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Lattice group in Euclidean space whose points are integer n-tuples
^{n}} whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice (or grid lattice) and the three-dimensional
Integer_lattice
Periodic set of points
in the plane whose coordinates are both integers, and its higher-dimensional analogues the integer lattices Z n {\displaystyle \mathbb {Z} ^{n}} . Closure
Lattice_(group)
2-dimensional integer lattice
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
Square_lattice
24-dimensional repeating pattern of points
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
Leech_lattice
Number in {..., –2, –1, 0, 1, 2, ...}
factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Complex number whose mapping on a coordinate plane produces a triangular lattice
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
Eisenstein_integer
Complex number whose real and imaginary parts are both integers
considered within the complex plane, the Gaussian integers constitute the 2-dimensional square lattice. The conjugate of a complex number a + bi is the
Gaussian_integer
Family of tetrahedra on an integer lattice
vertices of a Reeve tetrahedron are integer lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the
Reeve_tetrahedra
High-area shapes can shift to hold many grid points
includes at least ⌈ A ⌉ {\displaystyle \lceil A\rceil } points of the integer lattice. Equivalently, every bounded set of area A {\displaystyle A} contains
Blichfeldt's_theorem
Characterization by prime factors of sums of two squares
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
Sum_of_two_squares_theorem
Fourier transform of a real-space lattice, important in solid-state physics
)n} with an integer n {\displaystyle n} ) at every direct lattice vertex. One heuristic approach to constructing the reciprocal lattice in three dimensions
Reciprocal_lattice
Sequence of end-to-end vectors across points of a lattice
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,
Lattice_path
Type of crystal structure
{\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
Diamond_cubic
How many integer lattice points there are in a circle
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
Gauss_circle_problem
Algebra of formal sums
uniquely expressed as an integer combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group
Free_abelian_group
Algorithm in computational number theory
\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
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Theory of a class of elliptic curves
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
Complex_multiplication
Set whose pairs have minima and maxima
"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
Lattice_(order)
Integer that divides another integer
} of non-negative integers into a partially ordered set that is a complete distributive lattice. The largest element of this lattice is 0 and the smallest
Divisor
Triangle with integer side lengths
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_triangle
Lattice formed by all integer partitions
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative
Young's_lattice
Mathematical operation
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
Lattice_reduction
Curved triangle with constant width
provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing
Reuleaux_triangle
Natural number
ring of integers yields the Leech lattice. Conway and Sloane provided constructions of the Leech lattice from all other 23 Niemeier lattices. Twenty-three
23_(number)
Optimization problem in computer science
In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability
Lattice_problem
Natural number
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
124_(number)
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
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
Minkowski's_theorem
Natural number
chains of length six using horizontal and vertical segments of the integer lattice. 147 (disambiguation) Sloane, N. J. A. (ed.). "Sequence A005902 (Centered
147_(number)
set of integer numbers. For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped of the lattice) and the
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
Cryptographic primitives that involve lattices
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
Lattice-based_cryptography
Geometry and crystallography point array
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of
Bravais_lattice
Graph whose embedding in a Euclidean space forms a regular tiling
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
Lattice_graph
including integer lattices in real vector spaces, orders in algebraic number fields, and fractional ideals in integral domains. Formally, a lattice is a kind
Lattice_(module)
Czech mathematician (1897–1970)
developing Jarník's algorithm, he found tight bounds on the number of integer lattice points on convex curves, studied the relationship between the Hausdorff
Vojtěch_Jarník
One of the five 2D Bravais lattices
below. Square lattice (see dots in a diagonal square centered) Hexagonal tiling Close-packing Centered hexagonal number Eisenstein integer Voronoi diagram
Hexagonal_lattice
Notation system for crystal lattice planes
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
Miller_index
Mathematical proof technique
technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular
Vieta_jumping
Lattice in 8-dimensional space with special properties
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
E8_lattice
Geometric arrangements of points, foundational to Lie theory
coordinates are half-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions
Root_system
Application of geometry in number theory
a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle \mathbb {R} ^{n},} and the study of these lattices provides fundamental information
Geometry_of_numbers
Mathematical object
corresponding integer sublattice L ( I ) ⊆ Z n {\displaystyle {\mathcal {L}}(I)\subseteq \mathbb {Z} ^{n}} . An ideal lattice is an integer lattice L ( B )
Ideal_lattice
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Type of plane curve
points of the integer lattice. If the curve has length L {\displaystyle L} , then according to a theorem of Vojtěch Jarník, the number of lattice points that
