For screen

39 Pages·2011·0.44 MB·English

Geometry & Topology15(2011)2235–2273 2235
Rigidity of spherical codes
HENRY COHN
YANG JIAO
ABHINAV KUMAR
SALVATORE TORQUATO
Apackingofsphericalcapsonthesurfaceofasphere(thatis,asphericalcode)is
calledrigidorjammedifitisisolatedwithinthespaceofpackings. Inotherwords,
aside from applying a global isometry, the packing cannot be deformed. In this
paper,wesystematicallystudytherigidityofsphericalcodes,particularlykissing
configurations. OnesurpriseisthatthekissingconfigurationoftheCoxeter–Todd
latticeisnotjammed,despitebeinglocallyjammed(eachindividualcapisheldin
placeifitsneighborsarefixed);inthisrespect,theCoxeter–Toddlatticeisanalogous
totheface-centeredcubiclatticeinthreedimensions. Bycontrast,wefindthatmany
otherpackingshavejammedkissingconfigurations,includingtheBarnes–Walllattice
andallofthebestkissingconfigurationsknowninfourthroughtwelvedimensions.
Jammingseemstobecomemuchlesscommonforlargekissingconfigurationsin
higherdimensions,andinparticularitfailsforthebestkissingconfigurationsknown
in 25 through 31 dimensions. Motivatedbythisphenomenon,wefindnewkissing
configurationsinthesedimensions,whichimproveontherecordssetin1982bythe
laminatedlattices.
52C25;52C17
1 Introduction
One of the key qualitative properties of a packing is whether it is jammed, that is,
whethertheparticlesarelockedintoplace. Jammingisofobviousscientificimportance
ifweareusingthepackingtomodelagranularmaterial. Furthermore,itplaysacentral
roleinstudyinglocaloptimalityofpackings,becauseonenaturalwaytotrytoimprove
apackingistodeformitsoastoopenupmorespace.
Jamming has been extensively studied for packings in Euclidean space. See, for
example,TorquatoandStillinger[44]andthereferencescitedtherein. However,ithas
been less thoroughly investigated in other geometries. In this paper, we investigate
jamming for sphere packings in spherical geometry, that is, packings of caps on the
Published: 23November2011 DOI:10.2140/gt.2011.15.2235
2236 HenryCohn,YangJiao,AbhinavKumarandSalvatoreTorquato
surfaceofasphere. Jamminghaspreviouslybeenstudiedforspherepackingsin S2
(seeTarnaiandGáspár[41]),butthereseemstohavebeenlittleinvestigationinhigher
dimensions.
Apackingofcongruentsphericalcapsontheunitsphere Sn(cid:0)1 in Rn yieldsaspherical
code(thatis,afinitesubsetof Sn(cid:0)1)consistingofthecentersofthecaps. Theminimal
distanceofsuchacodeisthesmallestangularseparationbetweendistinctpointsinthe
code. In otherwords,the cosineofthe minimaldistanceisthe greatestinnerproduct
betweendistinctpointsinthecode. Thepackingradiusishalftheminimaldistance,
becausesphericalcapsofthisradiuscenteredatthepointsofthecodewillnotoverlap,
excepttangentially. Asphericalcodeisoptimalifitsminimaldistanceisaslargeas
possible,giventhedimensionofthecodeandthenumberofpointsitcontains. (Note
thatthisnotionofoptimalityisdifferentfromrequiringthatnomorecapsofthesame
sizecanbeaddedwithoutcausingoverlap. Neitherofthesetwonotionsimpliesthe
other.)
Sphericalcodesarisenaturallyinmanypartsofmathematicsandscience(seeCohn[9]
foramoreextensivediscussion). Forexample,in R3 theymodelporesinpollengrains
orcolloidalparticlesadsorbingtothesurfaceofadropletinaemulsionformedbytwo
immiscibleliquids. Inhigherdimensions,theycanbeusedaserror-correctingcodes
foraconstant-powerradiotransmitter. Furthermore,manybeautifulsphericalcodes
ariseinLietheory,discretegeometry,orthestudyofthesporadicfinitesimplegroups.
Adeformationofasphericalcodeisacontinuousmotionofthepointssuchthatthe
minimaldistanceneverdropsbelowitsinitialvalue. Adeformationisanunjammingif
itdoesnotsimplyconsistofapplyingglobalisometries(thatis,thepairwisedistances
do not all remain constant). A spherical code is called rigid or jammed if it has no
unjamming. Itiscalledlocallyjammed ifnosinglepointcanbecontinuouslymoved
whilealltheothersareheldfixed.
