2
votes
0answers
20 views

Number of points in the diagonal of an $X \times Y$ square lattice box.

So assuming we have a $X \times Y$ lattice, Say for example a $3 \times 5$ like so * * * * * * * * * * * * * * * I need to find the number of points that each ...
0
votes
0answers
25 views

Lattice orthogonal polyhedra face-area sequences: Golyhedra?

Let $P$ be a polyhedron, all of whose vertices are at points of $\mathbb{Z}^3$, all of whose edges are parallel to an axis, with every face simply connected, and the surface topologically a sphere. ...
1
vote
1answer
55 views

Lattice definition as finite density infinite set of vectors closed under addition?

This question came up in the definition of lattices (in the crystallography/group theory sense, not the ordered set sense) in our condensed matter lectures, but I believe it's more appropriate here ...
3
votes
0answers
30 views

Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html) The same is true for lattices in $\mathbb{R}^n$, ...
2
votes
1answer
48 views

Question regarding polygons

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?
2
votes
1answer
73 views

Proof of the finite number of Bravais lattices?

I've been taught that there are a finite number of Bravais lattices in 1, 2 and 3 dimensions. I am wondering if there is a proof of this fact. Maybe this is obvious and I am only missing certain key ...
2
votes
1answer
371 views

Find point on line that has integer coordinates

Given a 2D line e.g. (3,10) -> (8.3,16.5), how can I find any point on that line that has has whole-number coordinates? I can easily iteratively walk along one ...
0
votes
1answer
53 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
2
votes
2answers
84 views

What is a good way to simplicize the integer lattice?

I have a function $f$ defined on the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all nonnegative integers). I want to extend the domain of $f$ ...
0
votes
0answers
24 views

Find the centroid of an inequality-bounded integer lattice.

Let $Ax \ge c$ be a system of $k$ linear inequalities that define a bounded region in $\mathbb{R}^n$. Suppose we assign each point on the integer lattice (i.e. all coordinates are integers) of ...
0
votes
2answers
122 views

Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
0
votes
0answers
81 views

Counting lattice points in a square region

We have a lattice which is spanned by the vectors $(a,b)$ and $(-b,a)$, where $a,b \in \mathbb{N}$. Now we have a square region centered at $(0,0)$ having sides of length $2r+1$ for $r\in \mathbb{N}$. ...
3
votes
2answers
245 views

Number of lattice points

Does there exist a formula for counting the number of lattice points not outside of a square , with the at most information available concerning the square are the position coordinates of the four ...
5
votes
0answers
185 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
1
vote
1answer
220 views

Integral point inside a quadrilateral

Given four points in $\mathbb{R}^2$, is there an efficient method to determine if the convex hull contains an integral point $(m,n)\in\mathbb{Z}^2$? If it helps, I can assume the convex hull is a ...
1
vote
0answers
277 views

How many amplitude levels does a squared QAM constellation have?

In the above question, I choose terminology from communication engineering, but I will precisely define all the involved quantities below. It is a geometric question and I am not sure if it is ...
3
votes
2answers
113 views

Smallest angle among two lines in $n \times n$ grid

Given an $n \times n$ grid, what is the minimum angle between any two distinct lines, each going through some grid point $p$ and at least one other grid point? My guess is the minimum is attained by ...
2
votes
1answer
48 views

Lower-Bounding angles in integer Lattices

Given an $n \times n$ integer grid I chose any two grid points $a,b$, draw a line $l$ through $a$ and $b$ and measure the angle between $l$ and a horizontal line. I can do this for any grid point pair ...
2
votes
1answer
247 views

Closest point to line in lattice

Given an $n$ x $n$ integer grid, I look at all possible lines through two grid points and I am interested in the minimum distance from any grid point (not on the line) to any line. I my guess is that ...
10
votes
1answer
216 views

Question about Pick's Theorem

Is there a Pick's Theorem for a general lattice in $\mathbb{R}^{2}$?
3
votes
1answer
82 views

Nonzero element of minimal length in a lattice?

Given two linearly independent vectors $a,b\in\mathbb{R}^2$ we form the lattice $L=\{ma+nb|m,n\in\mathbb{Z}\}$. Now, a proof starts with "choose a nonzero vector in $L$ of smallest length...". Why ...
3
votes
1answer
291 views

Which internal angles can a lattice polygon have?

I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why? I guess there will be some restrictions due to the discrete nature of the grid but I am ...
2
votes
0answers
130 views

The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
16
votes
5answers
958 views

Pick's Theorem on a triangular (or hex) grid

Pick's theorem says that given a square grid (that is, all points in the plane with integer coordinates) and a polygon without holes and non selt-intersecting whose vertices are grid points, its area ...