The sphere-packing tag has no usage guidance.

**44**

votes

**5**answers

1k views

### 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, ...

**26**

votes

**2**answers

989 views

### 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.
&...

**25**

votes

**6**answers

4k views

### 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 radius ...

**25**

votes

**2**answers

1k views

### 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 ...

**18**

votes

**5**answers

906 views

### 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 ...

**18**

votes

**1**answer

742 views

### 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 ...

**15**

votes

**2**answers

414 views

### 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 ...

**14**

votes

**0**answers

1k views

### 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**

votes

**6**answers

1k views

### 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. ...

**12**

votes

**2**answers

574 views

### 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 ...

**12**

votes

**2**answers

328 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 ...

**12**

votes

**2**answers

787 views

### 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 ...

**10**

votes

**1**answer

284 views

### 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 ...

**10**

votes

**0**answers

156 views

### 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
"...

**9**

votes

**5**answers

990 views

### 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 $\...

**9**

votes

**5**answers

1k views

### 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 ...

**7**

votes

**1**answer

296 views

### 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 (...

**6**

votes

**2**answers

336 views

### 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 $1$ ...

**6**

votes

**1**answer

185 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 ...

**6**

votes

**1**answer

148 views

### 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 ...

**5**

votes

**1**answer

240 views

### 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 ...

**5**

votes

**0**answers

557 views

### N-balls covering n-balls

This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...

**4**

votes

**1**answer

311 views

### Is there an “accepted” jamming limit for hard spheres placed in the unit cube by random sequential adsorption?

I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" $\theta_{spheres}...

**4**

votes

**1**answer

218 views

### Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...

**3**

votes

**1**answer

147 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_{\...

**3**

votes

**1**answer

196 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 ...

**3**

votes

**2**answers

145 views

### Techniques for showing optimality of given packing

There are some natural packing problems that have been asked in mathematics. Some of them are:
1)How many balls can be placed with in a cube?
2)How many equidistant points can be place on the ...

**2**

votes

**1**answer

373 views

### Packing a closed 3D surface with non-overlapping spheres starting with the largest possible one and then working the way down

Let's say, I have a closed 3D surface (say, the surface of a pebble). I want to pack it with spheres, but starting with the largest possible sphere, then the next largest possible non-overlapping ...

**2**

votes

**1**answer

337 views

### 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 ...

**2**

votes

**0**answers

141 views

### 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}{...

**2**

votes

**0**answers

147 views

### 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?

**2**

votes

**0**answers

181 views

### Computing the Volume of Closed 3-Manifolds and the Geometrization Conjecture

My question is whether or not if I generalize Theorem 2(i) of "Contact Graphs of Unit Sphere Packings Revisited" [2012] by K. Bezdek and S. Reid (arXiv link) which states
The number of touching ...

**1**

vote

**1**answer

203 views

### How to compute the number of regular spheres needed to fill a rectangular space

Computing the volume of a sphere is straightforward 4/3*pi*R^3
As is the volume of a rectangular space length*width*height (e.g. 10*10*6)
How might I go about determining how many spheres would fit ...

**1**

vote

**0**answers

230 views

### Which term is better for the so called “sphere packing”?

I'm working on sphere packings. When I write, I'm confused with basic definitions. I'm hesitating between the terms "sphere", "ball" or "oriented sphere".
For example, on the wikipedia page of circle ...

**1**

vote

**0**answers

396 views

### maximal minimum distance in a sphere packing

Hi everyone.
I need to pick a set of 65 points p(x,y,z) in a 3D space of 274625 points; as
the set picked should provide the maximum possible minimum Hamming distance.
(Consider the Hamming bound for ...

**0**

votes

**1**answer

867 views

### Optimal fitting of spheres in a cylinder

How to find the minimum height and width of a cylinder containing n identical spheres?

**0**

votes

**0**answers

89 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}...

**0**

votes

**0**answers

42 views

### 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 $D \subseteq \mathbb{R}^m$. ...

**0**

votes

**1**answer

167 views

### Generalized Sphere Kissing Problem [duplicate]

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...