All Questions
Tagged with rational-points mg.metric-geometry
16 questions
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
25
votes
3
answers
994
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
- 513
24
votes
3
answers
3k
views
Integer-distance sets
Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...
- 151k
17
votes
1
answer
1k
views
Totally rational polytopes
Define a convex polytope in $\mathbb{R}^d$ as
totally rational (my terminology)
if its vertex coordinates are rational, its edge lengths
are rational, its two-dimensional face areas are rational, etc.,...
- 151k
17
votes
1
answer
822
views
Is the perimeter of an ellipse with integer axes irrational?
Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...
- 151k
11
votes
1
answer
702
views
Schoenberg's rational polygon problem
"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a ...
- 151k
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
9
votes
2
answers
929
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
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
7
votes
0
answers
205
views
Lattice radial-step (ratchet) spirals
(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...
- 151k
6
votes
2
answers
297
views
Can every set of points with rational distance squares be isometrically embedded in $\Bbb Q^d$?
Suppose we are given a finite family of points $p_1,...,p_n\in \Bbb R^d$, so that any two points have a rational distance square, that is,
$$\|p_i-p_j\|^2\in\Bbb Q,\quad\text{for all $i,j\in\{1,...,n\}...
- 13.6k
6
votes
0
answers
118
views
Rational $d$-simplices
Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...
- 151k
5
votes
1
answer
393
views
Maximum number of general-position points with mutual rational distances?
Richard Guy has shown that there are six points in the plane—no three collinear,
no four cocircular—such that all interpoint distances are rational.
Guy, Richard. Unsolved Problems in ...
- 151k
3
votes
2
answers
203
views
Existence of lattices whose circles have bounded number of points
For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \...
- 315
3
votes
2
answers
185
views
Lattice-point-free buffers around circles
Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
&...
- 151k
1
vote
0
answers
96
views
Shortest paths stepping on rational points of height $h$
Q. Do shortest paths walking between rational points of height $h$
ever properly cross themselves?
Explaining this question takes a bit of definitional exposition.
First, I copy definitions from ...
- 151k