mathematics, the n-dimensional integer lattice (or cubic lattice), denoted ⁠ Z n {\displaystyle \mathbb {Z} ^{n}} ⁠, is the lattice in the Euclidean space ⁠...

combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed...

)n} with an integer n {\displaystyle n} ) at every direct lattice vertex. One heuristic approach to constructing the reciprocal lattice in three dimensions...

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...

the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as ⁠...

based on the integer lattice, hexagonal tiling and E8 lattice, respectively. It has no root system and in fact is the first unimodular lattice with no roots...

the Gaussian integers constitute the 2-dimensional integer lattice. The conjugate of a Gaussian integer a + bi is the Gaussian integer a – bi. The norm...

{\sqrt {3}}} ⁠ apart in the integer lattice; the edges of the diamond structure lie along the body diagonals of the integer grid cubes. This structure...

uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group...

factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical...

Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers...

\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle d\leq...

includes at least ⌈ A ⌉ {\displaystyle \lceil A\rceil } points of the integer lattice. Equivalently, every bounded set of area A {\displaystyle A} contains...

In combinatorics, a lattice path L in the d-dimensional integer lattice ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ of length k with steps in the set S,...

are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special...

"lattice" is suggested by the form of the Hasse diagram depicting it. Pic. 2: Lattice of integer divisors of 60, ordered by "divides". Pic. 3: Lattice...

each dimension, then L(P, t) is the number of integer lattice points in tP. More formally, consider a lattice L {\displaystyle {\mathcal {L}}} in Euclidean...

mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius...

lengths of line segments between pairs of points in the two-dimensional integer lattice. The number of representable numbers in the range from 0 to any number...

2*(8*7)/(2*1) = 56 All integer (can only be 0, ±1): Two ±1, six zeroes: 4*(8*7)/(2*1)=112 These form a root system of type E8. The lattice Γ8 is equal to the...

Square lattice Hexagonal tiling Close-packing Centered hexagonal number Eisenstein integer Voronoi diagram Hermite constant Rana, Farhan. "Lattices in 1D...

three integers h, k, and ℓ, the Miller indices. They are written (hkℓ), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal...

either is an integer or a half-integer (has a denominator of 2). If the lattice triangle has integer sides then it is Heronian with integer area. Furthermore...

integer n {\displaystyle n} , also called a factor of n {\displaystyle n} , is an integer m {\displaystyle m} that may be multiplied by some integer to...

certain average-case lattice problem, known as short integer solutions (SIS), is at least as hard to solve as a worst-case lattice problem. She then showed...

mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This...

There are 124 different polygons of length 12 formed by edges of the integer lattice, counting two polygons as the same only when one is a translated copy...

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of...

theory called the geometry of numbers. It can be extended from the integers to any lattice L {\displaystyle L} and to any symmetric convex set with volume...

wazir form a square lattice graph. Lattice path Pick's theorem Integer triangles in a 2D lattice Regular graph Weisstein, Eric W. "Lattice graph". MathWorld...