Questions tagged [sphere-packing]
The sphere-packing tag has no usage guidance.
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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 ...
52
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5
answers
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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, ...
32
votes
1
answer
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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 ...
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 ...
21
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5
answers
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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 ...
35
votes
3
answers
2k
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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.
&...
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 ...
13
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2
answers
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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 ...
33
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
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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 $|v_i|\geq\sqrt{n}$? ...
21
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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 ...
17
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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 ...
16
votes
6
answers
2k
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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. ...
9
votes
1
answer
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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 ...
5
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1
answer
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5
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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 ...
3
votes
1
answer
190
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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_{\...
3
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0
answers
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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}{...
1
vote
1
answer
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Optimal fitting of spheres in a cylinder
How to find the minimum height and width of a cylinder containing n identical spheres?