All Questions
Tagged with mg.metric-geometry nt.number-theory
86 questions
3
votes
0
answers
80
views
Equidistribution of Brillouin zones
Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...
- 12.3k
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
11
votes
5
answers
2k
views
Defining Euler's number via elementary euclidean geometry (and a dimension limit)
Let $B_n$ be a closed ball in euclidean space $\mathbb{R}^n$, and consider the largest cube $Q_n$ contained in $B_n$. Then, let $C_n$ be a cube of maximal size that is contained in $B_n$ and disjoint ...
- 1,942
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 ...
- 3,707
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 ${\mathbb R}^2$). ...
- 1,072
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:...
- 22.3k
7
votes
2
answers
244
views
approximate two different real numbers to order $\frac{1}{z^{3/2}}$
I took this result from Minkowski's book on Geometry of numbers:
Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and $\...
- 22.8k
10
votes
3
answers
683
views
Circles avoiding rational points of height $\le h$
Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates are ...
- 151k
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
30
votes
0
answers
747
views
Is there an Ehrhart polynomial for Gaussian integers
Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
- 156k
22
votes
1
answer
1k
views
Characterization of Volumes of Lattice Cubes
This is a cross post from Math SE that no one seemed able to solve.
Here is a problem that came up in a conversation with a professor after I made a false assumption about the geometry of $\mathbb{Z}^...
- 612
11
votes
3
answers
2k
views
Could a perfect squared square be split into two perfect squared squares?
This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...
- 1,375
6
votes
2
answers
249
views
Intersecting Sets of Pythagorean Triples with Common Hypotenuses
For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$.
Given any $N\in\mathbb{N}$, does there exist $r,s$ ...
- 93
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 ...
- 1,531
14
votes
2
answers
2k
views
Right triangle with edge lengths equal to regular unit polygon edge lengths
This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
- 68.9k
12
votes
1
answer
585
views
Heronian triangle with two sides that are prime
Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
- 189
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
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 ...
- 96.4k
5
votes
0
answers
232
views
4D polytope analogues of the icosahedron/Rogers-Ramanujan continued fraction relationship?
The formula for the j-function which employs polynomial invariants of the icosahedron,
$$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$
where,
$$r^{-1}-...
- 12.6k
2
votes
2
answers
331
views
what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
9
votes
2
answers
449
views
Rational points on circular spirals
Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...
- 151k
28
votes
6
answers
2k
views
Patterns among integer-distance points
Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...
- 151k
1
vote
1
answer
210
views
Pythagorean triples related to non-isometric equidistant plane quadruples
QUESTION Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
$ x^2 + (x-u)^2\ =\ A^2$
$ x^2 + (x+u)^2\ =\ B^2$
?
...
- 7,500
6
votes
3
answers
1k
views
Consecutive Integer Squared Square
Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.
- 61
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
7
votes
3
answers
510
views
Proto-Euclidean algorithm
Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining
$$q_i = \left\lfloor \frac{r_i}{r_{...
- 9,720
2
votes
0
answers
362
views
Rational integer solutions of a linear Diophantine equation of cyclotomic integers
I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.
In particular, suppose my lattice vectors are $a$ and $b$ and I want to know ...
- 86
14
votes
1
answer
837
views
Applications of the GCD metric
In the pre-MO era, I once realized that on the integers, the function
$$
d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ ,
$$
is a metric (all properties are easily verified; in fact ...
- 28.6k
9
votes
2
answers
928
views
Shortest irrational path
What is the shortest curve $\gamma$ in $\mathbb{R}^2$
from the origin $o=(0,0)$ to a rational point $p=(a,b)$
that (a) passes through no other rational point, and
(b) contains no point a ...
- 151k
3
votes
1
answer
316
views
Hausdorff dimension and Mertens function
Hello,
when one plots the Mertens function, it really looks like a fractal. So does anyone know the (approximate) value of the Hausdorff dimension of the set $\{(x,y),y=M(x),x\in\mathbb{R}^+\}$?
...
- 6,984
8
votes
4
answers
1k
views
Sequences of evenly-distributed points in a product of intervals
Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distributed" in a way which ...
- 3,612
14
votes
2
answers
1k
views
Polygonal billards programs
I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
(source)
It was a good exercise, but at this point I ...
- 22.8k
18
votes
2
answers
2k
views
Assistance with understanding parent/child relationships in Pythagorean Triples
I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained elsewhere - ...
- 283
7
votes
2
answers
726
views
Zeta function for curves in a manifold
Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...
- 15.1k
12
votes
3
answers
707
views
A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
- 139