All Questions
109 questions
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
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 ...
- 7,159
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
3
answers
935
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
20
votes
1
answer
452
views
Hidden points in polygons
Let $h(n)$ be the largest number of mutually invisible points that can be located in a
polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment
$xy$ contains a point ...
- 151k
17
votes
4
answers
1k
views
Can I build infinitely many polytopes from only finitely many prescribed facets?
Given a finite set of convex $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.
Question: Is it true that there are only finitely many different convex $(d+1)$-dimensional polytopes whose ...
- 13.6k
16
votes
4
answers
2k
views
Point sets in Euclidean space with a small number of distinct distances
It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
- 1,314
16
votes
2
answers
466
views
Does a certain points and lines configuration exist?
For which $n$ we may mark $n$ red and $n$ blue points on the Euclidean plane, not all on a line, so that any line which passes through two points of different colour contains another point?
For $n=...
- 109k
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
- 4,922
16
votes
0
answers
298
views
Realization spaces of 3-dimensional polytopes with fixed face areas
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
- 31.2k
15
votes
2
answers
2k
views
Partitioning a Rectangle into Congruent Isosceles Triangles
Is it possible to partition any rectangle into congruent isosceles triangles?
- 404
15
votes
2
answers
885
views
Lattice n-gons with ordered side lengths 1,2,3,...,n
Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8).
Are there other (infinitely many) polygons, such as this, lying entirely in the ...
- 3,707
15
votes
3
answers
1k
views
Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 1,284
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
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
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
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
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
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 ...
user6976
10
votes
2
answers
387
views
What is Kept Fixed for Flexible Spheres
For background to this question much recent exciting related things, see this videotaped lecture by Alexander Gaifullin.
Consider a triangulation $K$ of a two-dimensional sphere and consider maps ...
- 24.7k
10
votes
1
answer
277
views
Optimization of points on a plane
Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...
- 1,129
10
votes
1
answer
300
views
Optimal shape for stabbing balls in $\mathbb{R}^3$
I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region,
say, within a unit-radius sphere $S$.
I shoot a ray/path through $S$, hoping to ...
- 151k
10
votes
1
answer
535
views
Maximum number of Vertices of Hypercube covered by Ball of radius R
Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
- 185
9
votes
3
answers
1k
views
Generalization of Sylvester-Gallai theorem
The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
of points so that a line through every two of them passes through a
third unless they are all on a single ...
9
votes
1
answer
160
views
Hyperplane arrangements whose regions all have the same shape
Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal ...
- 2,753
9
votes
1
answer
665
views
Question about tetrahedron decomposition
Are there tetrahedra which can be subdivided into three non-overlapping parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedra ...
- 93
9
votes
0
answers
144
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
8
votes
1
answer
361
views
Inscribed $n$-polytope with $2^n$ vertices of maximal volume
The question is in the title:
Question: Which inscribed $n$-dimensional polytope (inscribed in the unit sphere) with $2^n$ vertices has the largest possible volume?
Is it the $n$-dimensional cube? ...
- 283
8
votes
3
answers
390
views
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
- 13.6k
8
votes
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
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
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
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
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
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
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
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
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
6
votes
1
answer
429
views
Bichromatic pencils
A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...
- 18.9k
6
votes
2
answers
268
views
Counting valid coordinates
We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in \...
- 323
6
votes
1
answer
388
views
Covering number estimates on closed Riemannian manifolds
Let $(M^n,g)$ be an $n$-dimensional compact and connected Riemannian manifold with sectional curvature bounded above and below by $c,C$. Is it possible/known how to express the external covering ...
- 195
6
votes
0
answers
247
views
An extension of Erdos' distinct distances problem based on circles of various radii
Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the ...
- 24.7k
6
votes
0
answers
1k
views
How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?
Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
- 13.4k
5
votes
4
answers
1k
views
Coloring Points in the Plane
Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...
- 5,275
5
votes
2
answers
523
views
Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.
Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries?
The answer is easy for $k=1$....
- 17.9k