7
votes
1answer
190 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 ...
5
votes
0answers
385 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 ...
9
votes
4answers
737 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. ...
34
votes
6answers
992 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, ...
20
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 ...
8
votes
5answers
740 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
663 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 ...