Editing Sphere packing (section)

==Hypersphere packing==
The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is [[circle packing|packing circles]] on a plane. In one dimension it is packing line segments into a linear universe.<ref>{{cite journal | last1 = Griffith | first1 = J.S. | year = 1962 | title = Packing of equal 0-spheres | journal = Nature | volume = 196 | issue = 4856| pages = 764–765 | doi = 10.1038/196764a0 | bibcode = 1962Natur.196..764G | s2cid = 4262056 }}</ref>

In dimensions higher than three, the densest lattice packings of hyperspheres are known for 8 and 24 dimensions.<ref>{{MathWorld |title=Hypersphere Packing|urlname=HyperspherePacking}}</ref> Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing.<ref>{{cite journal | last=Sloane |first=N. J. A. | title=The Sphere-Packing Problem | year=1998 | pages=387–396 | journal=Documenta Mathematica|volume=3 | arxiv=math/0207256|bibcode = 2002math......7256S }}</ref>

In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice|E<sub>8</sub> lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space,<ref>{{Cite journal|last=Viazovska|first=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 8|url=http://annals.math.princeton.edu/2017/185-3/p07|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=991–1015|arxiv=1603.04246|doi=10.4007/annals.2017.185.3.7|s2cid=119286185|issn=0003-486X}}</ref> and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions.<ref>{{Cite journal|last1=Cohn|first1=Henry|last2=Kumar|first2=Abhinav|last3=Miller|first3=Stephen|last4=Radchenko|first4=Danylo|last5=Viazovska|first5=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 24|url=http://annals.math.princeton.edu/2017/185-3/p08|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=1017–1033|arxiv=1603.06518|doi=10.4007/annals.2017.185.3.8|s2cid=119281758|issn=0003-486X}}</ref> This result built on and improved previous methods which showed that these two lattices are very close to optimal.<ref>{{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=Optimality and uniqueness of the Leech lattice among lattices | doi=10.4007/annals.2009.170.1003 | mr=2600869 | zbl=1213.11144 | year=2009 | journal=Annals of Mathematics | issn=1939-8980 | volume=170 | issue=3 | pages=1003–1050 | arxiv=math.MG/0403263 | s2cid=10696627 }} {{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=The densest lattice in twenty-four dimensions | doi=10.1090/S1079-6762-04-00130-1 | mr=2075897 | year=2004 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=10 | issue=7 | pages=58–67 |arxiv=math.MG/0408174 | bibcode=2004math......8174C | s2cid=15874595 }}</ref>
The new proofs involve using the [[Laplace transform]] of a carefully chosen [[modular function]] to construct a [[Rotational symmetry|radially symmetric]] function {{mvar|f}} such that {{mvar|f}} and its [[Fourier transform]] {{mvar|f̂}}&nbsp; both equal 1 at the [[origin (mathematics)|origin]], and both vanish at all other points of the optimal lattice, with {{mvar|f}} negative outside the central sphere of the packing and {{mvar|f̂}} positive. Then, the [[Poisson summation formula]] for {{mvar|f}}&nbsp; is used to compare the density of the optimal lattice with that of any other packing.<ref>{{citation|url=https://www.youtube.com/watch?v=8qlZjarkS_g |archive-url=https://ghostarchive.org/varchive/youtube/20211221/8qlZjarkS_g |archive-date=2021-12-21 |url-status=live|first=Stephen D.|last=Miller|date=4 April 2016|title=The solution to the sphere packing problem in 24 dimensions via modular forms|publisher=[[Institute for Advanced Study]]}}{{cbignore}}. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.</ref> Before the proof had been [[Scholarly peer review|formally refereed]] and published, mathematician [[Peter Sarnak]] called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."<ref>{{citation|last1=Klarreich|first1=Erica|author-link1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/|magazine=Quanta Magazine|date=30 March 2016}}</ref>

Another line of research in high dimensions is trying to find [[asymptotic]] bounds for the density of the densest packings. It is known that for large {{mvar|n}}, the densest lattice in dimension {{mvar|n}} has density <math>\theta(n)</math> between {{math|''cn'' &sdot; 2<sup>−''n''</sup>}} (for some constant {{mvar|c}}) and {{math|2<sup>−(0.599+o(1))''n''</sup>}}.<!--<ref>{{cite journal|last=Rogers|first=C. A.|title=Existence Theorems in the Geometry of Numbers|journal=Annals of Mathematics |series=Second Series|volume=48|issue=4|year=1947|pages=994–1002|jstor=1969390|doi=10.2307/1969390}}</ref>--><ref>{{Citation | last1=Cohn | first1=Henry | title=A conceptual breakthrough in sphere packing | doi=10.1090/noti1474 | mr=3587715 | year=2017 | journal=Notices of the American Mathematical Society| url=https://www.ams.org/journals/notices/201702/rnoti-p102.pdf| issn=0002-9920| volume=64 | issue=2 | pages=102–115 |arxiv=1611.01685| s2cid=16124591 }}</ref>  Conjectural bounds lie in between.<ref>{{Citation| last1=Torquato| first1=S.| last2=Stillinger |first2=F. H.| title=New conjectural lower bounds on the optimal density of sphere packings| journal=Experimental Mathematics| year=2006| volume=15| issue=3| pages=307&ndash;331 | url=https://www.tandfonline.com/doi/pdf/10.1080/10586458.2006.10128964|doi=10.1080/10586458.2006.10128964| arxiv=math/0508381| mr=2264469| s2cid=9921359}}</ref> In a 2023 preprint, Marcelo Campos, Matthew Jenssen, Marcus Michelen and [[Julian Sahasrabudhe]] announced an improvement to the lower bound of the maximal density to <math>\theta(n)\geq (1-o(1))\frac{n\ln n}{2^{n+1}}</math>,<ref>{{Cite arXiv |last1=Campos |first1=Marcelo |last2=Jenssen |first2=Matthew |last3=Michelen |first3=Marcus |last4=Sahasrabudhe |first4=Julian |date=2023 |title=A new lower bound for sphere packing |class=math.MG |eprint=2312.10026}}</ref><ref>{{Cite web |last=Houston-Edwards |first=Kelsey |date=2024-04-30 |title=To Pack Spheres Tightly, Mathematicians Throw Them at Random |url=https://www.quantamagazine.org/to-pack-spheres-tightly-mathematicians-throw-them-at-random-20240430/ |access-date=2024-04-30 |website=Quanta Magazine |language=en}}</ref> among their techniques they make use of the [[Rödl nibble]].