Convex_curve
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Number-theoretical function
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A004018 (Theta series of square lattice, r_2(n))". The On-Line
Sum_of_squares_function
Positive real number which when multiplied by itself gives 7
two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). 15 {\displaystyle
Square_root_of_7
Number, approximately 3.14
(optimal) upper bound on the volume of a convex body containing only one integer lattice point. The Riemann zeta function ζ(s) is used in many areas of mathematics
Pi
Integer side lengths of a right triangle
over all positive and negative integers. Any Pythagorean triangle with triple (a, b, c) can be drawn within a 2D lattice with vertices at coordinates (0
Pythagorean_triple
On centroids of sets of lattice points
conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently
Kemnitz's_conjecture
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Vector quantization algorithm minimizing the sum of squared deviations
O ( d n 4 M 2 ) {\displaystyle O(dn^{4}M^{2})} for n points in an integer lattice { 1 , … , M } d {\displaystyle \{1,\dots ,M\}^{d}} . Lloyd's algorithm
K-means_clustering
Topics referred to by the same term
reducing the number of random variables under consideration Lattice reduction, given an integer lattice basis as input, to find a basis with short, nearly orthogonal
Reduction
Process forming a path from many random steps
example is the random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) Z d {\displaystyle \mathbb {Z} ^{d}} . If, in
Random_walk
Image comprising exactly two colors, typically black and white
Binary images can be interpreted as subsets of the two-dimensional integer lattice Z 2 {\displaystyle \mathbb {Z} ^{2}} ; the field of morphological
Binary_image
Construction analogous to that of a dual vector space
matrix B {\textstyle B} . The dual lattice is the set of linear functionals on L {\textstyle L} which take integer values on each point of L {\textstyle
Dual_lattice
Special type of lattice
A lattice-ordered vector space is a distributive lattice. Young's lattice given by the inclusion ordering of Young diagrams representing integer partitions
Distributive_lattice
Shape with four equal sides and angles
{\displaystyle c} . The Gaussian integers, complex numbers with integer real and imaginary parts, form a square lattice in the complex plane. The construction
Square
Reconstruction of binary images from a small number of their projections
problem of reconstruction of binary images (or finite subsets of the integer lattice) from a small number of their projections. In general, tomography deals
Discrete_tomography
Integral lattice of determinant 1 or –1
The lattice is integral if (·,·) takes integer values. The dimension of a lattice is the same as its rank (as a Z-module). The norm of a lattice element
Unimodular_lattice
Tessellation of Euclidean space
are unit squares or unit cubes, and the vertices are points on the integer lattice. A rectilinear grid is a tessellation by rectangles or rectangular
Regular_grid
Exponential function of an exponential function
The maximal volume of a polytope in a d-dimensional integer lattice with k ≥ 1 interior lattice points is at most k ⋅ ( 8 d ) d ⋅ 15 d ⋅ 2 2 d + 1 ,
Double_exponential_function
American mathematician (born 1951)
as lattice walks, which usually take place on the integer lattice and are sometimes confined to the upper right quadrant. An excursion is a lattice walk
Ira_Gessel
Relation of an integral polytope's volume to how many integer points it encloses
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
Ehrhart_polynomial
Natural number, composite number
(n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-15. Grünbaum
14_(number)
Set of distances defined from a set of points
numbers in the distance set of the two-dimensional integer lattice: they are the square roots of integers whose prime factorization does not contain an odd
Distance_set
Generalization of algebraic integers
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
Hurwitz_quaternion
Matrix form in linear algebra
normal form is an analogue of reduced echelon form for matrices over the integers Z {\displaystyle \mathbb {Z} } . Just as reduced echelon form can be used
Hermite_normal_form
Cryptography secured against quantum computers
algorithms rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm problem, or the elliptic-curve
Post-quantum_cryptography
Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle d}
Doignon's_theorem
Mathematical formula expressing equality
curve, algebraic surface, or more general object, and ask about the integer lattice points on it. The word Diophantine refers to the Hellenistic mathematician
Equation
Combinitorics of Polyhedra
convex polytopes does not form a convex subset of the four-dimensional integer lattice, and much remains unknown about the possible values of these vectors
Polyhedral_combinatorics
Classification of crystalline materials by their three-dimensional structural geometry
integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors. These lattices are classified by the space group of the lattice
Crystal_system
Periodic spatial graph
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
Laves_graph
Simple Lie group; the automorphism group of the octonions
(−1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be
G2_(mathematics)
Property of being an even or odd number
the face-centered cubic lattice and its higher-dimensional generalizations (the Dn lattices) consist of all of the integer points whose coordinates have
Parity_(mathematics)
Method to solve optimization problems
including the integer decomposition property and total dual integrality. Other specific well-known integral LPs include the matching polytope, lattice polyhedra
Linear_programming
Centered figurate number representing an octahedron
is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers
Centered_octahedral_number
Natural number
J4, J3, and J1). 6 is the smallest integer which is not an exponent of a prime number, making it the smallest integer greater than 1 for which there does
6
Exploring properties of the integers with complex analysis