Forexample,intheface-centeredcubicpackingofballsin R3,thekissingconfiguration
(thatis,pointsoftangencyonagivenball)consistsoftheverticesofacuboctahedron.
Thiscodeislocallyjammed,butitisnotinfactjammed(seeConwayandSloane[19,
page 29] or Proposition 3.5 below). However, it can be deformed into an optimal
spherical code, namely the vertices of a regular icosahedron, and the icosahedron is
thenarigidcodewithahigherminimaldistancethanthatofthecuboctahedron.
As this example shows, deforming a spherical code is one way to improve it. Some
optimal codes are not jammed; for example, the best five-point codes in S2 consist
oftwoantipodal pointsandthreepointsorthogonalto them,andthethreepoints can
movefreelyaslongastheyremainseparatedbyatleastanangleof (cid:25)=2. Furthermore,
computerexperimentssuggestthatanoptimalcodecanhaverattlers,thatis,pointsnot
Geometry & Topology,Volume15(2011)
Rigidityofsphericalcodes 2237
incontactwithanyotherpoint,althoughnosuchcasehaseverbeenrigorouslyanalyzed.
However,despitetheseissues,rigidityisapowerfulcriterionforunderstandingwhena
codecanbeimproved.
Note that whether a configuration is jammed depends on the ambient space. For
example,theverticesofasquarearejammedin S1 butnotin S2.
For infinite packings in Euclidean space, there are more subtle distinctions between
differenttypesofjamming(seeBezdek,BezdekandConnelly[7]andTorquatoand
Stillinger[43])basedonwhatsortsofmotionsareallowed. Forexample,areallbut
finitely many particles held fixed? Are shearing motions allowed? However, these
issuesdonotariseforpackingsincompactspaces.
Nevertheless, jamming seemsto bea moresubtle phenomenonon spheresthan itis in
Euclideanspace. InEuclideanspace,thereisanefficientalgorithmtotestforjamming
(see Donev, Torquato, Stillinger and Connelly [21]) but on spheres we do not know
such an algorithm. The difficulty is caused by curvature, which complicates certain
arguments. Forexample,inEuclideanspaceeveryinfinitesimalunjammingextendsto
anactualunjamming,aswewillexplaininSection2,butthecorrespondingprocedure
doesnotworkonspheres.
Oncewehavedevelopedthebasictheoryofrigidityforsphericalcodes,wewilldevote
therestofthispapertoapplyingittoanalyzespecificcodes. Wewillfocusprimarily
onkissingconfigurations(that is,sphericalcodeswithminimal angle atleast (cid:25)=3,or
equivalentlythepointsoftangencyinEuclideanspacepackings),becausetheyforma
rich classof spherical codes andinclude many of themost noteworthy examples. An
optimalkissingconfigurationisonewiththelargestpossiblesizeinitsdimension.
Asmentionedabove,theface-centeredcubickissingconfigurationisnotrigid,butwe
willprovethatalloftheotherbestconfigurationsknowninuptotwelvedimensionsare
rigid. Alongtheway,wewillproducewhatmaybethefirstexhaustiveenumerationof
theseconfigurationsinuptoeightdimensions,aswellasacompletelistoftheknown
examplesinninethroughtwelvedimensions(althoughwesuspectthatmoreremain
to be discovered). Above twelve dimensions, the calculations become increasingly
difficult to do, even by computer, but we analyze certain cases that are susceptible
toconceptualarguments. Inparticular,weshowthatthekissingconfigurationofthe
Coxeter–Todd lattice K is not rigid, while that of the Barnes–Wall lattice ƒ is,
12 16
althoughbothlatticesareconjecturedtobeoptimalspherepackingsintheirdimensions.
