# Questions tagged [circle-packing]

The circle-packing tag has no usage guidance.

51
questions

5
votes

1
answer

254
views

### Is the maximal packing density of identical circles in a circle always an algebraic number?

There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful.
My original ...

22
votes

1
answer

1k
views

### Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered.
The greedy circle packing of a ...

2
votes

0
answers

79
views

### Another variant of the Malfatti problem

We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...

2
votes

1
answer

100
views

### The problem of finding the smallest number of copies of a certain shape that can be placed into a space to make fitting another copy impossible

Packing problems often ask for the largest number of some identical shape that can fit in a given space without overlapping, if they are placed optimally.
I'm interested in the opposite question:
Q. ...

4
votes

0
answers

128
views

### Can a convex frame hold all circles of radius $1/n$ immobile?

Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile.
By "immobile", I mean no circle can move without overlapping other circles or the frame, ...

16
votes

1
answer

748
views

### Are there general principles that allow us to easily determine whether coins in simple arrangements in a frame can move?

Circular coins in a frame may all be stuck in their positions; for example:
Another possibility is that they can all move simultaneously; I claim the following examples:
It is not always obvious ...

15
votes

1
answer

1k
views

### Conjecture: If equal size circular coins are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move

Earlier I conjectured that if circular coins of any sizes are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move. A counter-example using coins of ...

40
votes

2
answers

3k
views

### Conjecture: If circular coins of any sizes are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move

Suppose some circular coins (not necessarily the same size) are in a frame. The coins may be immobile, as in this example:
On the other hand, they may be free to move, as in these examples (in which ...

11
votes

0
answers

214
views

### Can a billiard rack be a square, for every number of balls?

A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack.
Can the rack be a square, for ...

4
votes

1
answer

212
views

### Harmonic functions as limits of harmonic functions on graphs?

I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...

2
votes

0
answers

91
views

### Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...

6
votes

0
answers

78
views

### Implications of combinatorial results towards discrete function theory on circle packings

Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex ...

2
votes

4
answers

415
views

### Can we almost cover any shape in the plane by disjoint/tangent disks of prescribed radii?

This is a cross-post.
Let $(a_n)_{n \in \mathbb{Z}}$ be some given, strictly increasing sequence of positive numbers, such that $\lim_{n \to -\infty} a_n=0,\lim_{n \to +\infty} a_n=+\infty$.
Let $\...

7
votes

1
answer

511
views

### Packing equal-size disks in a unit disk

Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but ...

2
votes

1
answer

123
views

### On covering a disk by non-overlapping subdisks

I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school.
A disk $D$ of radius $1$ contains disks $D_i$ ($i \ge 1$) of ...

16
votes

1
answer

493
views

### Can solutions to Thomson's problem have pentagons?

Thomson's problem asks for the minimum-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/...

6
votes

0
answers

157
views

### On cutting disks from planar regions

Question: Given a planar region $R$ of unit area and an integer $n$, to cut $n$ circular disks (their sizes need not be equal) such that the highest fraction of $R$ is taken off.
A simple greedy ...

16
votes

4
answers

1k
views

### Squaring a square and discrete Ricci flow

Is this a theorem?
Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares....

10
votes

2
answers

1k
views

### Proofs of circle packing theorem

Circle packing theorem is a famous result stating that for every connected simple planar graph $G$ there is a circle packing in the plane whose intersection graph is $G$ https://en.wikipedia.org/wiki/...

1
vote

0
answers

205
views

### What is the nearest Ford circle for any point in $\mathbb R^2$

I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^...

1
vote

1
answer

388
views

### Densest safe disk packing

Inspired by current regulations regarding the minimal distance to be kept among people to prevent spreading of the COVID-19 virus and the maximal number of people in a group that is not subjected to ...

5
votes

1
answer

768
views

### Packing circles with radii 1, 2, 3, ..., n in a rectangle

For each positive integer n, let $a_n$ be the area of the smallest rectangle whose area is a whole number, and inside which it is possible to pack all n circles of radii 1, 2, 3, ..., n respectively (...

4
votes

0
answers

119
views

### Do kissing numbers with distance $d$ always grow polynomially or exponentially in dimension?

Let $A_d(n)$ be the largest number of points that can be packed on the $n$-unit sphere, such that every point is at least $d$ apart. Compare with, for instance, https://arxiv.org/abs/1507.03631
When ...

30
votes

5
answers

9k
views

### Six yolks in a bowl: Why not optimal circle packing? [closed]

Making soufflé tonight, I wondered if the six yolks took on the
optimal circle packing configuration.
They do not. It is only with seven congruent circles that the optimal
packing places one in the ...

1
vote

0
answers

201
views

### Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:
Let three $(A)$, $(B)$, $(...

2
votes

1
answer

230
views

### Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof:
Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...

9
votes

2
answers

453
views

### Density of a saturated random packing of congruent circles

The problem of the expected density of a saturated random packing of unit circles in the plane can be described as follows.
In a circular region $C$ of a large radius pick a point at random and draw ...

1
vote

0
answers

248
views

### Interior and boundary vertices of weighted graphs

Xu He's article Rigidity of Infinite Disk Patterns and I have a problem with a statement he makes on page 7.
He considers weighted embedded planar graphs $G=(V, E)$ with weight function $\Theta: E \...

1
vote

2
answers

196
views

### Descartes' theorem and Circle Packing [closed]

There's something I am missing comparing Descartes' theorem for three isometric circles here and this wiki post on circle packing of 3 circles here.
From my calculation:
$$
r_{ext} = \frac{r_{int}}{{...

15
votes

2
answers

1k
views

### Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?

I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B ...

7
votes

1
answer

221
views

### Optimal stacking of split logs

Consider firewood logs as unit-radius cylinders of the same length.
Each log is split into $k$ pieces by equiangular sectors
meeting in the circle center:
$k=2$ leads to semicircles, $180^\circ$;
$k=3$...

4
votes

0
answers

192
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

0
answers

135
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 ...

14
votes

1
answer

652
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

1
answer

382
views

6
votes

1
answer

278
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 ...

2
votes

0
answers

288
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

1
answer

271
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

0
answers

864
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

1
answer

152
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 ...

17
votes

2
answers

768
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 ...

15
votes

0
answers

477
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

0
answers

87
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

0
answers

303
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

0
answers

246
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 ...

4
votes

1
answer

460
views

### Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...

12
votes

5
answers

2k
views

### Computing the centers of Apollonian circle packings

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to ...

1
vote

1
answer

218
views

### Constant hole density on the area of a circle

I need to create about 100 (small) holes in a distributor plate (hole diam = 0.5 mm; plate diameter = 100 mm). The sm. holes should be distributed in such a way that the density (hole/area) is nearly ...

20
votes

1
answer

3k
views

### A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...

1
vote

1
answer

321
views

### Settling a circular argument: room for one more?

By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller ...