All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
3
votes
0
answers
98
views
Convex region $C$ with least kissing number of copies of $C$
Given a 2D convex region $C$, let us define its kissing number $K_0$ to be the largest possible number of copies of $C$ that can be arranged around a central copy of $C$ (call this $C_0$) and touching ...
- 5,979
5
votes
0
answers
139
views
On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
- 5,979
1
vote
2
answers
153
views
Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
- 5,979
3
votes
1
answer
473
views
On 4 random points in a rectangle [closed]
Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
How would I attack this problem? The goal is to ...
- 133
2
votes
1
answer
426
views
Minkowski sum, zonotopes, convex hull
For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, ...
- 29
3
votes
1
answer
484
views
On some infinite planar arrangements with triangles
Background: Given a convex region C. One can define a graph corresponding to a planar arrangement of non overlapping congruent copies of C - each unit C is a node and an edge connects it to another ...
- 5,979
6
votes
1
answer
928
views
To find the Largest Regular n-gon contained in a given convex region
Given a general convex region C, to find the largest regular polygon that is contained in it (shared boundaries allowed). Basically, one needs to find that particular value of n for which a regular n-...
- 5,979
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
2
votes
1
answer
504
views
Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
- 5,979
6
votes
1
answer
680
views
Napkin Folding Problem / Rumpled Ruble Problem
I am an outsider to this group. I'm a journalist and am working on a piece about theoretical math/geometry. Simply put, when a napkin is folder in such a way to increase its perimeter is that strictly ...
- 69
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
10
votes
2
answers
375
views
How many small dots can be drawn in a region such that no three are "collinear"?
When people draw dots on paper, they are actually not points, but small regions filled with ink. Suppose that each dot has disc-shape with fixed radius $r\ll 1$ and must be drawn inside (1) a square ...
- 909
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
6
votes
1
answer
631
views
On covering convex 2D regions with rectangles
Given a convex 2D region $C$ and a positive integer $N$. We need to cover $C$ with $N$ rectangles such that the sum of the areas of the $N$ rectangles is the least – no further constraints on the ...
- 5,979
2
votes
4
answers
997
views
Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
- 1,826
6
votes
1
answer
178
views
Point distributions in unit square which minimize E[1 / distance]
Choose $n$ points $p_1,\ldots,p_n$ in the unit square $[0,1]^2\subset\mathbb{R}^2$
such that $D:=\mathop{\sum}\limits_{1\le i<j\le n}\frac{1}{dist(p_i,p_j)}$ is minimized, where $dist(p_i,p_j)$ is ...
- 71
3
votes
0
answers
310
views
Upper bound on the number of lattice points on the intersection of a hyperplane and a sphere
Let $R>0$, $\overrightarrow{\alpha} \in \mathbb{R}^{d}$. Consider the intersection $T$of $RS^{d-1}$ and the hyperplane $\overrightarrow{\alpha} \cdot \overrightarrow{x} = n$. What is the best known ...
- 461
3
votes
0
answers
137
views
Aperiodic tile with rational area
Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
- 745
9
votes
1
answer
338
views
Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
- 151k
2
votes
0
answers
246
views
Recover unknown vectors with dot-product queries
Suppose there are $n$ unknown unit vectors in $\mathbb{R}^d$,
$V=\{v_1,\ldots,v_n\}$, no two identical.
Your task is to determine the vectors in $V$.
The only tool at your disposal is to query a ...
- 151k
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
3
votes
2
answers
179
views
Number of bitangents to convex polytopes
Let me state my question prior to defining terms:
Q. Let $P_1$ and $P_2$ be disjoint convex polytopes
in $\mathbb{R}^d$ of $n$ vertices each.
What is the maximum number of distinct bitangent
...
- 151k
2
votes
0
answers
69
views
Polygons such that $n^2 $ times magnification of a polygon could be covered by exactly $n^2$ original polygon
While studying about covering problems in combinatorics, I got to a simple question:
What polygons can be covered exactly, without any area that is covered twice or area that is outside the covered ...