about the origin in the plane with radius r, the problem asks how many integer lattice points lie on or inside the circle. It is not hard to prove that the
Analytic_number_theory
Mathematics of varieties with integer coordinates
therefore are the primary consideration; but integral solutions (i.e., integer lattice points) can be treated in the same way as an affine variety may be
Diophantine_geometry
Natural number
(Number of ways of embedding a connected graph with n edges in the square lattice)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. v t e
222_(number)
Doughnut-shaped surface of revolution
any coordinate. That is, the n-torus is Rn modulo the action of the integer lattice Zn (with the action being taken as vector addition). Equivalently,
Torus
Geometric object with flat sides
differs, in terms of integer lattice points, from a t {\displaystyle t} -dilate of P {\displaystyle {\mathcal {P}}} only by lattice points gained on the
Polytope
Rational number equal to an integer plus 1/2
from the integers to the half-integers: f : x → x + 0.5 {\displaystyle f:x\to x+0.5} , where x {\displaystyle x} is an integer. The densest lattice packing
Half-integer
Number of paths between grid corners, allowing diagonal steps
{\displaystyle m} and n {\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in
Delannoy_number
Mathematical term
a lattice word (or lattice permutation) is a string composed of positive integers, in which every prefix contains at least as many positive integers i
Lattice_word
Type of metric geometry
3D balls of radii 1 (red) and 2 (blue) are regular octahedrons: the number of integer lattice points enclosed form the centered octahedral numbers
Taxicab_geometry
Crystallographic concept
parallel lattice planes that, taken together, intersect all lattice points. Every family of lattice planes can be described by a set of integer Miller indices
Lattice_plane
Quasiparticle of mechanical vibrations
determined up to addition of constant vectors (the reciprocal lattice vectors and integer multiples thereof). For example, in the one-dimensional model
Phonon
apply types of double counting. One by Gotthold Eisenstein counts integer lattice points. Another applies Zolotarev's lemma to ( Z / p q Z ) × {\displaystyle
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Power series derived from a discrete probability distribution
variable taking values (x1, ..., xd) in the d-dimensional non-negative integer lattice {0,1, ...}d, then the probability generating function of X is defined
Probability generating function
Probability_generating_function
Mathematical problem
; Mahoney, Carolyn R. (1996), "Unit-distance graphs, graphs on the integer lattice and a Ramsey type result", Aequationes Mathematicae, 51 (1–2): 48–67
Hadwiger–Nelson_problem
Type of subatomic particle
subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin 3/2, etc.) and obey the Pauli exclusion principle
Fermion
Natural number
number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72. It is equal to the sum of its preceding smaller highly totient
72_(number)
Reversible block cellular automaton
two-dimensional infinite grid of cells, which may be identified with the integer lattice. As in Conway's Game of Life, at any point in time each cell may be
Critters_(cellular_automaton)
Three linked but pairwise separated rings
of ropelength, the shortest representation using only edges of the integer lattice, the minimum length for the Borromean rings is exactly 36 {\displaystyle
Borromean_rings
Tiling of the plane with 60° rhombi
to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way
Rhombille_tiling
Type of group in group theory
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm
Arithmetic_group
Simple curve of Euclidean geometry
Distance – Separation between two points Gauss circle problem – How many integer lattice points there are in a circle Inversion in a circle – Study of angle-preserving
Circle
Function in algebraic graph theory
natural numbers to each vertex, a graph coloring is a vector in the integer lattice. Since two vertices i {\displaystyle i} and j {\displaystyle j} being
Chromatic_polynomial
INTEGER LATTICE
INTEGER LATTICE
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Boy/Male
Arabic, Muslim
To Wait
Boy/Male
Norse
Son's army.
Surname or Lastname
English
English : variant of Fretter, an occupational name for a maker of ornaments (especially for the hair) consisting of jewels set in a lattice network, from an agent derivative of Middle English frette, Old French frete ‘interlaced work’.
Boy/Male
Muslim
To wait
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
INTEGER LATTICE
INTEGER LATTICE
Girl/Female
Hindu, Indian, Marathi
Goddess of Vegetation
Boy/Male
Hindu, Indian
Name of Lord Shiva
Boy/Male
British, English, Hindu, Indian, Kannada, Telugu
Beloved
Surname or Lastname
English
English : habitational name from any of the many places called Newbury, named with the Old English elements nēowe ‘new’ + burh ‘fortress’, ‘fortified town’ (see Berry 1 and Bury).Thomas Newberry emigrated from Devon, England, to Dorchester, MA, in 1634. Among his descendants were a number of very successful manufacturers and entrepreneurs, including the brothers Oliver (1789–1860) and Walter (1804–68) Newberry, whose prosperity was linked with the growth and development of Chicago.
Girl/Female
Hindu, Indian, Sikh
Clever
Boy/Male
Australian, French, Greek, Latin
From Rome
Girl/Female
Muslim/Islamic
Happy
Boy/Male
Tamil
Vedatman | வேதாதà¯à®®à®¨
Lord Vishnu
Boy/Male
Australian, Basque, German, Greek
Gracious Gift; Given by God
Boy/Male
Sikh
Victory of good company
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
v. t.
To deposit or inter in a chapel; to enshrine.
imp. & p. p.
of Inter
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
v. t.
To inter.
n.
One who makes an entrance or beginning.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
n.
One who gathers the vintage.
n.
One who makes an index.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
n.
One who inters.
p. pr. & vb. n.
of Inter
v. t.
To inter again.
v. t.
To place in a tomb; to bury; to inter; to entomb.
n.
One who intends.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
v. t.
To inter with funeral rites; to bury.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
v. t.
To inhume; to bury; to inter.