The sphere-packing tag has no wiki summary.
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Generalized Sphere Kissing Problem
I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...
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2answers
269 views
Are there locally jammed arrangements of spheres of zero density?
I know of a remarkable result from a paper of
Matthew Kahle (PDF download), that there are arbitrarily low-density
jammed packings of congruent disks in $\mathbb{R}^2$:
In 1964 Böröczky used
a ...
3
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2answers
123 views
Techniques for showing optimality of given packing
There are some natural packing problems that have been asked in mathematics. Some of them are:
1)How many balls can be placed with in a cube?
2)How many equidistant points can be place on the ...
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0answers
71 views
Formal definitions for a few lattice packing invariants
I'm a bit hesitant to post this question here because it regards definitions that are likely trivial, and please note that I'm cross-posting this question from Math.Stackexchange (posted nine days ...
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0answers
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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 ...
2
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1answer
158 views
Is there an “accepted” jamming limit for hard spheres placed in the unit cube by random sequential adsorption?
I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" ...
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2answers
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Kissing Number of Spheres in Non-Euclidean Geometry
There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ ...
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0answers
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Computing the Volume of Closed 3-Manifolds and the Geometrization Conjecture
My question is whether or not if I generalize Theorem 2(i) of "Contact Graphs of Unit Sphere Packings Revisited" [2012] by K. Bezdek and S. Reid (arXiv link) which states
The number of touching ...
2
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1answer
166 views
Packing a closed 3D surface with non-overlapping spheres starting with the largest possible one and then working the way down
Let's say, I have a closed 3D surface (say, the surface of a pebble). I want to pack it with spheres, but starting with the largest possible sphere, then the next largest possible non-overlapping ...
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0answers
647 views
How many unit balls can be put into a unit cube?
Here a unit ball is a ball of diameter 1, and a unit cube is a cube of edge length 1.
A famous counterintuitive fact is that, as the dimension increases, the volume of the unit ball tends to zero ...
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1answer
385 views
Optimal fitting of spheres in a cylinder.
how to find the minimum height and width of a cylinder containing n identical spheres?
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2answers
682 views
The kissing number of a square, cube, hypercube?
How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.
...
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0answers
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Which term is better for the so called “sphere packing”?
I'm working on sphere packings. When I write, I'm confused with basic definitions. I'm hesitating between the terms "sphere", "ball" or "oriented sphere".
For example, on the wikipedia page of circle ...
2
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1answer
258 views
What are some properties of Delone sets that come from Barlow packings of spheres?
Given a Barlow packing of $\mathbb{R}^n$ by balls with at most a finite number of different radii, the centers of the balls will form a Delone set in $\mathbb{R}^n.$
For a highest density sphere ...
4
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1answer
188 views
Inter-Kissing Number for Spheres of Different Sizes
What is the maximum number of spheres that can be placed in 3D such that all inter-touch?
One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the
middle, so this ...
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0answers
269 views
maximal minimum distance in a sphere packing
Hi everyone.
I need to pick a set of 65 points p(x,y,z) in a 3D space of 274625 points; as
the set picked should provide the maximum possible minimum Hamming distance.
(Consider the Hamming bound for ...
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4answers
723 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. ...
32
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6answers
962 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, ...
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6answers
2k 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 radius ...
10
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1answer
245 views
Identifying lattices
I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known ...
11
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2answers
460 views
What is the largest possible thirteenth kissing sphere?
It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...
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5answers
680 views
Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?
The Koebe–Andreev–Thurston theorem states that any planar graph can be represented
"in such a way that its vertices correspond to disjoint disks, which touch if and only if
the corresponding vertices ...
12
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2answers
648 views
Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
6
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5answers
515 views
covering by spherical caps
Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...
