Questions tagged [sphere-packing]
The sphere-packing tag has no summary.
101 questions
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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, ...
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Can cut-and-project produce dense sphere packings?
To build aperiodic tilings, you can take a periodic tiling in a higher dimensional
and use a "cut-and-project" to produce a tiling in a lower dimensional space. See app for Penrose tiling ...
- 781
1
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0
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87
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Packing real vectors at a fixed minimal angle
Let $\alpha$ be an angle contained in $(0,\pi/2]$ -- to fix ideas, let $\alpha = \pi/3$.
Then with $d > 1$ a positive integer, what is a minimal bound $M_d(\alpha)$ for the maximum number of ...
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8
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Flat torus circle packings and Groebner algorithm
We are given the following packing of the flat torus with 25 circles below :
Our goal is to compute the minimal polynomial of the corresponding circle diameter $l$.
Using classical trigonometric ...
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27
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2
answers
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What was the error in the proof about kissing numbers in Proofs from the Book
Apologies if this is a bit off-topic or if I'm just not doing the right searches. I asked the question on MSE a few days ago and didn't get an answer.
I understand from the paper by Bill Casselman The ...
22
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1
answer
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Can you see through a cannonball packing?
More precisely, in a regular sphere packing, either the
HCP or FCC lattice packing,
does there exist a line $L$ disjoint from every sphere,
i.e., not touching any sphere?
If so, one could "look ...
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1
vote
0
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129
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How many non-overlapping, mutually non-antipodal unit spheres can be placed on the surface of four dimensional unit sphere?
If by linear or semi-definite programming this number could be shown to be less than 24, then this could be a route to showing that the 24-cell (consisting of 12 antipodal pairs of spheres) is the ...
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7
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1
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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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5
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0
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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 $r<2$.
It appears to me that for $L \in [\sqrt{2},2]$
and "small" $r$,
the optimal packing ...
- 153k
18
votes
3
answers
745
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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5
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1
answer
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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 ...
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3
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2
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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 ...
- 88.3k
2
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1
answer
705
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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 ...
- 161k
0
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1
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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 ...
- 1,173
6
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0
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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-...
- 1,173
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2
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527
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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 ...
- 2,859
1
vote
1
answer
134
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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 ...
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2
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1
answer
529
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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 ...
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1
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1
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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 ...
CommunityBot
- 1
3
votes
1
answer
182
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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 $...
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35
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6
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Covering a unit ball with balls half the radius
This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
- 2,835
23
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1
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789
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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 $|v_i|\geq\sqrt{n}$? ...
- 7,394
8
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1
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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 ...
- 14k
34
votes
3
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 ...
- 23.5k
0
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0
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90
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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 $\Bbb R^n$ ...
- 571
3
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1
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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 ...
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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 ...
- 201
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1
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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:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...
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0
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0
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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
$$
F_r(\mathbb{R}^{mk},...
- 2,000
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0
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Why Densest packing of equal spheres in three dimensions is not 88.86? [closed]
I placed four spheres of radius R at vertices of a tetrahedron of edge length 2R .When I calculated density I got 88.86.Actualy I wanted to calculate what is the maximum number of earth that can be ...
- 67.4k
6
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1
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Sphere packing processes during biological development
Within the context of mathematical biology, a sphere packing problem occurred to me. I must note that unlike the typical sphere packing problems, the variant I consider involves minimising the average ...
CommunityBot
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1
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Simple non-asymptotic upper-bound for packing number of a hamming cube
Looking for a simple upper-bound for the packing number of hamming cube, I'm led to consider the following. Fix $p \in (0,1/2]$. For a positive integer $n$, define $S_n(p) := \sum_{i=1}^{\lfloor np\...
CommunityBot
- 1
3
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1
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Choosing maximum number of separated points on a sphere surface
The following problem came up in one of my research works. Suppose that $C$ denotes the positive face of the $d$-dimensional unit sphere surface, i.e.
$$C := \{\mathbf{x} \in \mathbb{R}^d: x_1 >0,\...
- 4,131
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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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1
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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 $...
- 11.4k
16
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1
answer
604
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 ...
- 67.4k
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0
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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)?
...
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covering by spherical caps
Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius $\...
- 14.7k
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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 ...
- 17.3k
5
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35
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3
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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.
&...
- 40.8k
1
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1
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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 ...
- 17.3k
2
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0
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75
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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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1
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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 ...
- 17.3k
1
vote
1
answer
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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 ...
- 17.3k
8
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2
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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 ...
- 3,269
4
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0
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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 ...
- 1,203
12
votes
5
answers
2k
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Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?
The Koebe–Andreev–Thurston theorem states that any planar graph can be represented
"in such a way that its vertices correspond to disjoint disks, which touch if and only if
the corresponding vertices ...
- 2,835
26
votes
0
answers
416
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 $\mathbb R^3$ ...
- 153k
25
votes
1
answer
738
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 a ...
- 17.3k