sloane - the packing of spheres

8/13/2019 Sloane - The Packing of Spheres

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The Packing of SpheresWhat is the densest way to arrange identicl spheres In space?

There has been much progress on the problem, particularly in 24

dimensions, and the results can be applied to digital signaling

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FCE-CENTEREDCUBIC packig ofpere, ofte ee i fruit tad or i pieof caoba at war memoria, i tougtto be te deet packig of pere i tredimeioa pace I pite of ceturie of effort, owever, a proof of it maxima deitya ever bee give Eac pere i te packig "kie, or touce, oter pere aproof tat ti umber i maxima wa otgive ti 874 If te ceter of oe perei xed, te et of a poibe rotatio adreectio tat permute te urroudigpere i caed te ymmetry group of tepackig Te ymmetry group of te facecetered-cubic packig a 48 eemet tey

ca bet be udertood if te ceter of eacpere i tougt of a te vertex of te poyedro at te rigt, wic i caed a cuboctaedro ye of te ix quare face ofte cuboctaedro ca become te frot faceby a appropriate rotatio of te gure aboutte gree or te bue axi Eac et offour pere tat form a quare face (ay tefourt face) ca te ame oe of four coguratio if te etire gure i rotated aboutte red axi Fiay, eac coguratio (ay te tird oe) ca be reectedabout a vertia pae to give a ew coguratio J J Te tota umber of eemet i te ymmetry group i terefore 6 4 , or 48 aaogou ymmetry group,

wic decribe te rotatio ad reectioof a dee packig of pere i 4dimeioa pace dicovered by Jo Leec at teUiverity of Gagow, a bee importati te matematica teory of ite group

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1983 SCIENTIFIC AMERICAN, INC


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117 1983 SCIENTIFIC AMERICAN, INC


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bi pakig f phr rib abvh kiig ubr i i 1 [se illusraion on preceding pge] Grg-r prbab argu ha a aiiaphr b uz i, ahughh a ab prv i.

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"SPHERE PACKINGS c b crri out i o two imio w i tr Io imio t pr r i gmt of uit gt, ctr t itgr poit Tpr covr 00 prct of t i c pr i two otr Ti pcig i bviouy poib: it i c Zl. I t p t pr r circ tr rtr pcig of itrt I t Z2 pcig t pr r ctr t vry poit i p wo coorit r itgr i t D2 pcig t pr r ctr t trtgpoit of t Z2 pcig, i ccrbor fio If t coorit x of t Z2 pci r

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vr a i r vu ih phr i ia a h phr i a phr pak-ig ui h pa a h ph i h rigia paki g. gra,hvr, iaig h ph i h vrappig gurai a h b ui f hvrig prb. hr ii,fr xap, i i biv h b v-rig i giv b phr arrag a hvrx f ha i k a a brubi ai. f vap-pi g phr ar r a h pi,hvr, h pakig ha ru i a a hr k ph pak-ig, uh a h farubipakig. Mrvr, h ju habrubi pa kg v hvrig prb ha b prv.

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LAMINTED ATTICE PACKING 3

BOTTOMLAYER YR

HON O K NON

P YR

CONSTRUCTION of the aminated attice packing L3, which isequivaent to the face-enteredcubic packing D3, isdone by ttingtogether aers of spheres whose enters are aranged aording to

the hexagona attice packing L If the sphees are packed so that

the spheres in the thid hexagona ae ae diecty above e spherein the t aer, the aking is caed the hexagona cse packing.The heagona ose aking is just as ense as L3, bt te centes ofthe spheres in th"e paking do not saisf the denition f a attie.

re l ntned nentely theresltng arrangement f spheres wldhave he greatest pakng ensty pss-le sne every tetraheral ngra-tn wld e pake as ensely as ps-sle. Rgers' pper n fr theensest sphee pakng pssle s asen presely ths argment a straght-frwar mptatn n spheral trg-nmetry shws that at 779 f thevlme f a tetrahern s lle y thefr nnverlappng spheres tha t an eentere at ts vert exes. The exa res lts V[3 ars (13) ].) Unrt-nately hwever tetraherns nt tperfet ly tgethe r t ll spae. The strategy f pakng spheres nt teraheralarrangemens whenever pss le whhmght e alle a greey algrthmeventally fres a a me. t a er-

tan st age n he pre re the grwngngratn f spheres presents a srfae that annt arete mre sphereswtht wastng nterr spae. Henealthgh the greey algrthm generates an ptmal sphere pakng ver theshrt rage f say a few spheral dameters t trns t hat the algrthmgves rse t a pakng that s less ensen a glal sal e than the faeentered - pakng.

n rder t make frther prgress nthe hree spherepakng prlems

mathematans have fnd t nveent t spplement gemetr nttnwth an analyt representatn f thespheres n terms f ther retangar

rdnates I t s well knwn that any pntn the plane an e pee y tw -rnates a hrntal rnate x anda vertal rdnat y: the pnt s gen-erally wrtten as the rdered par (x,y).Fr example the pnt (34) refers t thepnt n he plane three nts t the rghtf the rgn alng the x axs an frnts ave the rgn alng the y axs.

