The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
0answers
137 views

How large do algebraic representations need to be for packing circles in squares?

(This question is inspired by Erich's Packing Center. I'm just asking about circles in squares to keep things simple, since I suspect any answer would apply just-as-well to the rest of the problems ...
3
votes
0answers
85 views

Modeling bubble rafts

If you go to images.google.com and search on "bubble rafts", you'll see various pictures of disk packings that in large patches look like the six-around-one dense packing of the plane by equal-sized ...
13
votes
1answer
343 views

Is the Ford disk packing optimal?

Given two unit-diameter disks tangent to a given line and to each other, determining a region bounded by two circular arcs and a line segment, is the Ford disk packing of that region the unique ...
17
votes
1answer
240 views
6
votes
1answer
151 views

Spirals in Apollonian circle-packings

Given mutually (externally) tangent circles $C_1,C_2,C_3$, let $C_n$ be the unique circle externally tangent to $C_{n-1}$, $C_{n-2}$, and $C_{n-3}$ for $n \geq 4$. Let $P_{\infty}$ be the point toward ...
3
votes
0answers
106 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
200 views

Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
4
votes
0answers
210 views

Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$.                 Is the ...
4
votes
1answer
108 views

Local rigidity of square disk packing

Is the packing of the plane by disks of radius 1/2 centered at the points of ${\bf Z} \times {\bf Z}$ "locally rigid" in the sense that no finite subcollection of the disks admits any joint ...
15
votes
2answers
323 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 ...
7
votes
0answers
291 views

Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane, with some density $\rho$ per unit area. View the points as disks of radius zero. Now the radii $r$ of all disks grows ...
1
vote
0answers
59 views

Are morphisms of intersection graphs of circle packings harmonic?

Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...
1
vote
0answers
213 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
0answers
161 views

A primal-dual (double) circle packing (coin graph) question

I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem). Q1- But I am not sure whether ...