Questions tagged [discrete-geometry]
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
1,819
questions
6
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Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
2
votes
1
answer
122
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On equal area planar sections of 3D convex bodies
This is an extension of On segments of equal area cut from planar convex regions by chords.
While the 3D analog of the above question would be about 3D pieces cut from a convex body $C$ by planes, ...
2
votes
0
answers
136
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On segments of equal area cut from planar convex regions by chords
Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...
0
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0
answers
20
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Smallest Semi-ellipses that contain a convex n-gon
Definition: Let us define a semi-ellipse as either of the two halves into which an elliptical region is cut by any line that passes through its center. Obviously, all semi-ellipses that come from any ...
2
votes
0
answers
60
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Degeneracy and the "Linear Degeneracy Testing" problem
The Affine Degeneracy problem is about deciding whether $n$ given points in $\mathbb{R}^d$ (or $\mathbb{Q}^d$) are "in general position". i.e. there is no $d+1$ tuple of points which lies in ...
0
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0
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46
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Optimal packing and covering of a triangle with squares
We continue from Another variant of the Malfatti problem.
Given a triangle T and a number n, how to cover it with n squares (of possibly different dimensions) such that the sum of the areas (...
16
votes
0
answers
560
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Snakes on a plane
A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...
3
votes
1
answer
185
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Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix
Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...
1
vote
1
answer
62
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Covering convex regions with disks optimizing on area and perimeter
Question: Are there planar convex regions $R$ and integers $n$ with the property: if $R$ is covered by $n$ disks of possibly different sizes such that (1) the total area of the covering disks is ...
2
votes
0
answers
76
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Another variant of the Malfatti problem
We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
4
votes
2
answers
189
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Algorithm for grouping tetrahedra from Voronoi diagram
I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
5
votes
0
answers
71
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Is the choosability/list chromatic number of a circular arc graph equal to its chromatic number?
In 2003, Prowse and Woodall proved that for graphs $C_n^k$ which are powers of cycles,
$$\chi_\ell(C_n^k) = \chi(C_n^k).$$
They conjectured that this equality holds for the broader class of graphs ...
1
vote
0
answers
41
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Does a substitution tiling being FLC depend on starting seed?
I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
1
vote
0
answers
63
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Detecting points inside the convex hull with inner products
Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space.
Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\...
4
votes
1
answer
266
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Billiard circuits in pentagons
A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...
1
vote
1
answer
106
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Recognizability/unique composition property for substitution tiling
This may be a very basic question, but I have not found an answer to it so far in my search. The question is whether there is an "algorithmic" way to check unique-composition/recognizability ...
1
vote
0
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131
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A comparison between packing and covering as classes of problems
We continue from Bounds for the Dispersal Problem in convex regions and Bounds for minimax facility location in a convex region
Let us consider the classes of problems:
Given a convex region $R$ and ...
0
votes
0
answers
61
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Bounds for the Dispersal Problem in convex regions
We add a bit to: Bounds for minimax facility location in a convex region
Two earlier posts: Cutting convex regions into equal diameter and equal least width pieces - 2 and Facility location on ...
1
vote
0
answers
139
views
Ways of proving that a framework is locally rigid
Given a (bar-and-joint) framework/linkage, I would like to know what are possible ways of showing that the framework is locally rigid. Also, what is known about the computational complexity of ...
1
vote
0
answers
85
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Bounds for minimax facility location in a convex region
An earlier question: Facility location on manifolds
A possibly related earlier post: Cutting convex regions into equal diameter and equal least width pieces - 2
The minimax facility location problem ...
3
votes
0
answers
136
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Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
8
votes
1
answer
419
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When can $L$ sets of the form $\{a,b,a+b\}$ partition $\{1,2,\dots, 3L\}$?
Firstly, this question has been posted to Math StackExchange with no complete answer so far.
Consider a set of the form $\{a,b,a+b\}$ where $a$ and $b$ are positive integers with $b > a$. I will ...
2
votes
2
answers
347
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Sphere tessellation with congruent regular hexagons except finitely many
Let $S^2$ be the 2 dimensional sphere embedded in $\mathbb{R}^3$. It is well known, using the Euler characteristic, that it is not possible to tessellate $S^2$ with all congruent regular hexagons.
My ...
8
votes
0
answers
105
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Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
12
votes
0
answers
160
views
Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
3
votes
2
answers
188
views
Random walk to visible lattice points
Consider a random walk from the $\mathbb{Z}^2$ origin $(0,0)$ to visible
(not blocked) lattice
points $p$, with a parameter $r$ a given radius of a circle centered
on $p$.
With $p$ the previous point, ...
3
votes
1
answer
353
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Illumination from visible lattice points with inverse square intensity
It is well known that the number of $\mathbb{Z}^2$ lattice points visible from
the origin is $6/\pi^2$, about $61$%.
