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Results tagged with sphere-packing
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user 20595
14
votes
0
answers
2k
views
How many unit balls can be put into a unit cube?
Here a unit ball is a ball of diameter 1, and a unit cube is a cube of edge length 1.
A famous counterintuitive fact is that, as the dimension increases, the volume of the unit ball tends to zero whi …
1
vote
0
answers
304
views
Which term is better for the so called "sphere packing"?
I'm working on sphere packings. When I write, I'm confused with basic definitions. I'm hesitating between the terms "sphere", "ball" or "oriented sphere".
For example, on the wikipedia page of circle …
- 2,581
7
votes
1
answer
341
views
For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complemen...
The fractal dimension of the 3D Apollonian packing is computed in this paper.
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (Hausdo …
- 2,581
1
vote
Accepted
For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complemen...
OK, I found the reference:
For those who care, it's recently proved in a much stronger form by Oh and Shah in The asymptotic distribution of circles in the orbits of Kleinian groups. The paper is ab …
- 2,581
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 …
- 2,581
3
votes
Is there a midsphere theorem for 4-polytopes?
I recently showed that:
The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique.
While Eppstein, Kuperberg and Ziegler 20 …
- 2,581
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 t …
- 2,581