34
votes
Accepted
Optimal sphere packings ==> Thinnest ball coverings?
Nope, this is false in three dimensions (where the body-centered cubic beats the face-centered cubic for covering) and eight dimensions (where $E_8$ is not even locally optimal). The Leech lattice is ...
- 16.8k
22
votes
Accepted
Can you see through a cannonball packing?
Yes. View the FCC packing as a series of stacked square packings, with spheres of unit radii centered at the points $(2a,2b,2\sqrt{2}c)$ and $(2a+1,2b+1,(2c+1)\sqrt{2})$ for all $a,b,c,\in\mathbb Z$:
...
- 3,068
20
votes
Accepted
Is there a short proof of the decidability of Kepler's Conjecture?
I don’t believe any short proof is known for the decidability of the Kepler conjecture, or indeed any proof other than Hales’s proof and its descendants. The issue is exactly what Hales explains in ...
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19
votes
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Illustrating that universal optimality is stronger than sphere packing
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered ...
- 16.8k
17
votes
Accepted
Construction of an optimal electron cage
Electrostatic potential is a harmonic function on any region without charges. It has no local minimum, in fact the value at the origin is the average of the potential over a sphere centred at the ...
- 54.2k
16
votes
Terrible tilers for covering the plane
Here is a better (possibly best) way of covering the plane with congruent regular pentagons:
The density of this covering is ${\sqrt5}/2 = 1.1180...$.
This covering is generated by the ...
- 7,276
15
votes
Accepted
Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
This is a tough question, and I don’t think there’s a definitive answer yet. For some mathematical details, see the following survey articles:
https://arxiv.org/abs/1611.01685
https://arxiv.org/abs/...
- 16.8k
14
votes
Accepted
Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks
For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking ...
- 7,194
11
votes
Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
In my understanding, the connection to modular forms came via a result of Cohn and Elkies in their paper "New upper bounds on sphere packings I," Ann. of Math. 157 (2003) 689-714, also on the arxiv.
...
- 2,175
10
votes
Kissing number lower bound vs. upper bound - precise meanings?
Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...
- 156k
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 ...
- 1,016
9
votes
Balls in Hilbert space
For convenience of notation, let me write the expectation $\mathop{\mathbb{E}}_i t_i$ to denote the average $(\sum_{i=1}^n t_i)/n$.
If I understand your construction correctly, you have disjoint balls ...
- 4,994
9
votes
Kepler conjecture: Are there only two most efficient packings or could there be more than two?
I am posting this as a community wiki post because I am just reiterating things said in the comments, but I think you are missing some basic points and so wanted to highlight those.
You should read ...
8
votes
Nonnegativity of coefficients of a modular form defined in terms of the Jacobi thetanull functions
Not an answer but a possible approach. Set $T=\theta_3$ and
$F_2(q)=\sum_{m\ge0}\sigma(2m+1)q^{2m+1}$, so that $T^4$ and $F_2$ form a basis of $M_2(\Gamma_0(4))$. It is easy to see that $f\in M_{10}(\...
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8
votes
Accepted
Kepler conjecture: Are there only two most efficient packings or could there be more than two?
The short answer is that classifying all maximum-density three-dimensional sphere packings, even the periodic ones, remains an open problem.
It will help to introduce some terminology. In a lattice ...
- 82.7k
7
votes
How many cones with angle theta can I pack into the unit sphere?
This is the problem of finding spherical codes. Putatively optimal solutions can be found at Neil Sloane's website.
For an upper bound, there's $d\leq\sqrt{4-csc^2[\frac{πn}{6(n-2)}]}$, where $d$ is ...
- 9,501
7
votes
Accepted
Sphere packing and kissing numbers in 3D
Fedja is right that the gap in the argument is why the kissing number should increase if you increase the density. The 12-sphere kissing configuration is not unique, and one can imagine rearranging ...
- 16.8k
7
votes
Accepted
Prospects for deep learning of non-lattice sphere packings
There’s definitely a lot of potential for finding great packings using computers. I don’t believe the known sphere packings up through 24 dimensions are all optimal, and a clever heuristic algorithm ...
- 16.8k
7
votes
Vectors that are almost orthogonal on average: lower bounds on dimension?
The Johnson Lindenstrauss Lemma states that there are $k$ vectors achieving $\epsilon$ provided $d\geq C \epsilon^{-2} \log k.$
If you actually want to bound the maximum absolute value of the inner ...
- 10.4k
7
votes
Accepted
Upper bound of the kissing number in n dimensions
It’s almost certainly true, and provable, that $\alpha=\sqrt{6}$, although I haven’t worked out the details rigorously. Kabatiansky and Levenshtein give an exact upper bound (not just an asymptotic ...
- 16.8k
7
votes
Optimal sphere packings in dimensions different fom 8 and 24
[Collecting some of the comments in a community wiki answer.]
It's not clear whether you're asking about upper bounds or lower bounds.
At first glance, it seems you are asserting that in dimensions ...
6
votes
Accepted
What high dimensional lattices have Voronoi cells that have this property?
I always consider Voronoi cells of lattices as closed sets. Write $V(v)$ for the Voronoi cell around a lattice point $v$.
I assume that by two neighboring Voronoi cells you mean two Voronoi cells ...
- 558
6
votes
Nonnegativity of coefficients of a modular form defined in terms of the Jacobi thetanull functions
Hmm, after a bit of googling I found that my conjecture is in fact proved in the 2018 Harvard honors thesis "Modular Magic" by Aaron Slipper. (The lucky google search that yielded this was &...
- 2,549
5
votes
Accepted
Sphere packing processes during biological development
It is unlikely that the solution is known generally (for all $N$), but there are partial results known for similar kernels, or for these kernels with additional constraints.
At first approximation ...
- 3,209
5
votes
How many cones with angle theta can I pack into the unit sphere?
A good reference for volumetric arguments for the maximum number of 'cones' or spherical 'caps' that one can fit, is a series of papers by Jon Hamkins. The density of a packing of these caps can be ...
- 3,209
5
votes
Question arise from kissing number in 2 dimension
$a^2+ab+b^2=(a+\omega b)(a+\omega'b)=N(a+\omega b)$, where $\omega,\omega'$ are $(-1\pm\sqrt{-3})/2$ and $N$ is the norm in the field ${\bf Q}(\omega)$, takes on all and only the values of the form $...
- 39.9k
5
votes
Accepted
The lattice handshake number ("nearly kissing" number)?
I'm happy to say that this question has been answered by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: https://arxiv.org/abs/1802.00886 !
I.e., ...
- 1,016
5
votes
Accepted
Perfect sphere packings (as opposed to perfect ball packings)
Question: Is this the only way to construct a perfect sphere packing in a finite vector space?
No. Take the linear space $V$ generated by the following vectors in $\mathbb{F}_2^8$:
$(0,0,0,1,1,1,1,0)$...
- 9,501
5
votes
Choosing maximum number of separated points on a sphere surface
Having scalar product above $\rho$ is equivalent to having spherical distance above $2\theta$, where $\theta=\cos(\rho)/2$. I consider first the case of the whole sphere, and I'll discuss the positive ...
- 3,669
5
votes
Accepted
Monotonic dependence on an angle of an integral over the $n$-sphere
That is technically a 2D question. We can assume that $v=e^{-it}, w=e^{it}\in\mathbb R^2$ ($0<t<\pi/4$). Then in the polar coordinates the integral becomes
$$
\int_0^1 \varphi(r)dr\int_0^{2\pi}|\...
- 61.9k
Only top scored, non community-wiki answers of a minimum length are eligible