The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
1answer
115 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 ...
3
votes
0answers
99 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?
6
votes
1answer
147 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 ...
7
votes
1answer
263 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
1answer
136 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 ...
0
votes
0answers
109 views

Generalized Sphere Kissing Problem

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 ...
15
votes
2answers
298 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 ...
3
votes
2answers
134 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 ...
5
votes
0answers
405 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 ...
2
votes
1answer
183 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" ...
5
votes
2answers
264 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 $1$ ...
2
votes
0answers
161 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 ...
2
votes
1answer
210 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 ...
13
votes
0answers
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 ...
0
votes
1answer
569 views

Optimal fitting of spheres in a cylinder.

how to find the minimum height and width of a cylinder containing n identical spheres?
23
votes
2answers
761 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. ...
1
vote
0answers
206 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 ...
2
votes
1answer
282 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 ...
4
votes
1answer
196 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 ...
1
vote
0answers
317 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 ...
12
votes
6answers
850 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. ...
35
votes
6answers
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, ...
21
votes
6answers
2k 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 ...
10
votes
1answer
261 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 ...
11
votes
2answers
484 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 ...
8
votes
5answers
801 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 ...
12
votes
2answers
695 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 ...
6
votes
5answers
630 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 ...
1
vote
1answer
194 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 ...