The stane etween say he pnt(34) n the plane an any ther pnt(x,y) an e allated frm te Pythag-rean relatn amng the ses f a r ghtrangle The sqare f the stane e-tween the tw pnts s eqal t thesqare f the dstane etween themalng the x axs (x 3 ) pls thesqare f the stane etween themalng the y axs (y ) Sne a rle sy enn the set f all pnts

the plane eqstant frm a entralpnt ab) any pnt (x,y) n the rmferene mst satsfy the eqatn(x - a) (y b) R where R s theras f the rle. If the ras fthe rle s eqal 1 an the enters at the rgn 00) the eqatn smh smpler all pnts (x,y) n thermfeene mst satsfy the eqatn X y 1

mlarly any pnt n threemen-snal spae s speed y three rd-nates x, an : mre s ggestvely thepnt an e wrtten X[XX) The

srfae f a sphere f rads eneredat the rgn s made p f all the pntsX[XX) sh that Xt X X 1the eqatn arses mh s t des

n tw d mensns frm the gemetrentn f a sphere and fr tw ap-platns f the Pythagrean relatn.

In mre than three mensns gemetr nt tn s f lttl e vale and nes egn t thnk exlsvely n termsf rnates Fr example a "pntn frdmensnal spae s a mathemat al et that req res fr d stnnmers n rder t e speed namgsly sh a pnt s wrttenX[XX4)' If a per sn's hegt weghage and nme are sent t namgsly p k t the ame f tat persn frm a lst the fr qantte ane regarded as spefyng a pnt n afrmensnal spae.

frdmensnal sphere s deney analgy wt the dentns fr therle and the sphere n tw and three

mensns. ll the ns X[XZXX4)n the "srae f he sphere are fnat sme dstane R frm a entral pnta[aaa4)' The sm f th sqaresf the dstane alng eah ndependent rdnate axs etween any pntXtXXX4) n the sphere and the en-tral pnt a[aaa4) mst always eeqal t R

There has een a great deal f nn-sense wrtten n sene tn and else-where at the myster es f the frthmensn. n mathematal dsssnsne mst nt assme as the physst

des that the frth mensn represents tme. rthermre ne mstav the temptatn t refy the sme-what metaphra erms "srfae

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O MNON H MNON

y 'X2 + y = 1

00

POINT IN TWO IMENSIONS is speied y assigning a vaue o wo oordinaes, say and . A ire of radius around the origin 00) is e se of a poins () ha saisfy heequaion 2 2 lef In hree dimensions hree oordines, say X Xz and X3, are neededto speif a pin The surfae of a sphere of radius around e origin 000) is he se of apoints 2,X3 that saisfy he equaion X2 2 X32 In dimensions a poin is speied b rdinates, XX2,n' The surfae of a dimensiona sphere f radius entered on the rigin 00 . . . ,)ishe se of points (XX, )suh a2 X22 . . 2

"pont, "sphere and so on, whch areapple d to mathematcal ojects whosealerc propertes are otherwse perfectly strahtforward The terms arejsted ecase the ojects they referto are constrcted y analoy wth thealerac propertes of ordnary crclesand spheres; t st not e spposed,however, that the ojects represent realeometrc oects n some nverse wder than or own To repeat, n mathematcs fordmensonal space con

ssts of ponts wth for c

ordnates nstead of three and the same holds forany nmer of dmensons)

Once the concept of a spere s n derstood as a relaton amon coord

nates, there are several prolems, seemnly nrelated to sphere pakn, thatcan e r ed ced to one of the threespherepackn prolems The prncpalapplc atons of hhdmensonal spherepackn are to prlems n dtal commncatons, partclarly n the constrcton of snals for se on a nosychannel and n the desn of analoe

todtal cnvertes It mst e admtted, however, that most dscssons ofthese applcatons have een theoretcal; only qte recently have spherepackns een sd to desn practcalsystems s commncatons systemsecome more sophstcated frther applcatons can e expected

In dtal commncatons one of themajor ams of the desner s to constrct a lst of dstnct cded symols, orcode word, thatcan e transm tted wt hmaxmm relalty and mnmpower Each code word mht e repre

sented as, say, an ehtdt symol,each dt of whch can take on one ofve dstnct vales 0, /2, , /2 or t rst t wold seem that the sys

te wold provde for , or 390,,derent code words, t the derenceetween many pars f these coe wordss so small that sc a system wold ehhly s ject to random errors n tansmss on or to electrcal nterference Forexample, the derence etween thecode word (1 ,1 ,, 1, 1,,,1 ) nd the codeword (, 1,,1 ,, 1 , 1 , 1 /) s far too smllIf oth code word s were n se, t s l ely they wold ofte n e confsed Another way to pt the same pont s tat f

the derence etween two code wordswere as small as te d erence etween(1, 1 ,, 1 , 1 , 1 , 1 ,) and (1 , 1 , 1 , 1 ,, 1 , 1 , 1/) ,a very lare amot of power wolde needed to arantee that tese twocode words cold e dstnsed nthe presence of ackrond nose

