42
votes
Accepted
Understanding sphere packing in higher dimensions
There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition ...
- 15.9k
32
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 ...
- 15.9k
18
votes
Accepted
Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?
Here's an attempt at answering your Question A:
Currently, one of the most powerful methods for proving upper bounds on sphere packing densities is the linear programming bound of Cohn and Elkies (...
- 946
18
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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18
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 ...
- 15.9k
15
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 ...
- 6,990
15
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 ...
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13
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/...
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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.
...
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10
votes
Double kissing problem
Using global nonlinear optimization one can obtain a configuration of $19$ spheres, that touch at least one of the central unit spheres and have almost no overlap. In fact, if one takes their radii to ...
- 10.3k
8
votes
Double kissing problem
This problem might be small enough to be solvable by a global optimization algorithm such as SobolOpt or VNS. See New Formulations for the Kissing Number Problem for more information about this ...
- 66.2k
8
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 ...
8
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 ...
- 4,547
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 ...
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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 ...
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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 ...
- 8,284
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 ...
- 15.9k
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 ...
- 15.9k
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 ...
- 408
5
votes
Three-dimensional Apollonian spirals
According to the discussion in Coxeter (1968), the tangent points lie asymptotically on a concho-spiral, so the distribution is not uniform on the sphere, but is uniform on a circle.
By the way, the ...
- 2,429
5
votes
Accepted
Minimizing deep holes in sphere packings
The problem you are asking about is sometimes known as the packing-covering problem, since it asks for a configuration with a fixed packing radius that minimizes the covering radius, irrespective of ...
- 5,758
5
votes
Is there a midsphere theorem for 4-polytopes?
In a recent paper of Padrol and me, we studied several generalizations of this problem. http://arxiv.org/pdf/1508.03537v1.pdf
Regarding Q1, Yoav already mentioned Schulte's work, and Gil mentioned ...
- 2,429
5
votes
Is there a midsphere theorem for 4-polytopes?
In the paper "Analogues of Steinitz's theorem about non-inscribable polytopes" by E. Schulte, which comes out of the collection "Intuitive Geometry" from 1987, the author seems to prove a negative ...
- 5,758
5
votes
Best non-lattice sphere packings
I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. ...
- 5,758
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., ...
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 $...
- 36.4k
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 ...
- 3,094
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 $\mathbb{F}_2^8$:
$(0,0,0,1,1,1,1,0)$...
- 8,284
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 ...
- 2,419
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}|\...
- 53k
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