# Questions tagged [sphere-packing]

The sphere-packing tag has no usage guidance.

95
questions

3
votes

2
answers

800
views

### Kepler conjecture: Are there only two most efficient packings or could there be more than two?

Today I attended a talk by Terence Tao, attended by (I'm guessing) probably at least a couple of thousand people, in which among other things he said it had been proved that no packing of spheres in ...

5
votes

1
answer

191
views

### Sphere packing and modular forms in known dimensions (maybe 2)

Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions (with other mathematicians after ...

2
votes

1
answer

570
views

### Kissing number lower bound vs. upper bound - precise meanings?

According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds
It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$:
$$
K_L < K < K_U.
$$
I ...

4
votes

0
answers

133
views

### Does this code have a name?

Hamming-distance-4 binary codes have a very direct relationship to sphere packings. That's because we can identify the codewords with the cosets of $\mathbb{Z}^n/(2\mathbb{Z})^n$, and Hamming-distance-...

12
votes

2
answers

452
views

### Nonnegativity of coefficients of a modular form defined in terms of the Jacobi thetanull functions

Question
Let
\begin{align*}
\theta_2(q) & = \sum_{n=-\infty}^{\infty} q^{(n+1/2)^2}
\\
\theta_3(q) & = \sum_{n=-\infty}^{\infty} q^{n^2}
\\
\theta_4(q) & = \sum_{n=-\infty}^{\infty} (-1)^n ...

2
votes

1
answer

112
views

### Packing problem over discrete space

Let $q$ be an positive integer and $F = \{0,1,\dots,q-1\}$, we define Hamming distance in $F^n$ between $x = (x_1, \dots, x_n), y = (y_1, \dots, y_n) \in F^n$ is the number of indices $i$ such that $...

1
vote

1
answer

92
views

### Probability density function for the polar sine of uniformly distributed points on the sphere

If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?
More generally, for $k<n$ if I ...

7
votes

1
answer

423
views

### Optimal sphere packings in dimensions different fom 8 and 24

After the groundbreaking work of Viazovska, now we have a proof for the optimal density of sphere packings in dimensions 8 and 24. Both packings emerge from very particular algebraic lattice ...

10
votes

0
answers

485
views

### Kissing the Monster, or $196,560$ vs. $196,883$

The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...

23
votes

1
answer

692
views

### Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks

Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? ...

1
vote

1
answer

138
views

### Packing number in finite-dimensional normed spaces

I am working on a paper and quoted the following result from these lecture notes.
Where can I find a reference to this result either in a book or a paper, that I can cite?
(I looked on the course ...

0
votes

0
answers

79
views

### Integral over $S^{n-1}$ [duplicate]

What is the values of the following integral:
$$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$
where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ ...

3
votes

1
answer

132
views

### 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

91
views

### 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 ...

5
votes

1
answer

342
views

### 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
answers

83
views

### 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

1
answer

146
views

### 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
vote

0
answers

259
views

### 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 ...

2
votes

1
answer

223
views

### 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

377
views

### 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

148
views

### 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
answers

185
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 $r<2$.
It appears to me that for $L \in [\sqrt{2},2]$
and "small" $r$,
the optimal packing ...

8
votes

1
answer

258
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

520
views

### 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
votes

0
answers

119
views

### 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)?
...

11
votes

2
answers

912
views

### 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

247
views

6
votes

1
answer

406
views

### 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

423
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

46
views

### 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 ...

13
votes

1
answer

723
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

793
views

### 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

1k
views

### 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 ...

4
votes

0
answers

119
views

### 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 ...

26
votes

0
answers

353
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 $\mathbb R^3$ ...

25
votes

1
answer

660
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

2k
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 ...

3
votes

1
answer

117
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

94
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 ...

3
votes

0
answers

45
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

0
answers

80
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

0
answers

274
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 $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...

8
votes

1
answer

706
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

0
answers

189
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 $\epsilon$ ...

8
votes

1
answer

750
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

1
answer

304
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

604
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

1
answer

579
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

0
answers

235
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

1
answer

1k
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).
...