There s a eneral mathematcal relaton etween the dstsaltyof code word s and the power need ed totransmt them relaly The relaton wasrst formlated n 98 y lad EShannon, ten of the Bell TelephoneLaoratores, n s paper Mathemati-cal Theory o/Communication As my ds

tnshed colleae avd Slepan assad, "proaly no snle wor thscentry has more profonly alteredman's nderstandn of commncaton than Shnnon's paper What Shannon showed s that ven some xed,nte amont of power there alwaysexsts a system of code words that cane transmtted essentally wtot errorThe only qalcaton s that te ate atwhch the cde words are transttedcannot exceed a crcal treshold calledthe capacty of the tansmsson channel Unfrtnately Shannon's theorem

s onconstrctve t proves that schsystems for encodn a sal exst, tt ves no nt of how they ht eesned Althoh many snaln sys

tems have een onstrcted, schemestha perform as w ell a s Shannon's theorem promses have stll not en fond

One way to desn a snaln systetht comes close to meet the

standards of Shannon's theorem s torepresent each snal as a pon t n dmensonal space For exale, consderany seqene of eht nmes n thesnaln system descred aove Phys

cally each of te nmes correspondsto a voltae lev el on a transmsson l ne,and so each code word can e plottedon a twodmensonal raph as a seresof et dstnct plses whose hehts speced for each of eht ntervalsalon the tme axs Mathematcally,however, a sne pont n ehtdmensonal space can represent the same nformaton let the rst nmer n eachseq ence e the vale of the rst coordnate of the pont, the second nmer ethe vale of the second coordnate andso on Snce a pont n eh dmens ons

s determned y xn the vales of alleht coordnates, every code word nthe system can e represnted as a ds_tnct pont n ehtdmensonal space

When the ode words are represented as ponts, two mportant featres ofa system of code words can e ven aeometrc nterpretaton Frs, rememer that the code words mst e relaly dstnshale from one anotherIn ehtdmeonal space ths seststhat the ponts representn the possle code words mst e separated y acertan mnmm "dstance How welldoes the Pythaorean dstance etweetwo ponts n eht densons measre the dstnshalty etween twocode words? In order to determne thePythaorean dstance the derence etween the two vales of each coordnateof the two ponts mst e sq ared Smallderences etween coordnate vales(tha s, derences less than 1) are therey r ed ced, whereas lare d erences(that s, derences reater than 1) aremaned Becase a small derence nthe voltae levels etween two snals smch more lkely o lead to confsonetween code words than a lare der

ence, the Pythaorean dstance s a reasonale m easre of d stnshalty

For example, consder the two codewords(1,,,, 1, 1,) and (1/, /,1/,/2,/,/2,/2,/2) The sq are of thedstance etween the two ponts s thesm of eht sqares, each of the form( 1/), and so the dstance etweenthem s v Accordn to the Pythaorean measre f dstance, the two codewords (,,1,,,1,1,1) and (0,0,1,1,,,, ), whch der only n the rst tocoordnates, are also nts apart,and so they are jst as easy to dstn

sh as (,,1,,,1,,1) and (1/2,/2,1 /2,/,/2,/,/, 1/)

A second mportant featre of anysystem of code words s that the power



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te qae o te itance etween tepoint epeentin te coe w ineitieniona pace an te oiin,,,,,,,)

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powe can teeoe e euce to teeoetic poe o placin point inie a eion o pace wie containinte not to e to o coe toete I tepoint t e at eat a itance o, ay,V apat, te poe i equivalent tot pole o nin e enet pac

in o pee woe a iu i al tat itance, A cloely eate poe i to n a et o coe wo tat alave te ae eney Ti poe iequivaent to te pole o placin aany point a poie on te uace o an ieniona pee wiecontainin te not to e too cloetote Tat pol i in tun aenealize veion o te iinnue pole

It tun out tat in a pace o eitienion tee i an exteely enepacin cale te pacin it wa icovee in te lat ti o te 1 9t cen

tuy y te uian ateatic ian Aexane N oin an Zolotaean y te ni awye an aateuateatician Too Goet Tecente o te pee in te pacinae al te point woe cooinate aeequa to wole nue o to woenue plu a a Te u o t

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f te rt vte eve te e ritei te vue f te e vte eve (AD). e trii er require E

fr e vte ue i rrti t tequre f te vte te tt ereee t trit e e r i te uf te qure f te tree irete vteite it te e r e u iequ t te qure f te ite fr terii t te it i treeiei ett rereet te e r. u te re f iiii trii er i equivet t te re f i te ittt rereet e r e t te rii ie. O te ter te ee titiui e r fr e ter trete te requireet tt e iti e rereeti te e r t eer teter t e iiu ite

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12,2,12,12), whee thenme of mns sins s even.

