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13
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0answers
974 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 ...
5
votes
0answers
402 views

N-balls covering n-balls

This question is a follow-on question from: Covering a unit ball with balls half the radius The questions are these: Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
3
votes
0answers
97 views

Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?
2
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0answers
159 views

Computing the Volume of Closed 3-Manifolds and the Geometrization Conjecture

My question is whether or not if I generalize Theorem 2(i) of "Contact Graphs of Unit Sphere Packings Revisited" [2012] by K. Bezdek and S. Reid (arXiv link) which states The number of touching ...
1
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0answers
205 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 ...
1
vote
0answers
308 views

maximal minimum distance in a sphere packing

Hi everyone. I need to pick a set of 65 points p(x,y,z) in a 3D space of 274625 points; as the set picked should provide the maximum possible minimum Hamming distance. (Consider the Hamming bound for ...
0
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0answers
109 views

Generalized Sphere Kissing Problem

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...