All Questions
10 questions
3
votes
1
answer
366
views
Illumination from visible lattice points with inverse square intensity
It is well known that the number of $\mathbb{Z}^2$ lattice points visible from
the origin is $6/\pi^2$, about $61$%.
See, e.g.,
What fraction of the integer lattice can be seen from the origin?.
I am ...
- 151k
4
votes
1
answer
108
views
Closed cobounded additive submonoid of $\mathbb{R}^n$
Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more ...
- 143
11
votes
1
answer
499
views
Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
- 151k
3
votes
1
answer
349
views
Covering points with a shortest lattice spiral
Let $S$ be a finite set of lattice points in $\mathbb{Z}^2$.
My question is, roughly:
Q. How can a shortest lattice spiral that passes through
every point of $S$ be found?
A lattice spiral (my ...
- 151k
10
votes
1
answer
673
views
A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
- 151k
41
votes
2
answers
2k
views
Can we find lattice polyhedra with faces of area 1,2,3,...?
I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...
- 151k
7
votes
2
answers
453
views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
- 679
3
votes
0
answers
135
views
Lattices achieving best density
Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
- 800
49
votes
4
answers
4k
views
What fraction of the integer lattice can be seen from the origin?
Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is visible from the origin if the
segment from $(0,0)$ to $(x,y) \in Q$ passes ...
- 151k
7
votes
2
answers
907
views
Is there a 3d equivalent of this picture?
This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
The picture below is the image of a regular ...
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