Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [sphere-packing]

The tag has no usage guidance.

22
votes
1answer
408 views

Is there a short proof of the decidability of Kepler's Conjecture?

I've believed that the answer is "yes" for years, as suggested in various sources with reference to Tóth's work. For example, the Wikipedia article for Kepler Conjecture says: The next step toward ...
14
votes
2answers
405 views

The Disco Ball Problem

Let me first give some of a background as to where I got this problem. I had a math teacher ask me a few months ago: "How many 1 unit by 1 unit squares could one fit on a sphere with a radius of 32 ...
5
votes
0answers
141 views

Question related to high dimensional kissing number

I have a question related to the kissing number in $n$ dimension. Suppose we have many non-overlapping $n$-dimensional balls of radius $1/2$. We place one of the $1/2$-radius ball centered at the ...
3
votes
1answer
102 views

Question arise from kissing number in 2 dimension

I'm considering an extended problem of kissing number in $\mathbb{R}^2$. Suppose I have a given disc $\mathcal{D}$ of radius 1/2 and infinitely many discs all of radius 1/2 and all these discs and ...
2
votes
1answer
52 views

Maximal Vertex Degree of MSTs in Euclidean Spaces

Are there any Euclidean spaces, in which the maximal vertex degree of MSTs (Minimum Spanning Trees) of a finite set of points and edge weights equal to Euclidean distance, isn't equal to the kissing ...
0
votes
0answers
78 views

Sphere volume covering (disc covering in 3 dimensions)

http://mathworld.wolfram.com/DiskCoveringProblem.html Disc covering is a well known problem - find the placement and size of the smallest N discs that will fully cover a larger unit disc. Is there ...
3
votes
0answers
37 views

Bound on local packing density of 2D Delaunay cell

What is the history of the result that in a packing of the plane by unit disks, no Delaunay cell can be occupied by disk-sectors whose total measure exceeds $\pi/\sqrt{12}$ times the area of the cell? ...
6
votes
0answers
68 views

Packing points in a lattice

Let $L$ be the square or triangular lattice in the plane, with nearest neighbors having distance 1. Has anyone studied the problem of finding the maximum (okay, supremum) density achieved by a subset ...
1
vote
0answers
184 views

Sphere packings with antipodal (unequal) spheres

Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
5
votes
1answer
414 views

Randomly covering a sphere

Let $S$ be the $n$-dimensional unit sphere in the Euclidean space. Further, let $X_1,\ldots,X_k$ and $Y_1,\ldots,Y_m$ be iid $S$-valued random variables with common (unknown) distribution $\mu$. With $...
2
votes
0answers
44 views

Packing net of simplex

For given $d$, we can define the simplex as follows, $S=\{(x_1,x_2,\cdots,x_d):x_1\geq x_2\geq \cdots\geq x_d\geq 0,\sum x_i=1\}$. We can define the distance on $S$ as $L_1$ distance. An $\epsilon$ ...
6
votes
1answer
304 views

Upper bound of the kissing number in n dimensions

In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere. Let $\tau_n$ be the kissing number ...
2
votes
1answer
182 views

Packing number of $\ell_1$ ball in $\ell_{\infty}$ metric

Consider the $d$-dimensional $\ell_1$ ball $\mathbb B_d=\{x\in\mathbb R^d: \|x\|_1\leq 1\}$, where $\|x\|_1=\sum_{i=1}^d{|x_i|}$. I'm interested in the maximum size of the (finite) subset $S\subseteq\...
2
votes
1answer
238 views

What high dimensional lattices have Voronoi cells that have this property?

Which high-dimensional lattices (particularly $Z_n^*,D_n,D_n^*,A_n,A_n^*$), exhibit the following property shown in the attached diagram? Two 2D lattices are shown, with the lattice points in red, the ...
11
votes
1answer
197 views

The lattice handshake number (“nearly kissing” number)?

Update: I'm happy to say that this question has been made essentially obsolete by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: https://arxiv.org/...
6
votes
0answers
140 views

Positive-definite lattice with O(n,n) Gram matrix generated by minimal vectors

Consider a positive-definite $2n$-dimensional lattice with minimum norm $\mu$. It is sometimes possible to find a generating set of minimal vectors for the lattice such that the Gram matrix takes the ...
2
votes
1answer
212 views

New Perfect 2-bit Error Correction Code - Are there any other?

