All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
14
votes
3
answers
966
views
Can a tangle of arcs interlock?
Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining ...
- 151k
14
votes
4
answers
453
views
Smallest containing simplex
Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...
- 6,819
14
votes
2
answers
878
views
Sets of evenly distributed points in the Euclidean plane
Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can $P$...
- 19.6k
14
votes
2
answers
533
views
Double kissing problem
Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls?
An easy ...
- 141
14
votes
2
answers
540
views
Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
$M(M(a,b),M(a,c))=...
- 5,103
14
votes
1
answer
819
views
The geometry of crinkled aluminum foil
I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...
- 151k
14
votes
1
answer
280
views
How many distances are required to calculate all distances among $n$ points in the Euclidean plane?
I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
- 523
14
votes
3
answers
2k
views
Optimal wireframe sphere
Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...
- 151k
14
votes
1
answer
643
views
Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from an ...
- 45k
14
votes
0
answers
270
views
Regular $n$-gon with diagonals: bounds on area of largest cell?
Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
- 3,567
14
votes
0
answers
479
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
- 532
13
votes
5
answers
1k
views
Packing obtuse vectors in $\mathbb{R}^d$
I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...
- 151k
13
votes
3
answers
835
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
- 15.5k
13
votes
2
answers
3k
views
How many squares can be formed by using n points?
How many squares can be formed by using n points on a 3 dimensional space?
Like using 4 points, there is 1 square be formed
Using 5 points, still 1 square
Using 6 points, 3 squares can be formed
- 241
13
votes
3
answers
421
views
Maximal distance between $2d+1$ points on the $(d-1)$-sphere
If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
- 13.6k
13
votes
2
answers
919
views
Acute triangulation
Assume that $S$ is a finite 2-dimensional simplicial complex equipped with a metric $d$
such that each triangle is isometric to a plane triangle (so $(S,d)$ is a polyhedral space).
Is it possible ...
- 45k
13
votes
2
answers
1k
views
Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
- 151k
13
votes
2
answers
572
views
The most number of points that realize only $k$ distinct distances
For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$
in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances,
$\|p_i-p_j\|$.
Example. For points in the plane ...
- 151k
13
votes
1
answer
3k
views
What nets fold to polyhedra?
There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
- 1,314
13
votes
3
answers
388
views
Intersecting cylinders around a sphere
Intersecting $n$ unit-radius cylinders, each with axis through the origin,
produces a shape circumscribed about a unit-radius sphere:
My question is:
For each $n$, which arrangement of cylinders ...
- 151k
13
votes
1
answer
430
views
Detecting a hidden convex body with line probes
Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...
- 151k
13
votes
0
answers
378
views
Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
13
votes
0
answers
573
views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
- 3,068
12
votes
7
answers
683
views
Can a tangle of arcs of ellipses interlock
This is a variation on an earlier question resolved by user35353: Can a tangle of arcs interlock? In that question, the arcs were restricted to circular arcs, and user35353's proof that one arc can be ...
- 151k
12
votes
4
answers
2k
views
Longest path through hypercube corners
Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known,
where path length is measured by Euclidean distance in $\mathbb{R}^d$?
The unit hypercube spans from $(0,0,\ldots,0)$ ...
- 151k
12
votes
1
answer
373
views
A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
- 5,979
12
votes
2
answers
5k
views
The Gauss circle problem on a hexagonal lattice
Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...
- 197
12
votes
3
answers
418
views
'Trapping' 3D regions with sheets of paper
Given a square sheet of paper, how does one create a bag (a closed surface) with it such that the 3D region contained within this closed surface has maximum volume (operations allowed include ...
- 5,979
12
votes
1
answer
872
views
Tiling by regular simplices
The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional ...
- 123
12
votes
2
answers
2k
views
Distribution of pairwise distances
I am seeking results that describe the distribution of the set of
Euclidean distances between pairs of $n$ points in
a unit square in the plane.
For example: All the distances could be short (a tight ...
- 151k
12
votes
2
answers
11k
views
Covering a polygon with rectangles
I am trying to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about ...
12
votes
1
answer
614
views
Covering the unit sphere by open hemispheres
Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...
- 219
11
votes
4
answers
608
views
What is the right way to think about / represent general tilings?
For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
11
votes
2
answers
455
views
Dodecahedral rolling distance
Let a dodecahedron sit on the plane,
with one face's vertices on an origin-centered unit circle.
Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal.
For any $p \in \mathbb{R}...
- 151k
11
votes
3
answers
3k
views
polyhedra with equilateral pentagons faces
In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...
- 111
11
votes
1
answer
406
views
Thinnest 2-fold coverings of the plane by congruent convex shapes
It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...
- 151k
11
votes
2
answers
1k
views
Which (semi)regular polyhedra are combinations of two others?
The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and $...
- 151k
11
votes
2
answers
797
views
Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...
- 2,136
11
votes
1
answer
403
views
Smallest sphere containing three tetrahedra?
What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?
- 111
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
11
votes
1
answer
424
views
Needle probing for a convex body
Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...
- 151k
11
votes
1
answer
807
views
Soft question: mathematics about truchet tiles
It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...
- 191
11
votes
1
answer
712
views
Polygons uniquely inducing arrangements
A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines intersect ...
- 151k
11
votes
1
answer
652
views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is often stated briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
- 13.6k
11
votes
1
answer
266
views
Metric conditions on configurations of points with only finitely many solutions
There is an old puzzle, which I believe I learned from Nob Yoshigahara, that asks for all configurations of four (distinct) points in the plane such that the six pairwise distances assume only two ...
- 82.7k
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
10
votes
6
answers
700
views
Tiling with similar tiles
Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...
- 5,979
10
votes
2
answers
523
views
When does every point in a polytope lie along a chord between its edges?
Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...
- 329
10
votes
5
answers
960
views
Is this an instance of any existing convex pentagonal tilings?
Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
- 151
10
votes
5
answers
834
views
Tessellating $\mathbb{R}^n$ by bricks.
Let us call the $\ell_1$-product of intervals $[0,k_1]\times...\times [0,k_n]$ a brick of size $k_1+...+k_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point ...
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