We particularly focus our attention on 25 through 31 dimensions, because of the
remarkablysmallincreasesintherecordkissingnumbersfromeachdimensiontothe
next(seeTable3inSection7fortheoldrecords). Thebestconfigurationspreviously
Geometry & Topology,Volume15(2011)
2238 HenryCohn,YangJiao,AbhinavKumarandSalvatoreTorquato
Dimension Kissingnumber Dimension Kissingnumber
1 2 17 5346
2 6 18 7398
3 12 19 10668
4 24 20 17400
5 40 21 27720
6 72 22 49896
7 126 23 93150
8 240 24 196560
9 306 25 197040
10 500 26 198480
11 582 27 199912
12 840 28 204188
13 1154 29 207930
14 1606 30 219008
15 2564 31 230872
16 4320 32 276032
Table1: Thebestlowerboundsknownforkissingnumbersinuptothirty-two
dimensions. Numbersinboldareknowntobeoptimal(seeSchütteandvan
derWaerden[39],Levenšte˘ın[29],OdlyzkoandSloane[36]andMusin[34]).
knownwerenotevenlocallyjammed,butweseenosimplewaytodeformthemsoas
toincreasethekissingnumber. However,inSection7weshowhowtoimproveonthe
knownrecords. Wegiveasimpleargumentthatshowshowtobeatthem,aswellasa
more complicated construction that makes use of a computer search to optimize the
resultingbounds.
The new records are shown in Table 1. It is taken from Conway and Sloane [19,
page xxi, Table I.2(a)], with three exceptions: the entry for R15 was out of date in
thattable(see[19,Chapter5,Section4.3]),theentriesfor R13 and R14 comefrom
ZinovievandEricson[46],andtheentriesfor R25 through R31 arenewresultsinthe
presentpaper. SeealsoNebeandSloane[35]. Forthebestupperboundsknowninup
totwenty-fourdimensions,seeMittelmannandVallentin[32].
Figure1showsalogarithmicplotofthedatafromTable1,normalizedforcomparison
with 32 dimensions. Onecanseethelocalmaximacorrespondingtotheremarkable
E ,Barnes–WallandLeechlatticesindimensions 8, 16 and 24,respectively. Note
8
also that the growth rate of the known kissing numbers drops dramatically after 24
dimensions.
Geometry & Topology,Volume15(2011)
Rigidityofsphericalcodes 2239
log(cid:28) (cid:0) n log(cid:28)
n 32 32
3
2
1
0 n
1 8 16 24 32
Figure1: Aplotof log(cid:28) (cid:0) n log(cid:28) ,where (cid:28) denotesthecurrentrecord
n 32 32 n
kissingnumberin Rn
2 Infinitesimal jamming
We know of no efficient way to test whether a given spherical code is jammed. In
principle,itcanbedonebyafinitecalculation,atleastifthepointsinthecodehave
algebraic numbers as coordinates, by using quantifier elimination for the first-order
theoryoftherealnumbers(seeTarski[42]). (TheproofreliesonRothandWhiteley[38,
Proposition3.2].) However,quantifiereliminationisnotpracticalinthiscase.
On the other hand, there are much more efficient tests for a related concept called
infinitesimal jamming,using linear programming(see Donev, Torquato, Stillinger and
Connelly[21]). Givenacode fx ;:::;x g(cid:26)Sn(cid:0)1,imagineperturbing x to x C"y .
1 N i i i
Then
jx C"y j2D1C2hx ;y i"CO."2/;
i i i i
where h(cid:1);(cid:1)i denotestheinnerproduct,and
hx C"y ;x C"y iDhx ;x iC.hx ;y iChx ;y i/"CO."2/:
i i j j i j i j j i
Thus,topreservealltheconstraintsuptofirstorderin ",wemusthave hx ;y iD0 for
i i
all i,and hx ;y iChx ;y i(cid:20)0 whenever hx ;x i equalsthemaximalinnerproduct
i j j i i j
in the code. An infinitesimal deformation of the code fx ;:::;x g is a collection
1 N
of vectors y ;:::;y satisfying these constraints. It is an infinitesimal rotation if
1 N
thereexistsaskew-symmetricmatrix ˆ2Rn(cid:2)n suchthat y Dˆx forall i,anda
i i
code is infinitesimally jammed if every infinitesimal deformation is an infinitesimal
rotation. (Recallthattheskew-symmetricmatricesareexactlythoseintheLiealgebra
of SO.n/.) Notethatforaninfinitesimalrotation,
hx ;y iDhx ;ˆx iD(cid:0)hˆx ;x iD(cid:0)hx ;y i:
i j i j i j j i
Geometry & Topology,Volume15(2011)
2240 HenryCohn,YangJiao,AbhinavKumarandSalvatoreTorquato
Thus, an infinitesimal rotation does not change any distances, up to first order. The
converse is false (consider a square on the equator in S2, with an infinitesimal de-
formationmovingtwooppositecornersupandtheothertwodown),butitistruefor
full-dimensionalcodes:
Lemma2.1 Let y ;:::;y beaninfinitesimaldeformationofacode fx ;:::;x g
1 N 1 N
in Sn(cid:0)1 such that x ;:::;x span Rn. If hx ;y iChx ;y i D 0 for all i and j,
1 N i j j i
thenthedeformationisaninfinitesimalrotation.