- 103
1
vote
1
answer
209
views
Is a polytope with vertices on a sphere and all edges of same length already rigid?
Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...
- 13.6k
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
2
votes
1
answer
110
views
A questions concerning Laguerre/Voronoi tessellations
Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by
$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...
user128095
3
votes
2
answers
344
views
Is a vertex- and edge-transitive polytope already a uniform polytope?
I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...
- 13.6k
3
votes
0
answers
142
views
Can bellows make loops?
Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
- 4,600
1
vote
1
answer
648
views
How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
I don't need to fill the sphere with equidistant points. I just need less than a ...
- 11
18
votes
2
answers
573
views
Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
- 13.6k
2
votes
0
answers
98
views
8-partition of a planar convex body by 4 concurrent lines
It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...
- 151k
1
vote
1
answer
230
views
A possible characterization of the cube?
Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$,
and $|P|$ the sum of the Euclidean lengths of the edges of $P$.
Let $P_1, P_2, P_3$ be the perpendicular projections of $P$
...
- 151k
3
votes
2
answers
323
views
Minimum weight triangulation of lattice points in a circle
Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
- 151k
1
vote
0
answers
69
views
Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?
General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
- 13.6k
2
votes
1
answer
428
views
Name and Algorithms for a Sparsest Circle Packing
The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, ...
- 13.2k
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
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
1
vote
3
answers
229
views
Repeatedly halve and twist a planar shape: Limiting shape?
Consider the following iterative process.
Start with a planar region $R=R_0$ of $\mathbb{R}^2$.
I am thinking of $R$ as connected,
but it may become disconnected.
In the example below, $R$ starts as ...
- 151k
6
votes
3
answers
365
views
Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
- 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
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 ...
- 19k
5
votes
0
answers
508
views
Longest simple path through hypercube corners
This is a variation on a previously answered question,
Longest path through hypercube corners.
Here I am seeking the longest simple (non-self-intersecting) path through
the unit hypercube's vertices,
...
- 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
3
votes
1
answer
118
views
Question arise from kissing number in 2 dimension
I'm considering an extended problem of kissing number in $\mathbb{R}^2$.
Suppose I have a given disc $\mathcal{D}$ of radius 1/2 and infinitely many discs all of radius 1/2 and all these discs and ...
- 1,495
6
votes
0
answers
164
views
Sets of points avoiding small angles
(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...
- 151k
3
votes
1
answer
206
views
Separating points of shifts of a finite set in the plane
Let $A\subset \mathbb{R^2}$ be a finite set such that $|A|=k^2$. Let $x_i\in \mathbb{R^2}$, $i=1,2,3,4$, be four points in the plane in general position (no three lie on any line).
Let us form the ...
- 2,288
1
vote
0
answers
196
views
Squares as sum of squares
For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?
- 3,707
3
votes
0
answers
351
views
Properties of inverse Cayley-Menger matrices
in the online article A formula for the N-circumsphere of an N-simplex dated April 2013, G. Westendorp provides an interpretation of the entries of inverse of Cayley-Menger matrices $\hat{B}$, that ...
- 13.2k
3
votes
0
answers
214
views
Volume of intersection of a ball and cube with arbitrary position in $n$ dimension
Let $ A(n, r, x) = B^n_r(x) \cap [0,1]^n $ denote the intersection between an $n$ ball $B^n_r(x)$ with arbitrary radius $r$ and arbitrary center $x \in \mathbb{R}^n$ that intersects a unit $n$ cube $ [...
- 365
5
votes
2
answers
267
views
A Voronoi Iteration Game
Let $P_i$ be a set of points in the plane and $P_{i+1}$ the corners of the Voronoi diagram of $P_i$. Start with some $P_1$ and iterate away. What happens? Can you choose a $P_1$ so the iteration goes ...
- 4,829