# Tagged Questions

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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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
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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 ...
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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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### 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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### 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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### 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 ...
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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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### 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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### 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 ...
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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 $P_{\... 1answer 196 views ### Packing bounds for sumsets, or, very discrete balls Let$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 ... 2answers 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 ... 1answer 375 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 ... 1answer 338 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 ... 0answers 142 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}{... 0answers 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? 0answers 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 ... 1answer 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 ... 0answers 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 ... 0answers 401 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 ... 1answer 873 views ### Optimal fitting of spheres in a cylinder How to find the minimum height and width of a cylinder containing n identical spheres? 0answers 94 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}...
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 $D \subseteq \mathbb{R}^m$. ...
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 ...