Proof First,notethatifalinearcombination P ˛ x vanishes,then P ˛ y D0 as
i i i i i i
well,because
(cid:28) (cid:29)
X X
˛ y ;x D ˛ hy ;x i
i i j i i j
i i
X
D(cid:0) ˛ hx ;y i
i i j
i
(cid:28) (cid:29)
X
D(cid:0) ˛ x ;y
i i j
i
D(cid:0)h0;y iD0
j
forall j (andonlythezerovectorisorthogonaltoasetthatspans Rn). Thus,thereis
awell-definedlinearmap ˆ suchthat ˆx Dy . Furthermore,theidentity
i i
hx ;ˆx iDhx ;y iD(cid:0)hy ;x iD(cid:0)hˆx ;x i
i j i j i j i j
implies that ˆ is skew-symmetric, because it holds for a basis of Rn and hence
hu;ˆviD(cid:0)hˆu;vi forall u;v2Rn.
Everyinfinitesimallyjammedcodeisinfactjammed. Thisisnotobvious: onecannot
simplydifferentiateapurportedunjammingmotiontogetaninfinitesimalunjamming,
withoutdealingwithtwotechnicalities,namelywhetherthereisadifferentiableunjam-
mingandwhathappensifallthefirst-orderderivativesvanish. However,itistrue,as
pointedoutbyConnelly[13,Remark4.1]andbyRothandWhiteley[38,Theorem5.7]:
Theorem2.2 (Connelly,RothandWhiteley) Everyinfinitesimallyjammedspherical
codeisjammed.
Thecitedpapersdealwiththemoregeneralsettingoftensegrityframeworks,inwhich
movablepointscanbeconnectedbybars(withfixedlengths),cables(withspecified
maximum lengths) or struts (with specified minimum lengths), and they prove that
Geometry & Topology,Volume15(2011)
Rigidityofsphericalcodes 2241
infinitesimaljammingimpliesjamminginthissetting. Forthespecialcaseofspherical
codes,weconnecteachpointinthecodetotheoriginusingabar,andweinsertstruts
betweenneighboringpoints(thatis,thoseattheminimaldistance).
Wedonotknowwhethereveryjammedsphericalcodethatspanstheambientspace
is infinitesimally jammed. For tensegrity frameworks, the corresponding statement
is not true: if we place bars along the edges of a regular octahedron, and use two
additionalbarstoconnectitscenterwithapairofoppositevertices,thentheframework
is rigid, but flexing the center orthogonally to the two adjacent bars is a nontrivial
infinitesimal deformation. We have not found such an example for spherical codes,
but we expect that there is one. By contrast, infinitesimal jamming is equivalent to
jammingforperiodicpackingsinEuclideanspace(seeDonev,Torquato,Stillingerand
Connelly[21]). Specifically,ifweperturb x to xC"y and x0 to x0C"y0,then
j.xC"y/(cid:0).x0C"y0/j2Djx(cid:0)x0j2C2hx(cid:0)x0;y(cid:0)y0i"Cjy(cid:0)y0j2"2:
Thesecond-ordertermisalwaysnonnegative,sononnegativityofthefirst-orderterm
sufficestoproduceanactualunjamming. (Deformingtheunderlyinglatticecomplicates
theanalysis, butthe resultremainstrue; see[21, Appendix C].)Whatgoes wrongin
the spherical case is that xC"y is no longer a unit vector and must be normalized,
whichcausesthedistancestodecrease.
This is not merely a technicality: there seems to be no simple method to turn an
infinitesimalunjammingintoanactualunjamming. Nevertheless,inallourexamples,
wehavebeenabletoaccomplishthis(withsomeeffort).
Thelinearprogrammingalgorithmforinfinitesimalrigiditytestingworksasfollows.
