# Tag Info

### 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 ...
• 180k
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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $C_k$, ...
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### 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. ...
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### 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 ...
• 2,437
Accepted

• 20k

### 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 ...
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### 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 ...
• 23.3k
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 ...
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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 ...