The packn cold ecome the ass of a pactcal and ecent snalnscheme. If the scheme wee to incldeexactl y 20 code wods, the 20 pontsn the packn that ae eqdstantfom the on cold e chosen as thecode wods. In most pactal systems,howeve, the nme o dstnct codewods needed s some inteal powe of

2. Fo example, n med umdist ance telephony the dtal tansmisson systemknown as pulsecode modlation isnow n wde sevice. The voltae of tevoice sinal s meased evey 1/8,second, and the meased samples aeqantzed, o eplacd, y one o ,o 2, levels. ach quantzed voltaelevel s then expessed as an ehtdiitnay nme sch inay numesmake p what s called the souce code

It has een known since the wo oHay Nyqst o Bell Laoatoies inthe 1 920's tat the voice sinal can be

econstcted fom the sample valesalone. The voce sinal can e thouhtof as a qantty, sch as a pessue ovoltae, that vaies contnously withtme. aly in the 19th centuy eanBaptste oseph Foie poved tatte aph o any such qantity can beappoxmated to any desed deeeo accacy y speposin sne andcosne cves of appopate amplitdeand feqency The sine and osnecves (which, fo an atay uve,

may e innite n nume) ae calledthe Fouie components of the cuveon the aph

Sppose the aph o a qantity cane exactly eneate y speposin anite nme o Foe componentswhose feqencies ae no hhe thansome eqency W cycles pe secondThen wat Nyquist showed s that theap can also be peect l econstucted solely om the values t takes evey

1 W second Fo example, a voice s inal that has no feq uency componen tshihe tan ,00 ycles pe second cane peectly eonstucted om sampleso the sna taen evey 1/8, seond. It s theeoe enouh to sample thevoie sina and tansmit nly the smple val es, whic ae epesented by theode wods in e soue ode, insteado tansmittin te entie voice sinalI te sampe values we ansittedwithout bein ounded o, Nyquiststeoem sows tat an entie seondo speeh sinal, whih oesponds to

8, sample values, an be epesented as a sinle point in a space o 8,0 dimensions Tis shows the powe omathematcs

Fo ecient tansmission the soueode numes tat epesent te

sample values must be uthe encodedby anothe code, alled a cannel codeit is the channel ode that petai ns to thespheepain poble n excellentway to deive a annel code om te

QUANTIZING DAA from a continuously vaiable source is closely elated to the poblemof coering ace wit te leat ene arrangement of overapping spheres. For example, inorer to quantize data in two dimension the input data ae paired and each pair is teated as coorinae of a point in he plane. Each data point, such a the point A is then rounded oo a quantizing poin, uc as te point that lies in the ame preselected egion of the planea te aa point Te poblem i o coose the uantizing oints and the partiio of the planein a wa at minimize te aerage qantizing eor. If the daa points are uniformly distribute an e quantizing points are coen as the centes of suare, the average error is / ete way of qanizing can be derived from the best coering of the plane by cicles. hatcoering i generaed from te eagonal packing of circes lk iles by increasin theraiu of eac circe by jut enoug to include every poin in he pane inside or on the circumference of a ea one circe lred irles he radius of the coverin circles is the disance from e center of a circe to the eaest "deep hole in the packing. If he deep holesae connecte by te appropriae raight lines, te plane is patitioned into egula hexaos;for uniformly distributed data poin the averae uantizing error for uantizing points thatae centere on regular hexagons is v8 o about 8 which is slihtly ess tha

sphee packin is to encode ea paiof successive eihtd iit inay nubesin the soce code ach 1 d it binaynme that eslts s then assined tothe ent e pon t o one o 2, o ,3,spees in the pacin. ood annel code can then be onstcted bychoosin the ,3 cente poins tatlie clo sest to the oiin t the eeivinend o te telepone line the ode wodsthat coespond to te coodinates o

each cente point ae conveted bainto the nay nmbes o te soueode, and the voice s nal is econs tuted om the binay numbes.