I have recently looked through perfect error correcting codes and found the Hamming(7,3) and Golay(23,7). Using a computer program I have found a new 2 bit perfect error correcting code: Code(90, 2). ...
13
votes
1answer
451 views

Vectors that are almost orthogonal on average: lower bounds on dimension?

Let $v_1,\dotsc,v_k \in \mathbb{R}^d$ be unit-length vectors such that $$\sum_{1\leq i,j\leq k} |\langle v_i,v_j\rangle|^2 \leq \epsilon k^2.$$ What sort of lower bound can we give on $d$ in terms of $...
26
votes
1answer
888 views

Terrible tilers for covering the plane

Let $C$ be a convex shape in the plane. Your task is to cover the plane with copies of $C$, each under any rigid motion. My question is essentially: What is the worst $C$, the shape that forces the ...
5
votes
1answer
136 views

Extremal functions for the 'packing density in dimension one'

The $n = 1$ case of Theorem 3.1 of Cohn and Elkies's paper New upper bounds on sphere packings I amounts to the inequality $f(0) \geq 1$ for all ('admissible') functions $f$ on $\mathbb{R}$ satisfying ...
26
votes
1answer
827 views

Optimal sphere packings ==> Thinnest ball coverings?

It was proved by Kershner long ago that the thinnest (least density) covering of the plane by congruent disks can be obtained by enlarging the radii of the optimal circle packing to just cover the ...
17
votes
2answers
409 views

Construction of an optimal electron cage

I will describe the question first in 2D, but my interest is in $\mathbb{R}^3$. An electron $x$ will shoot from the origin along an initial vector $v$. You know the speed $|v|$ but not the direction. ...
9
votes
1answer
255 views

Best non-lattice sphere packings

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius. In dimensions $2, 3, 8,$ and $24$, it is known that lattice ...
0
votes
0answers
180 views

Finding good high-dimensional sphere coverings in Euclidean space

Suppose we want to cover the unit sphere $\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ with spherical caps $\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...
28
votes
2answers
2k views

Understanding sphere packing in higher dimensions

In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved. Admittedly it is very ...
19
votes
1answer
962 views

Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?

Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ...
11
votes
0answers
180 views

Electrons on a pancake ellipsoid

The problems of minimizing the potential energy of electrons on a sphere, or maximizing the smallest distance between the electrons, have been well-studied. E.g., see the earlier MO question "...
2
votes
0answers
232 views

What is the connection between the Riemann Xi-function and n-sphere? [closed]

Riemann's Xi-function is defined as $$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ At the same time we have the following formulas for n-sphere's area and volume: $$\begin{array}{...
14
votes
2answers
417 views

Double kissing problem

Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls? An easy ...
3
votes
1answer
214 views

Packing bounds for sumsets, or, very discrete balls

Let $D\subset \mathbb{F}_2^n$ with $D=-D$ and $0\in D$. Write $k D$ for the set of all sums of $k$ (not necessarily distinct) elements of $D$. (This is the "ball" in the title.) Now let $d(g,h)$ be ...
0
votes
0answers
55 views

Covering number of the range of a function

I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
3
votes
1answer
158 views

Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let $P_{\...
2
votes
0answers
235 views

Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?
6
votes
1answer
212 views

Minimizing deep holes in sphere packings

What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I ...
7
votes
1answer
311 views

For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
6
votes
1answer
164 views

Hiding $k$ disks inside a larger disk

Suppose one has $k$ unit-radius disks, and the goal is to hide them inside a disk of radius $R \gg k$. The detection probes are rays along a line. (Think of the disks as tumor cells, and the rays as ...
0
votes
1answer
202 views

Generalized Sphere Kissing Problem [duplicate]

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 ...
15
votes
2answers
521 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
162 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 ...
5
votes
0answers
671 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 ...
4
votes
1answer
258 views

Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...
4
votes
1answer
377 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" $\theta_{spheres}...
7
votes
2answers
422 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
210 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 ...
1
vote
1answer
478 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 ...
14
votes
0answers
2k 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 ...
1
vote
1answer
965 views

Optimal fitting of spheres in a cylinder

How to find the minimum height and width of a cylinder containing n identical spheres?
33
votes
3answers
2k 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
244 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
374 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 ...