leech lattice, correspond to the holes of the Leech lattice. Niemeier lattices also correspond to the 24 orbits of primitive norm zero vectors w of the even unimodular Lorentzian lattice II25,1, where the Niemeier lattice corresponding to w is w⊥/w. Chenevier, Gaëtan, Lannes, Jean, Formes automorphes et voisins de Kneser des réseaux de Niemeier, arXiv,1409.7616 Conway, J. H. Sloane, N. J. A. CS1 maint, Multiple names, authors list Ebeling, Wolfgang, Lattices and codes, Advanced Lectures in Mathematics, Braunschweig, vieweg & Sohn, ISBN 978-3-528-16497-3, MR1938666 Niemeier, Hans-Volker. Definite quadratische Formen der Dimension 24 und Diskriminate 1, on the classification of integral even unimodular 24-dimensional quadratic forms, Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik. Trudy Matematicheskogo Instituta imeni V. A. 1007/BF02940750, MR0005508 Witt, Ernst, gesammelte Abhandlungen, Berlin, New York, Springer-Verlag, ISBN 978-3-540-57061-5, MR1643949 Aachen University lattice catalogue

19.
No-three-in-line problem
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In mathematics, in the area of discrete geometry, the no-three-in-line problem asks for the maximum number of points that can be placed in the n × n grid so that no three points are collinear. This number is at most 2n, since if 2n +1 points are placed in the grid, then by the pigeonhole principle some row, the problem was introduced by Henry Dudeney in 1917. Paul Erdős observed that, when n is a prime number, when n is not prime, one can perform this construction for a p × p grid contained in the n × n grid, where p is the largest prime that is at most n. As a consequence, for any ε and any sufficiently large n, again, for arbitrary n one can perform this construction for a prime near n/2 to obtain a solution with n points. Guy & Kelly conjectured that for large n one cannot do better than c n with c =2 π233 ≈1.874. Pegg, Jr. noted that Gabor Ellmann found, in March 2004, an error in the paper of Guy and Kellys heuristic reasoning. The Heilbronn triangle problem asks for the placement of n points in a square that maximizes the area of the smallest triangle formed by three of the points. By applying Erdős construction of a set of points with no three collinear points, one can find a placement in which the smallest triangle has area 1 − ϵ2 n 2. Non-collinear sets of points in the grid were considered by Pór & Wood. They proved that the number of points in the n × n × n grid with no three points collinear is Θ. Similarly to Erdőss 2D construction, this can be accomplished by using points mod p, another analogue in higher dimensions is to find sets of points that do not all lie in the same plane. As of 2015, it is not known what the solution is for 6x6x6 grid. Similar to the 2n upper bound for the 2D case, there exists a 3n upper bound for the 3D case, though as seen above, the cap set problem concerns a similar problem in high-dimensional vector spaces over finite fields. A noncollinear placement of n points can also be interpreted as a drawing of the complete graph in such a way that, although edges cross. Erdős construction above can be generalized to show that every n-vertex k-colorable graph has such a drawing in a O × O grid, one can also consider graph drawings in the three-dimensional grid. Here the non-collinearity condition means that a vertex should not lie on a non-adjacent edge, for n ≤46, it is known that 2n points may be placed with no three in a line. The numbers of solutions for small n =2,3. are 1,1,4,5,11,22,57,51,156,158,566,499,1366. A puzzle with pawns, Amusements in Mathematics, Edinburgh, Nelson, Journal of Graph Algorithms and Applications

20.
Pick's theorem
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In the example shown, we have i =7 interior points and b =8 boundary points, so the area is A =7 + 8/2 −1 =7 +4 −1 =10 square units. Note that the theorem as stated above is valid for simple polygons, i. e. ones that consist of a single piece. For a polygon that has h holes, with a boundary in the form of h +1 simple closed curves, the result was first described by Georg Alexander Pick in 1899. The Reeve tetrahedron shows that there is no analogue of Picks theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points, however, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra, consider a polygon P and a triangle T, with one edge in common with P. Assume Picks theorem is true for both P and T separately, we want to show that it is true for the polygon PT obtained by adding T to P. Since P and T share an edge, all the points along the edge in common are merged to interior points. So, calling the number of points in common c. From the above follows i P + i T = i P T − and b P + b T = b P T +2 +2. Therefore, if the theorem is true for polygons constructed from n triangles, for general polytopes, it is well known that they can always be triangulated. That this is true in dimension 2 is an easy fact, to finish the proof by mathematical induction, it remains to show that the theorem is true for triangles. Integer points in convex polyhedra Picks Theorem at cut-the-knot Picks Theorem Picks Theorem proof by Tom Davis Picks Theorem by Ed Pegg, Jr. the Wolfram Demonstrations Project

21.
Reciprocal lattice
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In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, this first lattice is usually a periodic spatial function in real-space and is known as the direct lattice. The reciprocal lattice plays a role in most analytic studies of periodic structures. In neutron and X-ray diffraction due to the Laue conditions the momentum difference between incoming and diffracted X-rays of a crystal is a lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice, using this process, one can infer the atomic arrangement of a crystal. The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice, assuming an ideal Bravais lattice R n = n 1 ⋅ a 1 + n 2 ⋅ a 2 where n 1, n 2 ∈ Z. Any quantity, e. g. the electronic density in a crystal can be written as a periodic function f = f Due to the periodicity it is useful to write it in Fourier expansions. Mathematically, we can describe the lattice as the set of all vectors G m that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the lattice is the original lattice. Using column vector representation of vectors, the formulae above can be rewritten using matrix inversion. This method appeals to the definition, and allows generalization to arbitrary dimensions, the cross product formula dominates introductory materials on crystallography. The above definition is called the physics definition, as the factor of 2 π comes naturally from the study of periodic structures. The crystallographers definition has the advantage that the definition of b 1 is just the reciprocal magnitude of a 1 in the direction of a 2 × a 3 and this can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice. The direction of the lattice vector corresponds to the normal to the real space planes. The magnitude of the lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. Reciprocal lattices for the crystal system are as follows. The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a primitive cell of side 2 π a

22.
Regular grid
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A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes. Grids of this appear on graph paper and may be used in finite element analysis as well as finite volume methods. Since the derivatives of field variables can be expressed as finite differences. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element, a Cartesian grid is a special case where the elements are unit squares or unit cubes, and the vertices are integer points. A rectilinear grid is a tessellation by rectangles or parallelepipeds that are not, in general, the cells may still be indexed by integers as above, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a grid that is not regular appears on logarithmic scale graph paper. Cartesian coordinate system Integer point Unstructured grid

23.
Square lattice
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In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the version of the integer lattice, denoted as Z2. It is one of the five types of two-dimensional lattices as classified by their groups, its symmetry group in IUC notation as p4m, Coxeter notation as. Two orientations of an image of the lattice are by far the most common and they can conveniently be referred to as the upright square lattice and diagonal square lattice, the latter is also called the centered square lattice. They differ by an angle of 45° and this is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard. The square lattices symmetry category is wallpaper group p4m, a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a square lattice with a mesh size that is √2 times as large. Correspondingly, after adding the centers of the squares of a square lattice we have a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice. A pattern with 4-fold rotational symmetry has a lattice of 4-fold rotocenters that is a factor √2 finer. With respect to reflection axes there are three possibilities, None, centered square number Euclids orchard Gaussian integer Hexagonal lattice Quincunx Square tiling