Tee is a seond majo appliationo sphee pacn in dital omunications tat sall desie bieyememe tat in deivin te binaynumbes in te soue ode o tevoie s inal o the telepone i t is neessay to quantize te peise intensity ote sina to one o levels e ealwod is ull o awwad numbes sas 9 1 3 . . . , bt te wold o opues

and diital systes must ultiately deaonly wt ound numbes su as and1 . ny device tat ounds o a ontinuously vaiable qantity o soe set odisete values is al led an an alouetodii tal onvete, o q uant ize

Qantization an be caied out intwo o moe dime nsions as wel as alona sinle oodinate axis Iaine tate plane is divided into eions, notneessaiy onuent, and imaine tawitin eah eion one point as beenmaed ny su aay o points andeions an unction as a twodiensional quantize te input to te quantize is a pai o eal numbes that speiy some abitay point, and te outputis the peseleted quantizin point tatlies in the same eion o the plane aste abitay point hs any point in eplane i ond ed o to one o te q uantizn ponts The poess ompesesinput data a sinle i ndexin numbe ote quantizn point an be tansmittedin place o the peis e oodinate val ueso the data point.

Quantzin intodues eos, and soone ties to choose e quantizin pointsin sc a way as to mini mize the aveae

eo Fo example, i te input to tequantize is unioly distibuted, oin ote wod s i evey input val ue isequally pobable, it is staiowadto alculate te aveae eo o a nube o quanizin sees sineoodinae a xis is divid ed into eq ualseqmens o unit len and te q uanizin points ae enteed on ea seen,te aveae eo is /, o abou833 e same daa an be quantiedin two diensions te data points aepaied, and ea pai is onsideed apoint in e plane te plane is en

divided into squaes and te quntiinpoints ae eneed on ea squae, eaveae eo is stil /1 n te oteand, i te plane is divided into eu



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hexagons having th same area as squares, and if each quanizing poi isa he cner o a hexagon, he averagquantizion error can be reduce o50 or abou 002

Uemarkably i turns ou ha a corre. sponding iprovemen can alwasbe made even i the daa pois ar noevnly disribued. In a 6 docoral

disseration a Sanord Uniersi, P. Zador showed tha i is always possibleo reduce e aerage error by quanizing in a space o higer diesions. I ismor ecien o wait unil seeral dapoins ave been colleced and enquanize hem all a he same ime bregarding hem as a poi i -dimesional space ha it is o qanize emone a a ime along a sigl axis Iquanizig it pays o procrasiae

UnfortunaelY Zador's resul, likehannons eorem, is onconsrcie:The problem of ding goo mliimensional quanizers ven o niormy disributed daa is sil nsoleThere are seeral sphere packigs, owver, hat apear o give rise o excllquanizing schees Consider packing o spheres i two dimensions, a is,the packig o circles. I has been kownsince 40 hat circles can be packed aaximum densiy i he r arragedso ha each circle is surronded b sixohers [see ustraton on opposite pge].

Imagine ow hat eac circle in epackg is bounded by a in, exilemebrane and a te ierior of tcircle is inted. As e circls gt big

ger he membranes press agains oanoher o ll e rmaining space ith pane; i he inaion is uiformthroughou e plane, eac crcle willexpand o orm a reglr exago. As Ihave meniod, basig e qanizaion o uniorml disribted daa ohexagonal regions leads o miimum avrage error. A similar exansioof he eight-diensiona sers i e packing also leads o low qanizaion error, wic is een smallr ahe error in wo dimensions. Te geralproblem o quanizaion, wic reqires

hat space be broken up, or coered, b adiscree pariion, is closely relaed ohe problem o nding te bes coeingo space by speres.

The searc or dense spere packigsin mulidimensional spacs is grealsimplied by ocsing atenion o cerain kinds of packigs calld laicpackings ha hae a ighly reglarconguraion. Consider exagoapacking o circles I havejus describednoice hat he cners of an wo adjacen surrounding circles and he cenero he circle in he middleJorm an qiaeratriange In order o cacu edensiy o the packing it is sucient odermin he proporion o eac triang tha is coered b circs or pars ocirces. Because he riangles tsslla,

or compll ll, pn a casethe coguraio o ccs thi a riangle is lwas s h circle desi in oe rigle is qa o the nsito pacig rogou th pl appling mer gomtr o cansow e desi is ql o 0/6 orapproximael 6

Te oregoig calclaion wol oha been possibl i o repaig i

at lls e lan could a befoud It is noneeless eas to imagia er xis igl irglr sperpackigs ta a no repeaig nits.Sc packings re mc arder o sda ose a repea o only is edesi of an irreglar packig icor ipossible o deermie so ee coordiaes of ll cr poism o b seciabl diio of laice packig gares esedisaaags c be aoided A sperepacking is si o be a laic pckigi wener re ar wo spes oa as s cr a e oi(UhU2'" u) an e or a e poi(VhV2, v) ere ar so sprs in packig wi ceers ll poiof e orm (ul + bVhu2 + bV2U + b) were ad b ar aywole mbers Te cner cooriaeso laer srs are said o be gerad from cer coories oe s wo seres

Te sipes lic packing is cbic laic packig i wic e ceercoordies o ac sere i e ackig are egers; e cbic lice i aarbirar nmber of dimsios n is s

igaed Z Te odimesioal "cbic ic ZI i mae p of discre lisegmns, ac o i e, a arceered a e ieger pois o lie.e spers, or lie segmes, coer percn o e li, ad ec sroces wo oers; nc Z soles spere-packig roblem a e issgnmber problem oe imsio.

