Questions tagged [sphere-packing]

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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 ...
Michael Hardy's user avatar
5 votes
0 answers
116 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 ...
Seewoo Lee's user avatar
  • 1,911
2 votes
1 answer
513 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 ...
zeta's user avatar
  • 337
4 votes
0 answers
127 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-...
Veit Elser's user avatar
  • 1,045
12 votes
2 answers
431 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 ...
Dan Romik's user avatar
  • 2,480
2 votes
1 answer
102 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 $...
The Nguyen's user avatar
1 vote
1 answer
90 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 ...
Daniel S's user avatar
  • 111
7 votes
1 answer
344 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 ...
Johnny Cage's user avatar
  • 1,543
10 votes
0 answers
413 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 ...
Harry Wilson's user avatar
23 votes
1 answer
647 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 $|v_i|\geq\sqrt{n}$? ...
Mohammad Ghomi's user avatar
1 vote
1 answer
128 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 ...
ABIM's user avatar
  • 5,019
0 votes
0 answers
77 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 $\Bbb R^n$ ...
zoran  Vicovic's user avatar
3 votes
1 answer
119 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 ...
Vincent Granville's user avatar
2 votes
0 answers
90 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 ...
Ye Tian's user avatar
  • 161
5 votes
1 answer
330 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: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ ...
LayZ's user avatar
  • 115
0 votes
0 answers
69 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},...
Shiquan Ren's user avatar
  • 1,970
0 votes
1 answer
138 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 ...
Shiquan Ren's user avatar
  • 1,970
1 vote
0 answers
252 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 ...
abhishek gayari's user avatar
2 votes
1 answer
215 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\...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
285 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,\...
Probabilist's user avatar
2 votes
0 answers
133 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,...
W Sao's user avatar
  • 509
5 votes
0 answers
129 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 ...
Joseph O'Rourke's user avatar
8 votes
1 answer
252 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 $...
Nick Gill's user avatar
  • 11.2k
16 votes
1 answer
506 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 ...
Bruce Blackadar's user avatar
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)? ...
James Propp's user avatar
  • 19.4k
10 votes
2 answers
843 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 ...
VS.'s user avatar
  • 1,816
5 votes
1 answer
246 views

Wrapping juggling balls

...
Joseph O'Rourke's user avatar
6 votes
1 answer
393 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 ...
Aidan Rocke's user avatar
  • 3,827
1 vote
1 answer
398 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 ...
Archie's user avatar
  • 883
2 votes
0 answers
43 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 ...
Chris Jones's user avatar
13 votes
1 answer
717 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 ...
Diffycue's user avatar
  • 242
1 vote
1 answer
632 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 ...
Craig Feinstein's user avatar
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 ...
Thomas Hubregtsen's user avatar
4 votes
0 answers
117 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 ...
Alex Meiburg's user avatar
  • 1,193
25 votes
0 answers
343 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 $\mathbb R^3$ ...
Joseph O'Rourke's user avatar
24 votes
1 answer
646 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 ...
Dustin Wehr's user avatar
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 ...
A_Curious_Kid's user avatar
3 votes
1 answer
111 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 ...
neverevernever's user avatar
2 votes
1 answer
85 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 ...
Manfred Weis's user avatar
  • 12.6k
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? ...
James Propp's user avatar
  • 19.4k
6 votes
0 answers
78 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 ...
James Propp's user avatar
  • 19.4k
1 vote
0 answers
262 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 $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
TMM's user avatar
  • 713
8 votes
1 answer
676 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 $...
Christopher's user avatar
2 votes
0 answers
149 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 $\epsilon$ ...
gondolf's user avatar
  • 1,487
8 votes
1 answer
709 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 ...
Sebastien Palcoux's user avatar
2 votes
1 answer
297 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\...
Yining Wang's user avatar
3 votes
1 answer
566 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 ...
user3433489's user avatar
11 votes
1 answer
541 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/...
Noah Stephens-Davidowitz's user avatar
6 votes
0 answers
226 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 ...
Chaitanya Murthy's user avatar
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). ...
RobertB.'s user avatar