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Sphere packing is the problem of arranging non-overlapping spheres within some space, with the goal of maximizing the combined volume of the spheres. In the classical case, the spheres are all of the same sizes, and the space in question is three-dimensional space (e.g. a box), but the question can be extended to consider different-sized ...
Sphere packing finds practical application in the stacking of cannonballs. In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space.
Apr 30, 2024 · To squeeze together three-dimensional cubes, you stack them like moving boxes. Mathematicians can easily extend these arrangements, packing cubes in higher-dimensional space to perfectly fill it. Packing spheres is much harder.
- The Problem
- Higher Dimensions
- Finding Bounds
- Closing The Gap
- Where Next?
- About This Article
"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.
Jul 24, 2017 · How does sphere packing based on a integer lattice compare to the best packing in other high dimensions? Although optimal packings are not known in high dimensions, upper and lower bounds on packing density are known.
1 Statement of the problem. Consider Rn and let r > 0 be a strictly positive number. How can we ar-range a collection P of non-overlapping spheres of radius r in Rn such that the volume between them is minimized? This collection P is called a packing, or. a sphere packing.
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Feb 27, 2014 · Spheres in 24 dimensions pack extremely well in a highly symmetrical arrangement called the Leech lattice. Each sphere touches exactly 196,560 others in a pattern that turns out to be useful in generating error-correcting binary codes.
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