n wo diesios, ower,

sqar laice 2 s o e sesackig. Its pcki esiy is /4 orabo 4 fr les an e esi of

e exagol pckig of crles [see ustion Ol pg 118 I lik mre spere-packig esit of e ciclice is reliel low i is /6 orabot .5236 A mc dense amil oflace packigs c be geered romcbic laies sprs are ceera alere ois o laic i cckerboard fasion o costrc e wfamil of laics color e pois of acbic laic alrael re a blckand make cenrs o e sperescoincid wi t black poi Eqivalel, e cers of w pacig ar pois wi inegr coorinas a ad up to an e mbrI an arbitrary mb of dimsios tis packig is desigat n Dor xamp, orign 0 a

poi re ga cers of shrs,b e oin 0 is no becaus + + is a odd umber.

Te seqnce o atice packings DD so o i of cosierabl imporc for e preacking problem e ckig D is e facecnrdcic lice model consructdof PgPog blls sows a one rpaig i cell o e laice is a cube

wo is o a side aig a sr at tscnr; rais of eac spere is 2 Te dsi of t ackig can be ca-

Lo= EITATICE

L= IEGE

b= EAGAATICE

ATTICE= FACECETEECUIC ATICE J

L''

L =

L =

La= GETATICE Ea

L

L,.

L

b!

L

Ls =s

L

L'

Ls

L

L'9

L

bJ

t.a lace packngs of spbr in dmnsons a ult up y packng togtbrlays of a suital lainatd atc packingn tnxtsmallrdnson -1.Forxap, tb twonsonal bxagona packing can ult up ckn rows of circlswb cnt ar arangd according to t packng Siilaly layrs of spbr, acblay arrangd according to tb bxagonal lat

tc packg can sck to yld tb lattcpackng L3 tb dns packing known inth dnsons Jobn Hoton Conway o tbUnv of Carid and tb autbor bacontnud tb constructon and found all tlaminatd lattc n dnsons up to 5 Tb

laminatd lattc in dinsions 1 t andn dmnsos to ar unqu; tbr artwo lainatd lac in dinsion 11 tracb in dinsons 12 and n dision 5 and at last 5 in dnsion



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culat d by nd ing th fraction of the

volum of the cube that is ll ed ysphrs; t s qu to V1/6, r abut.7405 Althought s possibl thatdens-er sphr pckngs exst n thr d men-sns Crl redrch Gss prved n1 83 tht D s the ensest thredmensnl lttc pacing. It s ls nwnthat D nd D re the dnsst ltticepckngs n fr an ve dnsns.

Av v dmesns hwvr Ds nt th dnsest latt ce pckng nd ythe tm ne reches D thr r hgegps twn the sheres. The gs res rg tht t is ssie t slde -ther cpy f D t the hls wth-t vrlppng th sheres; the resttrns t t e the ttce. I 1934H F. lchfldt Stfr Uvrsty prvd tht s the densest lttcepcng n eght mensns nd heswd tht certain crss sects f cale n ae the ensest tticepackngs n six n see mensns. Inthes dmnsns enser packngs

whch wd necessry e nnltticepcngs hve een fnd snce.

In 965 Jhn Leech tn t the Ui-versity f Glasgw cnstrcte re-marale sphere packng n 24me-snal spc; hs cnst[ctn s tlnen the ilstrtn elw. The stdy fthe Leech lttce as the pckng s s-aly cld hs le t a deeper ner-standng f the prpeties f ther hgh-eriensn attices and t mpr-tt reslts n the tery f grps.The lttce s st certnly the dens-est sphre packg psse n 24 i-

esns. C. A. Rgers rging s he

dd fr sphere pckng n three n-sns g ns r the xmensty f pckngs ny ensnl spce; hs nd fr ny 24dmensnl sphre pckng is nly lghtlygreter thn te desity f the Leechlttce. Eh sphre n the lttce tchs 1 96560 thers n 1 979 . M.Olyk f ell Lrtries a Iprved that ths nmer slves the ss-

ngnmer prlem n 24 mensns.The same ethd f prf s svethe kssngnmer pre n eght -menns; there te swer s 24 tener f spheres tht tc ne spere te tce Tese tw rests weref nepeently y V I Leven-shte f t L. V Kelysh Isttte fApe Mthetcs scw. Incdental the rlem remas slvedin ll ther dmensns except ne twand three whre the answers re e-spectively tw sx n )

