All Questions
Tagged with reference-request discrete-geometry
174 questions
5
votes
2
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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 ...
- 5,259
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
4
votes
2
answers
482
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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 ...
- 12.3k
4
votes
2
answers
1k
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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 ...
- 7,276
3
votes
1
answer
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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 ...
- 1,502
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 $(d{-}1)$-...
- 6,210
1
vote
0
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193
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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
6
votes
1
answer
666
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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 ...
- 12.9k
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 ...
- 1,083
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
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 ...
- 91.8k
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 ...
- 23.3k
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 (...
- 28k
24
votes
2
answers
1k
views
A geometric Ramsey problem
The following problem seems like one to which the answer could well be known: if so, I'd be interested to have a reference.
How large does n have to be such that among any n points in the plane you ...
- 1,319
0
votes
1
answer
91
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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 ...
- 45k
3
votes
1
answer
853
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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
4
votes
0
answers
189
views
Slices of Simplices that are Simplices, Reference?
I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...
- 41
12
votes
2
answers
1k
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Helly theorem + Nerve
Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not ...
CommunityBot
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8
votes
0
answers
358
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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-...
3
votes
0
answers
142
views
Dimension of convex arrangements for hypergraphs
Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...
- 12.6k
5
votes
2
answers
333
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Characterization of combinatorial manifolds in terms of links
I need to reference the following result. Do you know a good source?
The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent:
a) $S$ is an $n$ manifold;
b) The link of ...
- 5,019