Questions tagged [sphere-packing]
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95
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Tetris-like falling sticky disks
Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...
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votes
3
answers
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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.
&...
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votes
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 ...
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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 ...
user88381
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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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27
votes
1
answer
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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 ...
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24
votes
1
answer
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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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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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23
votes
1
answer
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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 $|v_i|\geq\sqrt{n}$? ...
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Is there a midsphere theorem for 4-polytopes?
The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...
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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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2
answers
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What is the largest possible thirteenth kissing sphere?
It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...
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votes
2
answers
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Are there locally jammed arrangements of spheres of zero density?
I know of a remarkable result from a paper of
Matthew Kahle (PDF download), that there are arbitrarily low-density
jammed packings of congruent disks in $\mathbb{R}^2$:
In 1964 Böröczky used
a ...
- 149k
17
votes
2
answers
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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.
...
- 149k
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votes
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answers
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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 $\...
- 95.3k
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Optimal pebble-packing shape
Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes.
Q. ...
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votes
1
answer
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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 ...
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2
answers
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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 ...
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How many unit balls can be put into a unit cube?
Here a unit ball is a ball of diameter 1, and a unit cube is a cube of edge length 1.
A famous counterintuitive fact is that, as the dimension increases, the volume of the unit ball tends to zero ...
13
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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 ...
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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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2
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Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
- 149k
13
votes
1
answer
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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 $...
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votes
5
answers
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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 ...
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11
votes
2
answers
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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 ...
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votes
1
answer
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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/...
- 1,016
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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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10
votes
2
answers
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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 ...
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Identifying lattices
I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known ...
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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, and ...
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1
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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 ...
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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 ...
8
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1
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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 ...
- 25.5k
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votes
1
answer
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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 ...
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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 $...
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2
answers
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Kissing Number of Spheres in Non-Euclidean Geometry
There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ ...
- 1,381
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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 $...
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For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?
The fractal dimension of the 3D Apollonian packing is computed in this paper.
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
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1
answer
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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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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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1
answer
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Hiding $k$ disks inside a larger disk
Suppose one has $k$ unit-radius disks, and the goal is to hide them inside
a disk of radius $R \gg k$.
The detection probes are rays along a line.
(Think of the disks as tumor cells, and the rays as ...
- 149k
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votes
0
answers
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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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votes
0
answers
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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 ...
- 19.1k
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votes
0
answers
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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 ...
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Inter-Kissing Number for Spheres of Different Sizes
What is the maximum number of spheres that can be placed in 3D such that all inter-touch?
One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the
middle, so this ...
- 1,527
5
votes
1
answer
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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 ...
- 19.1k
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votes
1
answer
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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
...
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245
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5
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0
answers
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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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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 ...
- 149k