By Lemma 2.1, to test whether a full-dimensional code is infinitesimally jammed,
we need only check for each pair of points whether the distance between them can
be changed. In other words, in an infinitesimal deformation, are the maximum and
minimumof hx ;y iChx ;y i zeroforall i and j? Foreach i and j,thisgivesrise
i j j i
totwolinearprogrammingproblems,becauseweareimposinglinearconstraintson
theperturbationvectors y ;:::;y andmaximizingorminimizingthelinearfunction
1 N
hx ;y iChx ;y i. (Of course, when hx ;x i is maximal in the code, the definition
i j j i i j
of an infinitesimal deformation requires that hx ;y iChx ;y i(cid:20)0, so maximizing
i j j i
thislinearfunctionalistrivial. However,theothercasesarenontrivial.) Thecodeis
infinitesimallyjammedifandonlyiftheoptimainalltheselinearprogramsarezero. If
not,thensolvingthelinearprogramswillproduceaninfinitesimalunjamming,provided
that we also bound the coordinates of the perturbation vectors (to avoid unbounded
linearprograms).
Insomecasesweareaidedbysymmetry,becauseweonlyneedtocheckonerepresenta-
tivefromeachorbitoftheactionofthecode’ssymmetrygrouponpairsofpointsinthe
Geometry & Topology,Volume15(2011)
2242 HenryCohn,YangJiao,AbhinavKumarandSalvatoreTorquato
code. Forexample,ifthesymmetrygroupactsdistance-transitively,thenweonlyneed
tocheckonepairofpointsateachdistance. UsingtheapproachofDonev,Torquato,
StillingerandConnelly[21], wecanevenreducetosolvingonelinearprogram, atthe
costofrandomization. Specifically,considermaximizingthelinearcombination
X
ci;j.hxi;yjiChxj;yii/;
i;j
wherethecoefficients ci;j arechosenrandomlyfromtheinterval Œ(cid:0)1;1(cid:141). Withproba-
bility 1,thisapproachwillproduceaninfinitesimalunjammingifoneexists. Thus,if
theoptimumiszero,thenwecanbeconfidentthatthecodeisjammed,althoughthis
doesnotconstituteaproof.
For mostof theexamples inthispaper, wegiveshort conceptualproofsof jamming.
However, forsome caseswe mustrely oncomputercalculations. In thesecases, we
havegivenrigorous,computer-assistedproofsbyusingexactrationalarithmeticviathe
QSopt_exlinearprogrammingsoftwareofApplegate,Cook,DashandEspinoza[1]
andcheckingeverypairofpointsinthecode.
3 The kissing configurations of root lattices
We begin by proving that the root systems D (for n(cid:21)4) and E , E and E are
n 6 7 8
infinitesimally jammed, while A is not jammed (for n (cid:21) 3). Note that these root
n
systemsarethekissingconfigurationsofthecorrespondingrootlattices.
p
In this section, we will use spheres of diameter 2 instead of 1, because that is
standard for these root systems and makes the inner products integral. Note that the
theoryofinfinitesimaljamminginnowaydependsonthisnormalization.
Thefollowingelementarylemmawillplayakeyroleintheproofs:
Lemma3.1 Let C and D besphericalcodeswith C (cid:18)D andwiththesameminimal
distance. If C isinfinitesimallyjammedwithinthevectorspaceitspans,theninany
infinitesimaldeformationof D,theinnerproductsbetweenpointsin C areunchanged
(uptofirstorder).
Theinterestingcaseiswhen C islowerdimensionalthan D.
Proof Let x and y be points in C, and let u and v be their perturbations in an
infinitesimaldeformationof D. Wewrite uDuCCu? and vDvCCv?,where uC
and vC areinthespanof C while u? and v? areintheorthogonalcomplementofthe
span.
Geometry & Topology,Volume15(2011)
Rigidityofsphericalcodes 2243
The orthogonal projections to the span of C yield an infinitesimal deformation of
C. (Here we need C and D to have the same minimal distance, since otherwise the
conditions on which inner products can increase will differ.) Thus, because C is
infinitesimallyjammedwithinitsspan, hx;vCiChuC;yiD0. Furthermore, hx;v?iD
hy;u?iD0. Itfollowsthat hx;viChu;yiD0,asdesired.
Lemma3.2 The A rootsystemisinfinitesimallyjammed.
2
The A root system is a regular hexagon, and it is easy to show that every regular
2
polygonisinfinitesimallyjammed. Thissimpleobservationprovidesausefultoolfor
analyzingmoreelaborateconfigurationsviaLemma3.1.
Proposition3.3 The D rootsystemisinfinitesimallyjammed.
