# Questions tagged [sphere-packing]

The sphere-packing tag has no usage guidance.

85
questions

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### Would anybody be able to summarise Minkowski Successive Minima in slightly simpler terms? [closed]

I came across this definition in Lenny Fukshansky's paper "Revisiting the hexagonal lattice: on optimal lattice circle packing" and I can't seem to fully grasp the concept of successive ...

3
votes

1
answer

99
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### Lattice-like structure with maximum spacing between vertices

I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I ...

2
votes

0
answers

59
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### Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...

6
votes

1
answer

309
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### Monotonic dependence on an angle of an integral over the $n$-sphere

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...

0
votes

0
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42
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### packing numbers of the unit balls in Euclidean spaces and the dimensions

Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number.
The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is
$$
F_r(\mathbb{R}^{mk},...

0
votes

0
answers

82
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### packing numbers and configuration spaces of the torus

Let $S^1$ be the unit circle of radius $1$.
For any $k\geq 1$, let the $k$-dimensional torus $T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$ be the $k$-fold self-Cartesian ...

1
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0
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178
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### Why Densest packing of equal spheres in three dimensions is not 88.86? [closed]

I placed four spheres of radius R at vertices of a tetrahedron of edge length 2R .When I calculated density I got 88.86.Actualy I wanted to calculate what is the maximum number of earth that can be ...

0
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44
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### Geometric of sphere caps with infinite radius $R$ (asymptotic) in dimension $C$

Let $B_C(R)$ be the Ball in $R^C$ with radius $R$, the vectors $w_1,w_2,\dots,w_{C-1}\in R^C$ is the normal vector of hyper half plane, the area
$$\mathcal{K}=\{x:w_1^Tx\leq0, w_2^Tx\leq0,\dots,w_{C-1}...

2
votes

1
answer

119
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### Simple non-asymptotic upper-bound for packing number of a hamming cube

Looking for a simple upper-bound for the packing number of hamming cube, I'm led to consider the following. Fix $p \in (0,1/2]$. For a positive integer $n$, define $S_n(p) := \sum_{i=1}^{\lfloor np\...

3
votes

1
answer

143
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### Choosing maximum number of separated points on a sphere surface

The following problem came up in one of my research works. Suppose that $C$ denotes the positive face of the $d$-dimensional unit sphere surface, i.e.
$$C := \{\mathbf{x} \in \mathbb{R}^d: x_1 >0,\...

2
votes

0
answers

116
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### density of lattices

I'm looking for references pertaining to the remark at the bottom of p.18 of Conway-Sloane, "Sphere Packings, Lattices and Groups" (3rd ed), henceforth referred to as "SPLG".
First,...

5
votes

0
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74
views

### Packing cylinders in a sphere: Phase transition?

Let $S$ be a unit-radius sphere in $\mathbb{R}^3$,
and $c$ a cylinder of length $L$ and radius $\epsilon$.
It appears to me that for $L \in [\sqrt{2},2]$
and small $\epsilon$,
the optimal packing of ...

8
votes

1
answer

217
views

### Perfect sphere packings (as opposed to perfect ball packings)

I came across this question when I was discussing the rather wonderful Devil's Chessboard Problem with my colleague, Francis Hunt.
We realised that there is a nice connection to a packing question in $...

16
votes

1
answer

350
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### Balls in Hilbert space

I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...

6
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answers

116
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### Aperiodic packings of the plane with disks of multiple radii

Does there exist a finite set of radii such that some aperiodic packing of the plane by disks of those radii is believed to achieve the maximal packing density (not achieved by any periodic packing)?
...

10
votes

2
answers

565
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### Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...

5
votes

1
answer

222
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6
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1
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353
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### Sphere packing processes during biological development

Within the context of mathematical biology, a sphere packing problem occurred to me. I must note that unlike the typical sphere packing problems, the variant I consider involves minimising the average ...

1
vote

1
answer

275
views

### Prospects for deep learning of non-lattice sphere packings

I have been looking for litterature on results obtained by deep neural networks to find dense (and quite possibly non-lattice, perhaps even non-periodic) sphere packings, but I have not been too ...

2
votes

0
answers

37
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### Simplicial density function simultaneously defined for hyperbolic and spherical space (Kellerhals, 1998)

I am confused about the proof of Corollary 4.2 in "Ball Packings in Spaces of Constant Curvature and the Simplicial Density Function". The point of confusion is equation 4.3, where Kellerhals states ...

12
votes

1
answer

639
views

### Illustrating that universal optimality is stronger than sphere packing

I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...

1
vote

1
answer

342
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### Sphere packing and kissing numbers in 3D

When one looks at the way cannon balls and oranges are normally packed by the military and by groceries, it seems intuively clear that there is no way anybody can pack these any tighter. However, it ...

8
votes

2
answers

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### How many cones with angle theta can I pack into the unit sphere?

Given a unit sphere (radius 1), I would like to know how many cones I can pack into this unit sphere. Restrictions: The top of the cone needs to be in the center of origin. The bottom of the cone ...

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### Do kissing numbers with distance $d$ always grow polynomially or exponentially in dimension?

Let $A_d(n)$ be the largest number of points that can be packed on the $n$-unit sphere, such that every point is at least $d$ apart. Compare with, for instance, https://arxiv.org/abs/1507.03631
When ...

23
votes

0
answers

285
views

### Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...

23
votes

1
answer

581
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

2
answers

1k
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### 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 ...

3
votes

1
answer

106
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

1
answer

73
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### 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 ...

3
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answers

42
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### 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
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77
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### 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
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0
answers

199
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### 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 $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...

7
votes

1
answer

585
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### 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

0
answers

85
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 $\epsilon$ ...

7
votes

1
answer

526
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### 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

1
answer

251
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\...

3
votes

1
answer

449
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### 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

1
answer

452
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

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205
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### 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

1
answer

628
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### 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
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1
answer

622
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### 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 $...

27
votes

1
answer

1k
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### 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

1
answer

148
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### 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
...

29
votes

1
answer

1k
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### 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

2
answers

522
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

1
answer

357
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### 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 ...

2
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0
answers

350
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### 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}...

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2
answers

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### 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 ...

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answer

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### 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 ...

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204
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### 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
"...