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- "It's intuitively clear, though mathematically one has to work a little bit to see this, that there is a maximal possible [proportion of the volume you can fill with spheres]." The sphere packing problem is to find this highest proportion, also called the sphere packing constant.
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A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement.
The problem of sphere packing is best understood in terms of density: rather than trying to determine how many spheres can fit into a specifically sized box, the more interesting question is how much of 3-D space can be filled with spheres (in terms of volume).
Apr 30, 2024 · Mathematicians can easily extend these arrangements, packing cubes in higher-dimensional space to perfectly fill it. Packing spheres is much harder. Mathematicians know how to pack circles or soccer balls together in a way that minimizes the empty space between them.
- The Problem
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- Closing The Gap
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"Suppose you have a very big box and a supply of spheres," Viazovska explains. "To make the problem easier suppose the spheres are of equal size and also hard, so we cannot squeeze them. We put as many spheres as we can into the box." The question is, what's the largest number of spheres you can fit in? If the box is small, then the answer depends ...
Involving no computers and filling just a fewpages, Viazovska's proofs are as solid as proofs canget. They involve packing higher-dimensional spheres into higher-dimensional spaces, namely spaces of dimensions 8 and 24. The endeavour might seem both useless and impossible to get your head around, but it's neither. Higher-dimensional sphere packings...
When you're trying to find a number that attains some sort of amaximum, like being the highest packing density, but haven't got muchluck, one approach is to lower your bar and look only for an upperbound: in our case a number you can prove the packing constant can't exceed. Various upper bounds for packing densities have been known for some time, b...
Viazovska's work, which closed the gap for dimension 8 and waslater extended with the help of Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko to dimension 24, builds on thecentre piece of Cohn and Elkies' work. If you forget about the actual spheres in a sphere packing and only consider their centres, you're left with a configuration o...
The obvious next question is whether similar techniques can prove sphere packing constants in other dimensions. The answer, sadly is, no. "For other dimensions the method of Cohn and Elkies gives some bound, but the bound is not optimal," says Viazovska. "Everybody asks what is special about dimensions 8 and 24 — I don't know, it's a mystery. In th...
Marianne Freiberger is Editor of Plus. She interviewed Maryna Viazovska at the Royal Society's celebrationof the centenary of the election as a Fellow of Srinivasa Ramanujan. She would like to thank Alison Kiddle for her take on higher dimensions. You can also listen to the interview in a podcast.
First: How many spheres can fit into a given space? Like, packed optimally. Second: Given a random packing of spheres into space, how much volume does each sphere account for? I think that's just about as clear as I can make the question, sorry for anything confusing.
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is.
Jun 30, 2016 · In any dimension the sphere-packing problem is the question of how equal-size spheres can be arranged with as little empty space between them as possible.