4
Proof The minimal vectors of D have norm 2 and the possible inner products
4
betweendistinctminimalvectorsare 0, ˙1 and (cid:0)2. First,notethattheautomorphism
groupof D actstransitively onpairsofminimalvectors withagiveninner product,
4
sowithoutlossofgeneralitywecanconsiderjustonepairofpointsateachdistance.
(Thistransitivityfailsfor D with n>4,becausetherearetwoorbitsforinnerproduct
n
0,butthetrialitysymmetryof D collapsesthemtooneorbit.)
4
Furthermore, D contains A ,andbyLemma3.1thedistancesinacopyof A cannot
4 2 2
changebecause A isinfinitesimallyjammedwithinitsspan. Thistakescareofallthe
2
casesexceptforapairoforthogonalvectors.
Wenowhavetoshowthatif hx;yiD0 then hx;yi doesnotchangeinanyinfinitesimal
deformation. Againby thedistance transitivity ofthe automorphismgroup, wemay
assume x D.1;1;0;0/ and y D.1;(cid:0)1;0;0/. Let uD.1;0;1;0/, v D.1;0;(cid:0)1;0/,
w D .1;0;0;1/ and z D .1;0;0;(cid:0)1/ be other minimal vectors of D . Denote the
4
firstorderperturbationsof x;y;u;v;w;z by x0;y0;u0;v0;w0;z0. Notethat xCy D
uCv D wCz, and that hx;ui D hx;vi D hy;ui D hy;vi D 1. Thus, by the A
2
embeddingargument,wehave
hx;u0iChx0;uiD0; hy;u0iChy0;uiD0;
hx;v0iChx0;viD0; hy;v0iChy0;viD0:
Addingtheseequations,weget
hxCy;u0Cv0iChx0Cy0;uCviD0;
or(using xCy DuCv)
huCv;u0Cv0iChx0Cy0;xCyiD0:
Geometry & Topology,Volume15(2011)
2244 HenryCohn,YangJiao,AbhinavKumarandSalvatoreTorquato
Sinceweknowthat hu;u0iD0,etc.,weget(denotingthefirstorderchange hu;v0iC
hu0;vi in hu;vi by ı.u;v/)
ı.u;v/Cı.x;y/D0:
Similarly,wehave
ı.u;v/Cı.w;z/D0;
ı.w;z/Cı.x;y/D0:
Fromthesethreeequations,elementaryalgebraimplies ı.x;y/Dı.u;v/Dı.z;w/D0.
Thiscompletestheproof.
Thelengthyargumentforthelastcaseamountstoverifyingthatasquareembedded
within D cannotbeinfinitesimallydeformed. Notethatthiscannotsimplybesettled
4
usingLemma3.1,althoughthesquareisindeedjammedwithinitsspan,becausethe
minimal distance in the square differs from that in D . If that argument worked, it
4
wouldalsoproveinfinitesimaljammingfor D ,whichisnottrue. (The A and D root
3 3 3
latticesareisomorphictotheface-centeredcubiclattice,whosekissing configuration
isnotjammed.)
Corollary3.4 The D rootsystem(for n(cid:21)4)andthe E , E and E rootsystems
n 6 7 8
areinfinitesimallyjammed.
Proof Theseconfigurationshavenorm 2 andinnerproducts 0, ˙1 and ˙2,thesame
asin D . Wefirstdealwith E , E and E . Theirautomorphismgroupsactdistance
4 6 7 8
transitively,soitsufficestoconsiderasinglepairofpointsateachdistance. The D
4
root system embeds in each of these configurations (in fact, its Dynkin diagram is a
subdiagram),sowithoutlossofgeneralitywecanassumethepairofpointsisin D .
4
NowcombiningLemma3.1andProposition3.3completestheproof.
Thesameproofworksfor D with n>4,withoneexception,namelythattherearetwo
n
orbitsofpairsoforthogonalvectors,sothegroupdoesnotquiteactdistancetransitively.
Specifically,thestabilizerof .1;1;0;:::;0/ cannotinterchange .1;(cid:0)1;0;:::;0/ and
.0;0;1;1;0;:::;0/. However,inbothcases,thesevectorsarecontainedinacopyof
D (namely, the one in the first four coordinates), so we can complete the proof as
4
before.
The A root system is locally jammed, and for nD2 it is in fact jammed, but the
n
unjammingfor nD3 extendstohigherdimensions.
Proposition3.5 For n(cid:21)3,the A rootsystemisnotjammed.
n
Geometry & Topology,Volume15(2011)