72
votes

Accepted

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

The system doesn't try to minimise the radius of the enclosing circle, but its potential energy. We can idealise this as non-overlapping disks in a convex rotationally symmetric potential $V$ with $V(...

31
votes

Accepted

### 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

The following seems like a counterexample to the conjecture as originally stated, allowing different size coins. It doesn't seem like the big coin, with diameter $1-\epsilon$, can move right, up, or ...

28
votes

Accepted

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

Yes, the reason is the same as in the proof of Vitali covering theorem. It suffices that $U$ has finite measure, allowed to be unbounded.
If $D_i$ are the greedy-chosen (closed, but this is not ...

26
votes

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

Those packing rules only apply for rigid circles. Anyone who's ever cracked an egg knows that yolks are not rigid. As a result of that, you can clearly see that the sides of yolks are flattened as ...

23
votes

### 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

This is to confirm that the construction offered in Zach Teitler's answer works.
Indeed, look at this picture:
The square here is $[0,1]^2$. The Black circle is $C(R,1/2;R)$, with center $(R,1/2)$ ...

14
votes

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

Here's a better solution for $n=12$, with area approximately 2496:
Even better, with area approximately 2463:
Here's @MattF's suggestion, but it's worse in both dimensions:
@GerhardPaseman, if I ...

13
votes

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

No: according to the pictures in
https://en.wikipedia.org/wiki/Circle_packing_in_a_circle,
$f(r)$ is never $6$, with $f(\frac13) = 7$ but $f(\frac13 + \epsilon) = 5$.
(This result is attributed to the ...

12
votes

Accepted

### 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

This is to confirm that the suggestion by Edward H is valid.
The specific calculations have been done in a Mathematica notebook, whose image is below. I think the notebook contains all the necessary ...

12
votes

### Squaring a square and discrete Ricci flow

What does "tangent" mean? If two squares touch only at a vertex, are they tangent? According to the diagram provided, tangent seems to mean that two squares should meet along some (...

12
votes

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

What do you mean, "the yolks don't follow optimal packing"? Sure they do. The configuration with one yolk in the center has the exact same radius as the one with six yolks distributed along the edge.
...

12
votes

Accepted

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

Yes indeed, they are all algebraic. The idea is that we can describe the critical $r$ as a first-order formula in the language of fields, something like
$$\forall r_1\quad ((0 < r_1 \wedge r_1 \le ...

11
votes

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

Roberto De Leo, A conjecture on the Hausdorff dimension of attractors of real self-projective Iterated Function Systems, Experimental Mathematics 24, 270 (2015), doi:10.1080/10586458.2014.987884, ...

8
votes

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

With a ring configuration in a convex polygon and no coin touching two walls, they can always move.
Without loss of generality, assume that each wall has a coin on it. (Walls not touched by any coin ...

8
votes

Accepted

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

Such approximations have a long history, starting with
[1] Courant, R., Friedrichs, K. and Lewy, H. (1928) Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 32–74
...

8
votes

### Proofs of circle packing theorem

I can recommend Sariel Har-Peled's exposition in supplemental
Chapter 15 of his book
Geometric Approximation Algorithms.
Ch15 PDF download.
He emphasizes angles via a "whac-an-angle" game.
...

5
votes

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

Technically, there are best known configurations with pentagons and even with hexagons -- if we allow these polygons to loop around multiple faces. Such configurations may be read from the Wikipedia ...

5
votes

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

In addition to the excellent answers already added, it is important to note that in addition to Martin Hairer's description, this problem is distinct from circle packing in another way: the system ...

5
votes

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

Assuming that you're interested in random sequential addition / adsorption / deposition / packing (the random process described in the text of your answer) and not the process described in the paper ...

3
votes

### Proofs of circle packing theorem

Books are written on the subject, so, finding a proof (which are many by now) shouldn't be a problem. I also enjoyed greatly Rohde's tribute to Schramm that explains in very nice way some ideas that ...

3
votes

Accepted

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

It seems in the meantime, there are new results:
Bai, Zai-Qiao; Finch, Steven R. Precise calculation of Hausdorff dimension of Apollonian gasket. Fractals 26 (2018), no. 4, 9 pp.
claims a better ...

3
votes

Accepted

### Some inequalities on chain of circle packing

Draw the tangents $OP_i$ and $OQ_i$ from $O$ to the circle $(O_i)$ and let $\theta_i=\frac{1}{2}\angle P_iOQ_i$. These angles satisfy $\theta_i \in \left(0,\frac{\pi}{2}\right)$ and $\theta_1+\cdots+\...

3
votes

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

Following user @j.c.'s lead, here is another paper on RSA (Random Sequential Adsorption), which concludes with a density of $0.77$. From the abstract:
Hinrichsen, Einar L., Jens Feder, and Torstein ...

2
votes

Accepted

### Squaring a square and discrete Ricci flow

My question is answered in Lovász's book:
Lovász, László. Graphs and Geometry. Vol. 65. American Mathematical Soc., 2019.
p.82:
Theorem 6.2. Every planar map in which the unbounded country is a ...

Community wiki

2
votes

### Squaring a square and discrete Ricci flow

I just found this citation, not cited in the AMS Notices paper (but I cannot yet access the Israel J Math paper itself):
Schramm, Oded. "Square tilings with prescribed combinatorics." ...

Community wiki

2
votes

### Squaring a square and discrete Ricci flow

I don't know about having one vertex per square, but there is a similar very interesting construction with edges at squares. It does not answer your question but it will still surely interest you.
...

2
votes

### 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

On the sphere and for disks of radius $r$, your problem is equivalent to a covering problem with disks of radius $2r$.
Given a collection of $k$ centerpoints of open $r$-disks on the sphere, you can ...

2
votes

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

There is a stronger result. Suppose that $(b_n)$ is a sequence with the properties $b_n>0,\; b_n\to 0$ and $\sum b_n^2=\infty$.
Then for any region $D$, and for every $\epsilon>0$ there are ...

2
votes

Accepted

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

The answer is yes, in an arbitrary dimension $d$. Here is a short proof.
Suppose, contrary to the above claim, that for some Jordan measurable $\Omega$ the infimum of what is left is equal to $m > ...

1
vote

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

Let $\Omega \subset \mathbb R^d$ be a given open set of finite Lebesgue measure. Here is an argument that shows that we can in fact cover $\Omega$ up to a set of Lebesgue measure zero by pairwise ...

1
vote

Accepted

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

Proven by O. Wesler, “An infinite packing theorem for spheres,” PAMS Vol. 11, pp. 324-326, (1960).

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