The sphere-packing tag has no wiki summary.
15
votes
2answers
265 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
votes
2answers
120 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 ...
0
votes
0answers
69 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 ...
3
votes
0answers
353 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 ...
2
votes
1answer
153 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" ...
5
votes
2answers
232 views
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 $1$ ...
2
votes
0answers
151 views
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
votes
1answer
154 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 ...
10
votes
0answers
606 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 ...
0
votes
1answer
350 views
Optimal fitting of spheres in a cylinder.
how to find the minimum height and width of a cylinder containing n identical spheres?
23
votes
2answers
664 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.
...
1
vote
0answers
184 views
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
votes
1answer
256 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
votes
1answer
183 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 ...
1
vote
0answers
258 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 ...
9
votes
4answers
707 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
votes
6answers
958 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, ...
19
votes
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
votes
1answer
244 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
votes
2answers
451 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 ...
8
votes
5answers
659 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
votes
2answers
627 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 ...
5
votes
5answers
492 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 ...
