All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
99
votes
7
answers
20k
views
Can we cover the unit square by these rectangles?
The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \...
- 5,502
94
votes
5
answers
10k
views
Is there a dense subset of the real plane with all pairwise distances rational?
I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
- 236k
88
votes
2
answers
7k
views
65
votes
3
answers
3k
views
How many unit cylinders can touch a unit ball?
What is the maximum number $k$ of unit radius, infinitely long cylinders with mutually disjoint interiors that can touch a unit ball?
By a cylinder I mean a set congruent to the Cartesian product of ...
- 7,276
55
votes
6
answers
8k
views
Is it possible to partition $\mathbb R^3$ into unit circles?
Is it possible to partition $\mathbb R^3$ into unit circles?
- 1,414
52
votes
3
answers
5k
views
Is the "Napkin conjecture" open? (origami)
The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
- 1,306
51
votes
3
answers
3k
views
Can the sphere be partitioned into small congruent cells?
On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
- 7,276
51
votes
4
answers
7k
views
what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics
In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
- 10.7k
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
49
votes
5
answers
3k
views
If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we ...
- 19k
45
votes
1
answer
3k
views
two tetrahedra in $\mathbb R^4$
It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $\mathbb R^3$,
such that their union has diameter $1,$ then they must share a vertex.
I wonder whether ...
- 1,359
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
- 151k
43
votes
12
answers
2k
views
Can a discrete set of the plane of uniform density intersect all large triangles?
Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...
- 17.9k
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
38
votes
7
answers
5k
views
Shortest path connecting two opposite points on a cube
Is it true, that a path connecting two opposite points (i.e. such that the segment joining them passes through the centre of mass of the cube) on the surface of the $d$-dimensional unit cube (with $d&...
- 1,431
36
votes
2
answers
2k
views
Bodies of constant width?
In two-dimensional case one can generalize figures of constant width as figures which can rotate in a convex polygon.
Here is one example which can be used to drill triangular holes:
I would like to ...
- 45k
35
votes
3
answers
2k
views
The kissing number of a square, cube, hypercube?
How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.
&...
- 151k
34
votes
6
answers
8k
views
Covering a unit ball with balls half the radius
This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
- 151k
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
33
votes
3
answers
5k
views
Do bubbles between plates approximate Voronoi diagrams?
For example, soap bubbles:
Image from UPenn:
"A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours ...
- 151k
32
votes
5
answers
1k
views
Can every $\mathbb{Z}^2$ disk be pinball-reached?
Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be hit ...
- 151k
32
votes
5
answers
2k
views
Nonconvex manhole covers
One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...
- 5,275
31
votes
5
answers
1k
views
Fair cutting of the plane with lines
An infinite countable family $\cal{L}$ of straight lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:
$\bullet$ No circle intersects ...
- 7,276
30
votes
5
answers
16k
views
How to check if a box fits in a box?
How could I calculate if a rectangular cuboid fits in an other rectangular cuboid, it may rotate or be placed in any way inside the bigger one.
For example would, (650,220,55) fit in (590,290,160), ...
- 429
28
votes
5
answers
2k
views
Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
- 4,484
28
votes
3
answers
2k
views
Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
- 19k
27
votes
6
answers
2k
views
When shorter means smaller?
Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it ...
26
votes
7
answers
3k
views
What's that shape? Inferring a 3D shape from random shadows
Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$.
$P$ could be a polyhedron, or a smooth shape, or an arbitrary shape;
I'll assume below that $P$ is a (non-degenerate, perhaps non-...
- 151k
26
votes
0
answers
359
views
Can 4-space be partitioned into Klein bottles?
It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...
- 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
25
votes
3
answers
945
views
Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?
Question: Can I have an arbitrarily large finite family of lines $\ell_1,\dotsc,\ell_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$?
We can assume that all ...
- 13.6k
25
votes
1
answer
513
views
Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
- 151k
24
votes
3
answers
3k
views
Polyomino that can cover an arbitrarily large square but not the entire plane
https://userpages.monmouth.com/~colonel/nrectcover/index.html
For a polyomino with no holes that cannot tile the plane, we may ask what are the maximal rectangles and infinite strips that it can ...
- 359
24
votes
3
answers
3k
views
Can a unit square be cut into rectangles that tile a rectangle with irrational sides?
For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...
- 2,437
24
votes
3
answers
1k
views
Tetrahedron insphere iteration
I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...
- 151k
24
votes
1
answer
770
views
Given a group action on a simplex, can I always find a fundamental region that is a simplex?
Let $\Delta\subset\Bbb R^n$ be a simplex with $n+1$ vertices. Let $G\subset\mathrm{GL}(\Bbb R^n)$ be a finite group of linear symmetries of $\Delta$, i.e. linear transformations that fix the simplex ...
- 13.6k
24
votes
2
answers
754
views
Expected number of vertices of a hypercube slice -- is this new/interesting?
I am a (mostly) amateur mathematician, but my education and work have featured a lot of mathematics, and recently I bumped into a mathematical problem for which I can find no references, and I am ...
- 241
23
votes
3
answers
2k
views
Rolling-ball game
The analyses
in two recent MO questions
("recent" with respect to the original posting in 2011),
"Rolling a random walk on a sphere"
and
"Maneuvering with limited moves on $S^2$,"
suggest a Rolling-...
- 151k
23
votes
1
answer
524
views
Tying knots via gravity-assisted spaceship trajectories
Q.
Can every knot be realized as the trajectory of a spaceship
weaving among a finite number of fixed planets, subject to gravity alone?
To make this more ...
- 151k
22
votes
4
answers
2k
views
Non-chaotic bouncing-ball curves
I was surprised to learn from two
Mathematica Demos by
Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel
exhibits chaotic behavior:
(The Poincaré map ...
- 151k
22
votes
1
answer
696
views
Rational inscribed realization of the regular dodecahedron
While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...
- 10.7k
22
votes
2
answers
900
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...
- 151k
22
votes
3
answers
1k
views
Equilaterally triangulated surfaces with prescribed boundary
There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest ...
- 7,194
22
votes
1
answer
663
views
Voronoi cell of lattices with the same profile
Definition 1. Given a body $V$ in $\mathbb R^n$,
the function $p_V\colon \mathbb R_+\to \mathbb R_+$
$$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$
will be called profile of $V$.
Definition 2. Define ...
- 45k
22
votes
1
answer
886
views
Happy ants never leave compact domain?
I am curious if the following seemingly simple question has an easy answer?
Consider an ant population of $N$ ants that lives in $\mathbb R^2$. Each ant can be labeled by some coordinate $x\in \mathbb ...
- 124
21
votes
5
answers
1k
views
Is a rhombus rigid on a sphere or torus? And generalizations
If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...
- 151k
21
votes
3
answers
936
views
Cutting of a regular polygon into congruent pieces
Question. For which $N$ it is possible to cut a regular $N$-gon into congruent pieces such that the center of the regular polygon lies strictly inside one of the pieces? For $N=3,4$ there are trivial ...
- 691
21
votes
1
answer
975
views
Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...
- 3,527
21
votes
2
answers
1k
views
Forbidden mirror sequences
Let $\cal{M}$ be a finite collection of two-sided mirrors,
each an open unit-length segment in $\mathbb{R^2}$,
and such that the segments when closed are disjoint.
A ray of light that reflects off the ...
- 151k