All Questions
718 questions
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
35
votes
5
answers
3k
views
Tiling the plane with incongruent isosceles triangles
It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
...
- 151k
32
votes
1
answer
1k
views
Optimal sphere packings ==> Thinnest ball coverings?
It was proved by Kershner long ago that the thinnest (least density)
covering of the plane by congruent disks can be obtained
by enlarging the radii of the optimal circle packing to just cover
the ...
- 151k
28
votes
0
answers
828
views
Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...
- 151k
27
votes
1
answer
1k
views
Terrible tilers for covering the plane
Let $C$ be a convex shape in the plane.
Your task is to cover the plane with copies of $C$, each under any rigid motion.
My question is essentially: What is the worst $C$, the shape that forces the ...
- 151k
27
votes
3
answers
13k
views
Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...
- 151k
23
votes
4
answers
3k
views
Intrinsic metric with no geodesics
It seems that I have the needed example, but I want it to be simple and self-explaining...
Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing ...
- 45k
21
votes
5
answers
1k
views
Is there a midsphere theorem for 4-polytopes?
The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...
- 151k
19
votes
3
answers
2k
views
Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
- 151k
19
votes
3
answers
6k
views
What are the matrices preserving the $\ell^1$-norm?
So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
- 391
18
votes
1
answer
644
views
Egalitarian measures
A question I got asked I while ago:
If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
- 47.3k
17
votes
0
answers
731
views
Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
16
votes
3
answers
2k
views
Are infinite planar graphs still 4-colorable?
Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a ...
- 151k
16
votes
1
answer
3k
views
3-piece dissection of square to equilateral triangle?
At a workshop it was suggested that it likely remains an open problem
whether or not there is a 3- or 2 -piece
dissection
of a square to an equilateral triangle.
Can anyone confirm that this is ...
- 151k
16
votes
1
answer
589
views
(A question about)${}^3$ 3-dimensional convex bodies
Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...
- 7,276
15
votes
2
answers
589
views
Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
13
votes
1
answer
615
views
Find structure geometry of $A_1, A_2,...,A_n$ such that $\prod_{i<j} A_iA_j$ is maximum
In any triangle we have the well-known inequality:
$$\sin{A}\sin{B}\sin{C} \le \frac{3\sqrt{3}}{8} (1)$$
Signification of inequality (1): Let three points $A, B, C$ lie on a circle then $AB.BC.CA$ ...
- 1,044
13
votes
1
answer
921
views
Limiting shape for Brillouin zones
Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
- 12.3k
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
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
12
votes
3
answers
2k
views
Intuition for Levi-Civita connection via Hamiltonian flows
A Riemannian metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?
- 7,952
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
10
votes
1
answer
517
views
Monotonicity of Loewner ellipsoid?
Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?
I have just finished proving a ...
- 13.5k
10
votes
1
answer
1k
views
CAT(0) groups that does not act on CAT(0) cubical complex
CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
- 1,598
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 ...
- 5,979
9
votes
5
answers
2k
views
Feasibility of a list of prescribed distances in R^3
I am puzzled with the following problem:
Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every ...
- 555
9
votes
2
answers
594
views
Strengthened version of Isoperimetric inequality with n-polygon
Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
- 1,776
9
votes
1
answer
642
views
Twisted random walks
Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=...
- 151k
8
votes
2
answers
591
views
Cutting a spherical surface into mutually non-congruent pieces of equal area
Question: For what values of integer $n$ can the surface of a sphere be partitioned into $n$ convex and mutually non-congruent pieces of same area? (convexity could be viewed as geodesic convexity). ...
- 5,979
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
3
answers
3k
views
Reconstructing an Euclidean point cloud from their pairwise distances
I have a collection of points $P_1, ..., P_N$ in some Euclidean space $\mathbb R^m$ and the coordinates $x_1, x_2, ..., x_N$ respectively associated with them, where $x_i$ is the usual Cartesian tuple ...
- 1,890
7
votes
4
answers
2k
views
Do cotangent bundles have "bounded geometry"?
I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and $\mathbb{R}^n$...
- 71
6
votes
6
answers
3k
views
Circumference of Convex Shapes
Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
6
votes
3
answers
2k
views
Uniquely geodesic and CAT(0) spaces?
Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
5
votes
1
answer
406
views
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
- 867
5
votes
0
answers
177
views
Tiling with triangles of same circumradius and inradius
Consider a pair of positive real numbers $r$ and $R$ with $r<R/2$. Then we can form infinitely many triangles all with circumradius $R$ and inradius $r$.
For any such pair, the resulting triangles ...
- 5,979
4
votes
3
answers
347
views
Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
- 13.6k
3
votes
1
answer
328
views
LP Constraints for Connected Subgraphs of Fixed Size
Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
...
- 13.2k
3
votes
0
answers
76
views
A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
- 5,979
2
votes
1
answer
209
views
Cutting convex regions into equal diameter and equal least width pieces
The diameter of a convex region is the greatest distance between any pair of points in the region.
The least width of a 2D convex region can be defined as the least distance between any pair of ...
- 5,979
2
votes
1
answer
600
views
A geometric approach to the odd perfect number problem?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
- 3,054
1
vote
1
answer
144
views
On convex polygons contained in convex polygons
In what follows '$n$-gon' stands for '$n$-vertex polygonal region'.
Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it.
...
- 5,979
1
vote
2
answers
157
views
A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
- 5,979
97
votes
11
answers
13k
views
Is it possible to capture a sphere in a knot?
You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
- 8,688
96
votes
4
answers
5k
views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
- 4,316
94
votes
5
answers
9k
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
68
votes
2
answers
2k
views
Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
- 629
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
49
votes
5
answers
5k
views
Is Lebesgue's "universal covering" problem still open?
The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ...
- 6,215
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 ...
- 18.7k