30
votes

### Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

• Concerning question 2, you might want to take a look at Simulation of cubical particle packing under mechanical vibration (2016). The precise effect mentioned in the 2017 paper is not considered in ...

23
votes

Accepted

### Can squares of side 1/2, 1/3, 1/4, … be packed into three quarters of a unit square?

The standard simple proof that $\sum_{n=1}^\infty \frac1{n^2}$ converges is to round each $n$ down to the nearest $2^k$; this rounds each $\frac1{n^2}$ up to the nearest $\frac1{2^{2k}}$. In fact, ...

20
votes

### Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

I doubt that a mathematically rigorous explanation of the phenomenon discovered in that paper exists using today's technology. While mathematical statistical mechanics is a well-developed field of ...

17
votes

### Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

OK, posting then.
I prefer to think of triangles pointing to the right in the triangle pointing to the left. Let $\delta=e^{-\sqrt{\log 1/\varepsilon}}$. For each small triangle $T$, let $I$ be the ...

15
votes

### Can squares of side 1/2, 1/3, 1/4, … be packed into three quarters of a unit square?

Note: This answer is wrong. There are two problems:
The claim of Lemma 1 should presumably be read in the context of the global assumption in Paulhus's paper that $w\leq l$. This assumption ...

14
votes

### Dodecahedral rolling distance

Here are a few trivial lemmas. I won't use anything about the rolling motion, just that the distance is defined by gluing pentagons edge-to-edge:
The $dd$-circle of radius $k$, which I'll call $C_k$, ...

14
votes

### Can we cover the unit square by these rectangles?

I think the following result of Greg Martin related to this question deserves to be mentioned here. (It is already referenced in Chapter 3 of the book "Research Problems in Discrete Geometry" by P. ...

14
votes

### Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

The physical reason is that the cubically packed state has lower gravitational potential energy than the jammed random state. The overall process is analogous to annealing, although the reduction in ...

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

### Smallest sphere containing three tetrahedra?

Unclear this is best, but: $r=\sqrt{3}/2$.
JukkaKohonen's suggestion:
I made no attempt to optimize, but this certainly shows
the smallest sphere has radius strictly less than $\sqrt{3}/2$.

11
votes

Accepted

### Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

Ok I think this is an argument that the perimeter at least $\varepsilon^{-c}$ for some sufficiently small $c$. To avoid special cases where $\varepsilon$ is large, we use the convention that we count ...

10
votes

Accepted

### Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?

Certainly no. Consider metric spaces on $n$ points and all distance 1 and 2. There are $2^{n^2/2+o(n^2)}$ such spaces. But only polynomially many different covering and packing functions.

9
votes

Accepted

### Randomly covering a sphere

There's a simple way to get upper and lower bounds that are given by the solution to essentially the same combinatorial problem. In particular, we throw $k$ red balls and $m$ blue balls into some bins ...

7
votes

### Sequential addition of points on a circle, optimizing asymptotic packing radius

This is really a comment, but it's getting a bit long for the comment box.
I want to point out that in addition to what you call the "greedy process," there's another obvious attempt, which could be ...

Community wiki

7
votes

Accepted

### Thinnest covering of the plane by regular pentagons

The thinnest known covering of the plane with congruent regular pentagons is shown in my answer to: Terrible tilers for covering the plane. What you see there is probably not "rollable". The covering ...

7
votes

Accepted

### Does finite Hausdorff dimension imply finite packing dimension?

A construction used (repeatedy) in the paper
Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004.
For more information, see that paper.
We ...

6
votes

Accepted

### Sequential addition of points on a circle, optimizing asymptotic packing radius

I will show $\mu = \tfrac{1}{\log 4}$.
I first prove the upper bound $\mu \leq \tfrac{1}{\log 4}$. Fix a positive integer $r$. For $0 \leq k \leq r-1$, let $N_k$ be $2^{k/r} N$ rounded to the ...

6
votes

Accepted

### Can I cover a compact set by balls {B} such that {2B} has bounded overlap?

Let $r$ be less than half the distance from $K$ to the complement of $B_1(0)$. Start with a ball of radius $r$ centered at each point of $K$. By https://en.wikipedia.org/wiki/...

6
votes

Accepted

### On packing axisymmetric bodies in 3D

Your claim is false for axially-symmetric ellipsoids: when restricted to all have their axis of symmetry in the same direction, they cannot pack more densely than spheres (in terms of volume fraction)....

6
votes

Accepted

### Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

No. For $0 < \delta \leq 2$ let $E_\delta$ be the metric space
consisting of three points $A,B,C$ with
$d(A,B) = d(A,C) = 1$ and $d(B,C) = \delta$.
I claim that the $E_\delta$ for $1 \leq \delta \...

6
votes

### How many non-orthogonal vectors fit into a complex vector space?

Jan Nienhaus's answer treats the case $\epsilon<\frac{1}{D}$. Here's a generalization that works whenever $\epsilon<\frac{1}{\sqrt{D}}$:
$$ N \leq \frac{1-\epsilon^2}{1 - D\epsilon^2}\cdot D. $$
...

5
votes

### Dodecahedral rolling distance

Here are a few pictures and speculations to add to the the great ones from @j.c.
Here are the pentagons which arise from starting with the central green one and doing the reflections which do not ...

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 ...

5
votes

### Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

This should be seen as a comment, but it is too long for a comment:
One aspect, that has not been addressed in the answers, is the influence of the container geometry. In the experiment, a ...

5
votes

Accepted

### Exact bin packing the harmonic series: references?

The next $n$ for which there is an exact packing are $24 \leq n \leq 30$. This is because
$$
1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{1}{4} + \frac{1}{5} + \frac{1}{8} + \frac{1}{9} + \frac{...

5
votes

### Sequential addition of points on a circle, optimizing asymptotic packing radius

Thinking more about Christian's stingy process, I have a new conjecture for the optimal $\mu$. I motivate the conjecture by modeling the evolution of the distribution of empty interval lengths in the ...

5
votes

### Sequential addition of points on a circle, optimizing asymptotic packing radius

In this second answer, I want to discuss an upper bound on $\mu$ (not optimal, and I don't think this argument could give an optimal bound, even after fine tuning).
Suppose we have placed $N$ points ...

5
votes

Accepted

### Equation of state for hard rods

I don't know if there are any expressions that take such a simple form as the C-S equation of state. Note that spherocylinders have an additional geometrical parameter $L/D$ relating the length $L$ to ...

5
votes

### Coverage of balls on random points in Euclidean space

A partial answer: You might as well pick the ball centers first (at random of course) and then wonder what happens after that. If you want estimates rather than exact solutions, then there can't be ...

5
votes

### Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere

I'm not sure how to answer this question, but I'll suggest another approach to get an upper bound.
Considering the problem of packing equilateral $\pi/3$ triangles on a unit sphere, one may convert ...

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