All Questions
Tagged with reference-request discrete-geometry
174 questions
3
votes
0
answers
391
views
Dissection of a polygon into convex polygons
Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons.
I would be very grateful for any information on this problem.
Remark 1. There ...
- 935
5
votes
1
answer
398
views
Ham sandwich theorem for discrete measures - reference request
A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) ...
- 1,163
21
votes
5
answers
5k
views
What arrangement of unit cubes minimizes surface area?
For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...
- 7,831
1
vote
0
answers
61
views
Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances
To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view.
Examples are:
the diagonals of the convex hull of a set of points in the euclidean plane; ...
- 13.2k
6
votes
1
answer
587
views
Study of convex polytopes via commutative algebra
Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...
- 211
5
votes
2
answers
373
views
Genus of Tutte-Coxeter Graph
What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...
user48028
4
votes
2
answers
1k
views
Sphere - Symmetry and Triangulation [closed]
The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...
- 1,256
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
2
votes
2
answers
284
views
Three questions concerning lattice points on sphere surfaces
Pardon my ignorance of this topic.
Q1.
In which dimensions $d$ is it the case that, for every natural number $n$,
there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-...
- 151k
3
votes
2
answers
348
views
Request for some references exploring the connections of Riemann surfaces with medical imaging
I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical imaging/...
- 1,467
12
votes
2
answers
2k
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Fold-and-cut problem in three dimensions
The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...
- 851
10
votes
1
answer
426
views
Complexity of the union of randomly rotated unit cubes
It is a remarkable fact that the union of congrent cubes
has only at most near-quadratic combinatorial complexity,
$O^*(n^2)$ for $n$ cubes, known to be almost tight.
This contrasts with the union of ...
- 151k
6
votes
1
answer
666
views
Pick's Theorem for rational points of bounded height
I wonder if the various lattice-point theorems, such as
Pick's Theorem or
Minkowski's Lattice Theorem,
have been generalized to the collection of points
with rational coordinates no more than height ...
- 151k
6
votes
1
answer
276
views
Matching on sphere to create cycle with chords
Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like to ...
- 151k
3
votes
1
answer
292
views
Existence of Simple Closed Straightest Geodesics
There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...
- 133
15
votes
0
answers
477
views
Expanding disks lead to what packing of the plane?
Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...
- 151k
5
votes
0
answers
1k
views
N-balls covering n-balls
This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
- 151
3
votes
1
answer
1k
views
Regularity of Delaunay triangulation of a hypercube
First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations:
(A)
(B)
We say the lower triangulation is more "regular" than upper ...
- 339
13
votes
1
answer
933
views
Drawings of complete graphs with $Z(n)$ crossings
Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor \left\lfloor\frac{n-1}{...
- 6,101
11
votes
3
answers
1k
views
Combinatorial distance between simplicial complexes
Let $K_1$ and $K_2$ be two simplicial complexes.
I am seeking a measure of the distance between $K_1$ and $K_2$ when
viewed as combinatorial objects.
What I have in mind is something like this.
...
- 151k
4
votes
1
answer
323
views
What properties does generalized Delaunay triangulation have?
Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
- 19k
4
votes
1
answer
646
views
Combinatorial geodesics
[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics &...
- 9,720
17
votes
1
answer
1k
views
The optimal constant in Vitali covering lemma
Let me restate Vitali covering lemma.
Let $\{B_i\}_{i\in F}$ be a finite collection of balls in the $\mathbb{R}^n$. Then there is $S\subset F$ such that the balls $\{B_i\}_{i\in S}$ are disjoint and
...
- 293
1
vote
0
answers
193
views
Lattice-point enumeration question involving linear combinations of matrices
I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, \...
- 170
17
votes
1
answer
458
views
The sparsest planar net that captures every unit segment
Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...
- 151k
14
votes
1
answer
819
views
The geometry of crinkled aluminum foil
I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...
- 151k
1
vote
2
answers
204
views
Disks Packing Variant
Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...
- 57
0
votes
3
answers
512
views
The symmetry group of $\mathbb Z^d$
Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
- 8,512
3
votes
1
answer
356
views
Empty convex polytopes for random point sets
I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane
(the Happy-Ending Problem), and I know that there are higher-dimensional extensions.
A great source (...
- 151k
13
votes
1
answer
430
views
Detecting a hidden convex body with line probes
Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...
- 151k
4
votes
2
answers
506
views
Empty lattice simplex or White's theorem
White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...
- 12.3k
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
0
votes
1
answer
91
views
When do there exist locally regular embeddings of regular graphs?
I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more ...
- 2,041
16
votes
6
answers
2k
views
Optimal pebble-packing shape
Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes.
Q. ...
- 151k
52
votes
5
answers
2k
views
Tetris-like falling sticky disks
Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...
- 151k
2
votes
2
answers
676
views
Midpoint lattice polygons
Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of ...
- 151k
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
4
votes
2
answers
482
views
Generators of a 2D lattice
Dear MO_World,
I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are ...
- 23.2k
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 $P \...
- 151k
24
votes
1
answer
2k
views
Building a genus-$n$ torus from cubes
I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting object ...
- 151k
34
votes
6
answers
8k
views
Covering a unit ball with balls half the radius
This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
- 151k
5
votes
2
answers
441
views
Touching-tetrahedra graphs
Have the graphs representable by touching tetrahedra been explored?
Let $\cal T$ be a collection of tetrahedra in $\mathbb{R}^3$
with pairwise disjoint interiors.
Define a graph $G_{\cal T}$ to have ...
- 151k
13
votes
1
answer
3k
views
What nets fold to polyhedra?
There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
- 1,314
3
votes
1
answer
394
views
Min Bend Orthogonal Knots
I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...
- 151k
3
votes
1
answer
853
views
Transience of self avoiding random walks on $\mathbb{Z}^d$
I'm finishing up a masters thesis in computer science and want to say a bit in the introduction about self-avoiding walks. My thesis looks at a random process which arose in computer science and my ...
- 30.3k
14
votes
3
answers
2k
views
Optimal wireframe sphere
Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...
- 151k
10
votes
0
answers
1k
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Interpolating points with minimum curvature constraint
I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...
- 151k