Questions tagged [sphere-packing]

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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\...
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1answer
107 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,\...
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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,...
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Constraints on simplices in hyper sphere packings

Question: what is known about constraints on the side lengths of $n$-simplices that are defined by the centers of $n+1$ kissing $(n-1)$-spheres? The constraints on the curvatures $c_i$ of Soddy ...
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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 ...
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1answer
187 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
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1answer
314 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 ...
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111 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)? ...
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2answers
473 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 ...
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1answer
205 views

Wrapping juggling balls

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1answer
284 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 ...
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1answer
203 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 ...
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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 ...
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1answer
598 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 ...
2
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1answer
242 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 ...
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926 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 ...
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100 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 ...
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263 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 $\mathbb R^3$ ...
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1answer
552 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 ...
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2answers
998 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 ...
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1answer
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 ...
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1answer
71 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 ...
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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? ...
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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 ...
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191 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 $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
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1answer
529 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 $...
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65 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 $\epsilon$ ...
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1answer
445 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
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1answer
215 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\...
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1answer
394 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 ...
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1answer
375 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/...
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183 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 ...
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1answer
450 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). ...
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1answer
581 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 $...
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1answer
981 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 ...
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1answer
141 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 ...
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1answer
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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 ...
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2answers
508 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. ...
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1answer
315 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 ...
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296 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}...
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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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1answer
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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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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 "...
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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}{...
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2answers
457 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 ...
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1answer
226 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 ...
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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 $D \subseteq \mathbb{R}^m$. ...
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1answer
168 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_{\...
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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?
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1answer
218 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 ...