See, e.g.,
What fraction of the integer lattice can be seen from the origin?.
I am ...
9
votes
2
answers
475
views
Connected geometric thickness two
A graph $G = (V,E)$ has geometric thickness two if there exists an embedding $\varphi: V \rightarrow \mathbb{R}^2$ and a decomposition $E = E_1\cup E_2$ such that $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ ...
9
votes
4
answers
410
views
Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
4
votes
1
answer
280
views
Does Kalai's $3^d$ conjecture hold for simplicial spheres?
Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...
3
votes
0
answers
114
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Divide Euclidean space by surfaces
It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is
\begin{equation}
1 + n + C^2_n + \cdots + C^k_n
\end{equation}
Is there similar ...
4
votes
2
answers
157
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On the number of intersection points between a curve and a (horizontal, vertical) line inside a unit square
I am looking for a reference (name, source) of the following elementary geometric combinatorial problem:
Inside a unit square, given a smooth curve of length $L$. Then there exists a horizontal or ...
0
votes
0
answers
111
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Frameworks in general position that are locally rigid but not infinitesimally rigid
The classical theorem of Asimow and Roth says that for a generic framework (i.e., coordinates of the nodes are algebraically independent), local rigidity and infinitesimal rigidity are equivalent. I ...
2
votes
1
answer
94
views
The problem of finding the smallest number of copies of a certain shape that can be placed into a space to make fitting another copy impossible
Packing problems often ask for the largest number of some identical shape that can fit in a given space without overlapping, if they are placed optimally.
I'm interested in the opposite question:
Q. ...
4
votes
0
answers
109
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Advice on results for balls on regular $N$-dimensional grids
I have obtained some results regarding balls on regular $N$-dimensional grids. I would like expert opinion on wether the results are significant or interesting enough for (trying to) publish them in a ...
0
votes
0
answers
165
views
Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter. We ignore the case of $(m_1,\cdots,m_k)=n$.
I am ...
0
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0
answers
41
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3d convex body with max kissing number with translates
Ref: Convex region $C$ with least kissing number of copies of $C$
Given a convex body $C$, let us define its 'translate kissing number' $k_t$ as the largest possible number of translated copies of $C$ ...
1
vote
0
answers
37
views
The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
43
votes
0
answers
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Can a regular icosahedron contain a rational point on each face?
The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces?
For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
3
votes
0
answers
90
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
0
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0
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89
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Which polytopes can be folded to an edge?
While playing with bar-and-joint linkages, I noticed that the skeleton of a regular 3-dimensional cube can be folded to a single edge (this can be achieved by first flexing the cube to bring it to a ...
1
vote
0
answers
44
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Which rectangles can be cut into mutually non-congruent rectangles all of same diagonal length?
This earlier post asks, among other things, whether the plane can be tiled with mutually non-congruent rectangles all of which have same length of diagonal: Tiling the plane with mutually non-...
2
votes
0
answers
99
views
Minimum number of points on sphere which cannot be covered by three double caps
What is the minimum number of points on the sphere $S^d \subset \mathbb{R}^{d+1}$ which cannot be covered by $d+1$ double caps? A double cap is defined to be a set $\{x \in S^d: |\langle x,a \rangle| &...
3
votes
1
answer
100
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Does a matroid base polytope contain its circumcenter?
Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the ...
4
votes
0
answers
78
views
Which rectangles can be cut into finitely many rectangles all with same perimeter and different areas?
Ref 1: dividing a square into unique rectangles with the same perimeter
https://arxiv.org/ftp/arxiv/papers/1307/1307.3472.pdf
Ref 1 asks if a square can be cut into some finite number of rectangles ...
0
votes
0
answers
30
views
Convex 3d bodies for which all planar sections with max diameter have same diameter
Ref:
1.
A claim on planar sections of 3D convex bodies
On convex 3d bodies whose shadows are all of constant diameter
Given a 3D convex body $C$ and a specified direction $n$, we consider the planar ...
3
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0
answers
89
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On convex 3d bodies whose shadows are all of constant diameter [closed]
We add a bit to More on shadows of 3D convex bodies
By a shadow of a 3D body, we mean the orthogonal projection of it onto a 2D plane.
If all shadows of a convex 3D body have the same diameter, will ...
10
votes
1
answer
489
views
A projective plane in the Euclidean plane
Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...
0
votes
1
answer
48
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On largest convex m-gons contained in a given convex n-gon where m < n
This post is the inside-out variant of On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, how to find the max area convex m-gon Q ...
1
vote
0
answers
121
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The number of incidences between points and parabolas on $\mathbb{R}^2$
I was reading Adam Sheffer's book "Polynomial Methods and Incidence Theory" and I tried to solve the following exercise:
Exercise 1.1 Construct a set $\mathcal{P}$ of $m$ points and a set $\...