The Leech lttce hs een sens-ale t grp thersts fr the cstrc-

tin f certn nte smpe grs.These grps re the ing lcs fal grs havg nte er f el-eents; ther clssictn which hsjst een cmpeted hs preccpiedmy th emtcns fr mre t han 50years The simple grps pay ch these rle n grp thery that the premers py in ner thery a thecema elements play in chemistry.Sever imprtnt smpe grps havew ee cnstrcted y cserngthe set f al the rg rtatis a re-ecins f the Leech lttce that leave

the centr sphere xe a per te the

2 9 2 9 2 2 22 2 2

1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1

0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1

0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1

0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1

0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1

0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1

0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1

0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 10 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 10 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0

CNSTRUCTION OF THE LEECH LATICE, the densest known acking of spheres in2 4-dimensionl spa, is based on the 24-digit binary sequenes shown. The set of all possiblesums e 12 binarysequenes, where the additionisarried out mdulo , is madeup of 12,or 4,096, binary sequences called code words. (In addition modulo the sum of1 and 1 is equalto 0; te iit ordinarily carrid in binary arithmetic is ignored.)Th 12 code words ake upan ecet code for transmitting inormation that was devised by Marel J E. Golay a the'U.S. A SinalCorps Engineering Laboratories in 1949. The enters of the speres in theLeec cell have the form2C + 4Xor + 2C +4 Y where Cis a ode word of the Goly ode,

I is te int (1 ,1, . . . , 1 ) in 24 dimensionsand X and Y range over all points in 24 dimensionswhse cnates are all integers. The su of theoordintes of eah poin X mustbe evenandte sm o the oordinates of each poin Y ust be odd. Eah sphre has radius 2v, andthe spe closest to the origin have enters suh as ( 4,4,0,0, . . . ,0), ( 2,2,2,2,2,2,2,2,0,, . , 0 nd (t 3, I, I, . . . , I). ah sphere ouhes 196,560 others.

srrnng spheres. Ths st f era-tns s cle the symetry grp f thepckng; the nls symmtry grpf the fcecenteredcc pckng nthre dmensns s shwn n th ls-trtn n ges 1 6 and 1 7.

Th symmty grp f th Lech t-tce ws fnd n 968 y Jhn Hr-t Cnwy f the Unvrsty f Cam-rdg. Its rer r th nmer f

lemnts n the grp s mmnse -thgh nt ptclrly lrge n thectxt f grp thry: it is2 X 3 X 5 X 7 X 1 X 1 3 X 23 r83 1 55536308672000 Frm thsgrp wch s nt smle grp Cnwy cstrcted thre prvsly n-knwn smpe grps whs rer ec cse excty vdes the rder fth symmetr grp f the Leech lttic .In 1981 th Leech attce nled R-rt L. Gress Jr. f he Unverst fMchgn t cnstrct ne f th stnte simpe grs t e fn. It isther lrger thn Cnwys grps an

hs een nckmed the mnster: thenmr f eements n the grp is2 X 3 X X 7 X 1 } X 3 X 1 7X 1 9 X 23 X 29 X 3 X 4 X 4 7 XX 7 1 r 8080 1 742479451 2875886459904 961 7 1 07570057 54 36 800000000. Gresss cnsrctn s tt al strhtfrwar hwever ane f the fascntns f the eech lattce s tht ne feels there shd e amre irect cnnect etween t athe mnster smpe grp.

T

he Leech lttce s sch dnse pac -

ng tht ts inece s fet all w-er mensns. I t s t srprsng that aslce gd packing gvs gckg n spce f ne wer dmens; fr exaple ne slce thrghD exses srfce f hexgnalypcke spheres. Sitale crss sectnsf the Leech lttce wever gve riset the ensest knwn packngs n l dimensns less thn 24 except 1 0 a1 3 . Fr exmple ne eghtdmensnslic thrgh the Leech ttce expseste lattce

Snce dense packngs cn e lt

frm te Leech lt tce s t spe frthe tp dwn t s temptng t hwth Leech lttce mght e lt fr thettm p tht s frm en se p c gsn wer mensns. t trns t its cnstrctn cn e de n prticarly smpe way. egin wth te esestcng psse n ne dmens ZAt the ceter each nemensasphere in Z cnstrct wdimes-al sphere whse rs s 2. Nw cn-strct anther lyer f tw imesnaspheres etcl with the rst lyer at t nt the hles the rst layer astghty as psle. If an nnte ner

f yers re pce tgether thiswy the restng twensn at-tce s the dense hexgnl packg;cse f ths cnstrctn t can e



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called the laminate,d packing in two dimensions abbreviated Lz

I have already suggested how onegoes to three dimensions: a ball of radius1 !2 i s empl aced at t he cent er of eachcircle in Lz and idenical layers of ballsare tted into the holes of the rst layerin such a way as to form lattice Sincethis packing is e q uivaent to D the laminating procedure gives the best-known

packing in three dimensions as well asin two If the procedure is continued inhigher dimensions one dimension at atime the resulting lattice packings areextree y dense It as een known or along tie tat a L are eq uivaleto D4 and D ad ht 6 7 d arevalet resecve to 6 7 Hece te ite lttices aethe dens est at tice ings ossile np to eigt dimensins

