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 lim …

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 spac …

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 ba …

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 …

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 n …

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 …

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

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 stick …

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 c …

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 highes …

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 complet …

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 wiki …

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 subjec …

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 …

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 th …