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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 ...
Joseph O'Rourke's user avatar
34 votes
1 answer
3k views

Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area? The original problem, not requiring integer sides for rectangles, was proposed by Joe ...
Bernardo Recamán Santos's user avatar
11 votes
2 answers
444 views

The intersection of a circle and a rank 3 subgroup of the plane

Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...
user42355's user avatar
  • 1,531
9 votes
2 answers
598 views

Dissecting Ramanujan´s Cuboid: 1729 = 19 x 13 x 7

Consider the cuboid of dimensions 19 x 13 x 7 whose volume is 1729, the Hardy-Ramanujan number. What is the least number of smaller cuboids into which it can be dissected so that the resulting pieces ...
Bernardo Recamán Santos's user avatar
9 votes
0 answers
187 views

Cubing the cube - as 'perfectly' as possible

Ref: https://en.wikipedia.org/wiki/Squaring_the_square A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
Nandakumar R's user avatar
  • 5,979
8 votes
1 answer
417 views

Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin dodecahedra:...
John Baez's user avatar
  • 22.3k
8 votes
1 answer
2k views

Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...
Adam Sheffer's user avatar
  • 1,072
7 votes
1 answer
498 views

Is there a bicyclic irregular pentagon in integers?

Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well? If it does ...
shabo's user avatar
  • 71
6 votes
2 answers
544 views

On circles and ellipses drawn on an infinite planar square lattice

Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
Nandakumar R's user avatar
  • 5,979
6 votes
3 answers
1k views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
Matt Watson's user avatar
6 votes
2 answers
381 views

Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$. Define a blocking set for a lattice cube to be a set of points in ...
Joseph O'Rourke's user avatar
5 votes
1 answer
532 views

Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...
Igor Rivin's user avatar
  • 96.4k
4 votes
1 answer
438 views

Perfect squaring of rectangles

A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...
Nandakumar R's user avatar
  • 5,979
4 votes
0 answers
111 views

Advice on results for balls on regular $N$-dimensional grids

I have obtained some results regarding balls on regular $N$-dimensional grids. I would like expert opinion on wether the results are significant or interesting enough for (trying to) publish them in a ...
Luis Mendo's user avatar
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 ...
Joseph O'Rourke's user avatar
3 votes
1 answer
118 views

Question arise from kissing number in 2 dimension

I'm considering an extended problem of kissing number in $\mathbb{R}^2$. Suppose I have a given disc $\mathcal{D}$ of radius 1/2 and infinitely many discs all of radius 1/2 and all these discs and ...
neverevernever's user avatar
3 votes
0 answers
310 views

Upper bound on the number of lattice points on the intersection of a hyperplane and a sphere

Let $R>0$, $\overrightarrow{\alpha} \in \mathbb{R}^{d}$. Consider the intersection $T$of $RS^{d-1}$ and the hyperplane $\overrightarrow{\alpha} \cdot \overrightarrow{x} = n$. What is the best known ...
Martin Ortiz's user avatar
2 votes
4 answers
997 views

Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
VS.'s user avatar
  • 1,826
2 votes
0 answers
95 views

Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?

I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice. I know that it is exist for an square lattice https://mathworld.wolfram.com/...
Mihaela's user avatar
  • 31
1 vote
0 answers
52 views

'Self-similar and perfect' partitions of planar regions

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition. A classical example ...
Nandakumar R's user avatar
  • 5,979
1 vote
0 answers
100 views

Perfect 'cuboiding' of cubes and cuboids

We try to add a bit to ref 2 listed below. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special ...
Nandakumar R's user avatar
  • 5,979
1 vote
0 answers
77 views

Lattice packing

Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number. Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
user95246's user avatar
  • 237
1 vote
0 answers
196 views

Squares as sum of squares

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares? Of these n x n squares, which can be actually cut into n smaller squares?
Bernardo Recamán Santos's user avatar