What is lattice points: Definition and 16 Discussions

In geometry and group theory, a lattice in the real coordinate space



{\displaystyle \mathbb {R} ^{n}}
is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension


{\displaystyle n}
which spans the vector space



{\displaystyle \mathbb {R} ^{n}}
. For any basis of



{\displaystyle \mathbb {R} ^{n}}
, the subgroup of all linear 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 as a regular tiling of a space by a primitive cell.
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in percolation theory to study connectivity arising from small-scale interactions, cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "framework of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.

View More On Wikipedia.org
  1. Slimy0233

    I Can all possible periodic arrangements of lattices be arranged as a Bravais lattice just by taking a different motif?

    or Are all naturally occurring crystals with periodic arrangement of lattices Bravais lattices? From two days, I have been trying to understand Bravais lattices and what it's importance is and after a lot of research, I came to know that they are a periodic arrangement of lattice points with...
  2. P

    I Finite many Lattice Points in Sphere?

    Hello, I am wondering if in an n-ball the number of lattice points is finite. First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...
  3. J

    I Prove that a triangle with lattice points cannot be equilateral

    I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e) and of course: a, b, c, d, e∈Z Using the distance formula between each of the points and setting them equal: \sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...
  4. Kyueong-Hwang

    What are the lattice points for this?

    In picture, there is crystal structure with two atoms. Question is, what is the lattice points? Find a set of lattice points. I think green and pink points are both lattice points. Am i right? This is our full hw.
  5. F

    I Proof that lattice points can't form an equilateral triangle

    From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...
  6. K

    Lattice points and lattice basis

    Hi! I'm struggling in identifying the lattice points and atom basis. As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct? So if the points on the corners are the lattice...
  7. M

    Understanding Lattice Points in a Primitive Cubic Cell

    Hello, Suppose I have a primitive cubic cell with 8 atoms, one on each corner of the cube. I don't understand how this consists of only one lattice point? Doesn't each corner have a lattice point, thus the cell would consist of 8 lattice points??
  8. S

    How Does the Ratio of Lattice Points to Radius Behave as Radius Increases?

    Given a circle centered at the origin, how can one prove that the limit of the quotient of the number of lattice points on the circle over the radius goes to zero as radius goes to infinity?
  9. F

    Lattice Points on Circle: Determining the Number of Points on the Boundary

    Does any circle having irrational radius have no lattice points on its boundary ? Extended question: Is there any way to determine the number of lattice points lying on the boundary of a given circle ? *The centres of these circles are all (0,0) *
  10. S

    Continuum Conversion of Lattice Points via Taylor Series Expansion

    I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?
  11. M

    What is a Lattice Point in Geometry?

    What exactly is a lattice point (in relation to geometry)? I seriously doubt my simple minded explanation suffices... A lattice point is the meeting of the y and x integers on the Cartesian plane. And if that's in essence correct, is the way to find the number of lattice points found by...
  12. Y

    Counting Lattice Points in a Circle: A Math Contest Question

    In a math contest, the question goes somehow like this: A lattice point is a point wherein the value of (x,y) is an integer. Determine the total number of lattice points in a circle which has a radius of 6 and the its center is at the origin. Any one knows the solution or shortcut for this?
  13. L

    Lattice points : Convex region symm. about the origin

    Let R be a convex region symmetrical about the origin with area greater than 4. Show that R must contain a lattice point different from the origin. This is the 2-D case of Minkowski's theorem, right ? How about the n-dimensional version ? The n-dimensional version is : Given a convex...
  14. C

    Prove that the formula is valid for rectangles with sides parallel to the coordinate axes

    Let P be a polygon whose vertices are lattice points. The area of P is Z + \frac{1}{2}B - 1 . Z is the number of lattice points inside the polygon, and B is the number on the boundary. (a) Prove that the forumula is valid for rectangles with sides parallel to the coordinate axes. (b)...
  15. C

    Lattice Points and Equilateral Triangles: A Proof

    Hello all In a ordinary syatem of rectangular coordinates, the points for which both coordinates are integers are called lattice points . Prove that a triangle whose vertices are lattice points cannot be equilateral. Ok so I know that in a equilateral triangle the angle measures are...