Recently Conway and I continued telaminating rocess a it urther: weond all the aminate attices in up to25 d imensions and exaples o them foreach dimension u to 8. It turns outthat although there is a unique laminated lattice in every dimension up to 0there are two distinct ways to stack layers of t he O l at t ice . The t wo dimensional lattices have the same density btdierent kissing numbers There arethree 12dimensional and three 13dimensional laminated lattices but onlyone such lattice for each dimensionfrom through 2 The laminted latt ice Z4 is the eech lattice It is only inspaces having 1 1 2 and 1 3 d imensionswhere cross sections of the eech lat

tice are known that have a higher density than the lainated lattices in thosedimensions

The laminated lattices are built layer on layer by tting each new layer assnugly as possible into the deep holes ofthe preceding one. Hence t he investigation of laminated lattices is closely related to the covering problem: enlarge every sphere in any packing j ust enough toencompass the deep oles and by denit ion the packing must cover the space. In1966 eech conjectured that if each ofthe spheres in the eech lattice is en

larged by a factor of , the sphereswould cover all the oints in the space;suc a covering might well be the bestpossible covering of 2-dimensionalspace eech's conjecture however wasquite dicult to prove primarily because of the comexity of the deeholes in the eech latice

In the twodimensional lattice z it isclear there is only one kind of deep hole;the maximum distance from any pointin the lattice is alwas a point boundedby three circles and so tere is nothingto distinguish one dee ole from anyother In the eech lattice howeverConway Richard A Parker ad I classied 23 distinct kinds o deep hole beforewe were able to prove eech's conjeure Not surprisingly there turn out to

r-2

SS PSSB P SS PSSB P

N

0z

>fZwo

-2

-3

4 8 1 2 1 20 24 28S

32 44 48

DENSES T KNOW N P CKINGS of phere in pace of up o 48 dimenion are ploed according o a mehod uggeed by John Leech he "normalized deniy of he packing dependon he dimenion in which he packng done. I deniion i baed on he fac ha he deny of he 24-dimenonal Leech lace dvied by he volume of a 4dimenonal phere oun radu i eual o (The volume of an /dimenonal phere o radu i eual o ( . . 1/ ) if 1/ even or o ( ) ( . . . 1/) if 1/ odd.) hauoen n an arbrary pace of dmenon 1/ i called he cener deniy . The normalzeddeniy hown on he graph i eual o Log 1/(4 1/) 96 for laminaed lace packing hegraph ymmercal abou he normalzed deny of he Leech lace. The graph how haL3 L and L24 are uie cloe o he lea uper bound known for he deniy of any pheepackng. The lamnaed laice packng are he dene known packng in all dmenion up

o 3 excep dimenon 0 o 3 There an alernaive euence of laice called he euence whch ar a L 6 and rejoin he lamnaed euence a IS' The packng adener han he L packing n dimenon , nd 3 The " packng are alo laicepackng bu he dene known packng n 0 and 3 dimenon are nonlace packng all hee packing are conruced from code for he dgal ranmion of informaion.

be 23 d istinct ways to stack copies of theeech lattice together in 25imensionalspace: there are 23 dierent 25diensiona l aminated l at t ices In 26 dimensions the laminted lattices number atleast 75000.

In dimensions higher than 25 m uch essis known ermann M inkowski dem

onstrated in 1905 that there are latticeackings in any d imension n whose density is great er than 2 ". His argumentike others I have mentioned is nonconstructive In low dimensions the estimate is rather crude: when n i s equal to2 for example it states merely thatthere exist pac kings whose density exceeds about 6 10 whereas theeech lattice is known to be more than32000 times denser In the other direction Blichfeldt showed in 9 that orarbitrarily large numbers n the desitycannot exced about 2 5 . In spite ofmuch eort there was essentialy noimprovemen on this bound unti the1970's. G. A. Kabatiansky evenstein

and V M. Sidel 'nikov in Moscow thensowed that the density in very high dimensions n canot exceed about 2 599 "

E S. Barnes of the University of Adel aide A. Bos of N Pilis' Gloeiampenfabrieken in Eindhoven the N etherlands Conway eech nd I have con

structed a n uber of explicit ac kingsin igh dimensions but none of the isas dense as M inkowski's teorem proises. Recently Barnes and I constructedlattice packings rom the eech latticein dimensions up to 00000 The ensity of these packings is roughly 2 " which at rst glance seems almost asdense as the bound in Minkowski's theorem Indeed in dimension 65536 ourackings are aout 1 04 times densertan any lattice acking previousknown nfortunately the exponentilform of the bound can obscre te facttht we still fll rather short o thepromised goal The theorem guaranteestere remain packings to be discoveredthat are 04 ties denser than theones we have found

sloane - the packing of spheres

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