All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
8
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432
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What should a meaningful notion of curvature satisfy, in the absence of a smooth structure?
There are many generalizations of various curvatures to non-smooth metric spaces (e.g. Ollivier's Ricci curvature). Suppose I have a metric space $(X,d)$ and I want to define a notion of curvature ...
- 232
8
votes
2
answers
741
views
Lattice Stick Number vs. Stick Number of Knot
Can the lattice stick number of a knot be bounded
by the stick number of the knot?
The stick number
$S(K)$ of a knot $K$ is the fewest number of segments
needed to realize it by a simple 3D polygon....
- 151k
8
votes
1
answer
353
views
Are there any convex pentagonal rep-tiles?
A rep-tile is a shape that can tile larger copies of the same shape.
Question 1: Are there any convex pentagons that are also rep-tiles?
Remarks: 15 convex pentagonal tiles of the plane are known and ...
- 5,979
8
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1
answer
508
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'Fattest' polygons based on diameter and 'least width'
Definitions: The diameter of a convex region is the greatest distance between any pair of points in the region. The least width of a $2$D convex region can be defined as the least distance between any ...
- 5,979
8
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1
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417
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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
8
votes
1
answer
885
views
Maximal tetrahedra inscribed in ellipsoid
Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...
- 151k
8
votes
2
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489
views
Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,194
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2
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278
views
Symmetries of contractable subsets of $\Bbb R^n$
Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...
- 13.6k
8
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1
answer
591
views
Polyhedra that combinatorially shadow a sequence
Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...
- 151k
8
votes
2
answers
621
views
Generalization of Hamiltonian cycles to "Hamiltonian spheres"
One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be
called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$
...
- 151k
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
0
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149
views
Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
8
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0
answers
123
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Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?
A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...
- 13.6k
8
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0
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249
views
Approximating a general rectangle partition by a guillotine partition
There is a rectangle $R$ partitioned into some axes-parallel rectangles:
The goal is to construct another partition of $R$ into rectangles, using only guillotine cuts.
That is: cut $R$ into two ...
- 3,549
8
votes
0
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544
views
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
Let R be a region on a hypersphere. Each point A of the hypersphere
is associated with a vector pointing to A and with origin at
the centre of the hypersphere. So let me identify each point with a
...
- 1,207
8
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0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
7
votes
3
answers
866
views
Not quite regular polyhedra
Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...
- 1,314
7
votes
1
answer
293
views
Can a tangle of arcs interlock in plane?
This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here:
Can a (finite) collection of disjoint circle arcs in $\...
- 4,312
7
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1
answer
209
views
Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)
A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...
- 1,441
7
votes
2
answers
907
views
Is there a 3d equivalent of this picture?
This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
The picture below is the image of a regular ...
- 1,125
7
votes
1
answer
248
views
Decidability of completing Penrose tilings
Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
- 509
7
votes
3
answers
2k
views
Partitioning a rectangle into different isosceles triangles
After all the discussion raised by this old question, I am wondering about a somewhat complementary one:
For any given rectangle, does there exist a finite set of pairwise different isosceles ...
- 13.4k
7
votes
3
answers
1k
views
How can we count lines in an n-x-n rectangular array?
Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
- 79
7
votes
1
answer
318
views
Finding a short path using $(0.99n)!$ permutations
Suppose I have $n$ points $x_1,\dots,x_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a ...
- 4,049
7
votes
1
answer
938
views
Which knots' stick numbers are twice their crossing numbers?
Looking at a table of minimum stick numbers for knots (table here),
it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$
is realized by the trefoil $3_1$—it ...
- 151k
7
votes
1
answer
137
views
Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?
Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of $\triangle ...
- 151k
7
votes
1
answer
768
views
To minimize the Hausdorff distance between convex polygonal regions
Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given two convex polygonal regions P1 and P2 on the ...
- 5,979
7
votes
2
answers
453
views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
- 679
7
votes
3
answers
805
views
Wrapping a convex polyhedron with string
This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P \...
- 151k
7
votes
2
answers
393
views
Partitioning convex polygons into triangles of equal area and perimeter
This post is based on https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter, On a possible variant of Monsky's theorem and Cutting convex polygons into ...
- 5,979
7
votes
1
answer
311
views
Open covering of $S^n$ by sets not containing antipodal points
Given an $n$-dimensional sphere $S^n$ and an open cover such that none of the open sets contain antipodal points, does there exist a point on $S^n$ that belongs to at least $n+1$ open sets from the ...
- 71
7
votes
1
answer
439
views
Integral straight-line embeddings of planar graphs
Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
- 151k
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k
7
votes
1
answer
498
views
Is there a bicyclic irregular pentagon in integers?
Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well?
If it does ...
- 71
7
votes
1
answer
153
views
Above/below directed graph on cells of arrangement of lines
This question concerns the structure of a directed graph
built on the cells of an arrangement of lines.
My basic question is whether this graph has been
studied before, perhaps in another guise. I ...
- 151k
7
votes
1
answer
550
views
Approximating a real by a ratio of primes
Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$.
I seek smallest primes $p$ and $q$ such that
$$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$
Q. What upper bound $u(...
- 151k
7
votes
0
answers
316
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
7
votes
0
answers
162
views
Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
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0
answers
227
views
Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
- 151k
7
votes
0
answers
254
views
Set of unit vectors such that among any three there is an orthogonal pair
I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as:
Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
- 105k
7
votes
0
answers
201
views
Minimizing energy on $\mathbb{S}^2$ for absolutely monotonic type potentials
For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a ...
- 3,209
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0
answers
122
views
Discrepancy of the finite approximation of the Lebesgue measure
Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
- 109k
7
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0
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209
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Stabbing disks in space, or: Galactic alignment
I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...
- 151k
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
6
votes
3
answers
705
views
Minimum space dimension to place n-points knowing pairwise distances
Let $P$ be a set of $n$ points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those $n$ points with respect to ...
- 63
6
votes
2
answers
544
views
On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
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
4
answers
2k
views
Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
6
votes
1
answer
508
views
How many triangulations of a regular octahedron are there, without introducing new vertices?
It is easy to find three triangulations, each consisting of four tetrahedra. Are there more?
- 61