New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-space

0.21 MB·

by  Robert Thijs Kozma

New Lower Bound for the Optimal Ball Packing
4
1 Density of Hyperbolic 4-space
0 ∗
2

g
u
A
RobertThijsKozma(1)andJeno˝ Szirmai(2)

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(1)DepartmentofMathematics

] SUNYStonyBrook
G
StonyBrook,NY11794-3651USA
M
Email:[email protected]
.
h
t (2)BudapestUniversityofTechnologyandEconomics
a
m InstituteofMathematics,DepartmentofGeometry
[ H-1521Budapest,Hungary

Email:[email protected]
3
v
4 8/21/2014
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6
.
1 Abstract
0 Inthispaperweconsiderballpackingsin4-dimensionalhyperbolicspace.We
4
showthatitispossibletoexceedtheconjectured4-dimensional realizablepack-
1
ingdensityupper bound duetoL.Fejes-To´th(Regular Figures, 1964). Wegive
:
v seven examples of horoball packing configurations that yieldhigher densitiesof
Xi 0.71644896...,wherehoroballsarecenteredatidealverticesofcertainCoxeter
simplices,andareinvariantundertheactionsoftheirrespectiveCoxetergroups.
r
a
1 Introduction
LetX denotea space of constantcurvature,either the n-dimensionalsphereSn, Eu-
clidean space En, or hyperbolic space Hn with n 2. In discrete geometry, it is
≥
commonlyaskedtofindthehighestpossiblepackingdensityinX bycongruentnon-
overlappingballsofagivenradius[1],[6]. Euclideancasesarethebestexplored. For
example,thedensestpossiblelatticepackingsareknownforE2throughE8. Inhigher
dimensions,however,mostly onlyboundsare known. Furthermore,nosharpbounds
∗MathematicsSubjectClassification2010:52C17,52C22,52B15.
Keywordsandphrases:Coxetergroup,hyperbolicgeometry,packing,simplextiling.
1
exist for irregular packings in En when n > 3. One major recent developmenthas
been the settling of the long-standingKepler conjecture, part of Hilbert’s 18th prob-
lem,byThomasHalesattheturnofthe21stcentury. Hales’computer-assistedproof
waslargelybasedonaprogramsetforthbyL.FejesTo´thinthe1950’s[5].
ThedefinitionofpackingdensityiscriticalinhyperbolicspaceasshownbyBo¨ro¨czky
[3]. Forotherstandardexamplesseealso[6],[19]. Themostwidelyacceptednotion
ofpackingdensityconsidersthelocaldensitiesofballswithrespecttotheirDirichlet–
Voronoicells(cf. [3]and[14]). InordertoconsiderhoroballpackingsinHn,weuse
anextendednotionofsuchlocaldensity.
Let B be a horoball in packing , and P Hn be an arbitrary point. Define
B ∈
d(P,B) to be the perpendicular distance from point P to the horosphere S = ∂B,
where d(P,B) is taken to be negative when P B. The Dirichlet–Voronoi cell
∈
(B, )ofahoroballB isdefinedastheconvexbody
D B
(B, )= P Hn d(P,B) d(P,B ), B .
′ ′
D B { ∈ | ≤ ∀ ∈B}
BothB and areofinfinitevolume,sotheusualnotionoflocaldensityismodified
asfollows.LDetQ ∂HndenotetheidealcenterofBatinfinity,andtakeitsboundary
bdSeetttoehrbemeEintuhecelaiodcneoean-npv(oe∈nixn−tcoc1on)me-bCpaanlcl(trwifi)ict=hatcioceonnntoeefrQECuB∈clCniSd−e\1a(n{rQ)(n}.−∈T1hH)e-nnspQwaict∈eh.∂aLpHeetnxBaQCnnd−c1Bo(nrCns)−is⊂1ti(nrSg)
of all hyperbolic geodesics passing throug(cid:0)h BCn−1(r(cid:1)) with limit point Q. The local
densityδ (B, )ofB to isdefinedas
n
B D
vol(B n(r))
δ ( ,B)= lim ∩C .
n B r vol( n(r))
→∞ D∩C
ThislimitisindependentofthechoiceofcenterC forBCn−1(r).
Inthecaseofperiodicballorhoroballpackings,thelocaldensitydefinedabovecan
beextendedtotheentirehyperbolicspace.Thislocaldensityisrelatedtothesimplicial
densityfunction(definedbelow)thatwegeneralizedin[25]and[26]. Inthispaperwe
willusesuchdefinitionofpackingdensity(cf. Section3).
ThealternatemethodsuggestedbyBowenandRadin[2],[19]usesNevo’spoint-
wise ergodictheorem to assure that the standard Euclidean limit notion of density is
well-defined for Hn. First they define a metric on the space Σ of relatively-dense
packingsbycompactobjects,basedonHausdorffdistance,correPspondingtouniform
convergenceoncompactsubsetsofHn. Thentheystudythemeasuresinvariantunder
isometries of Σ rather than individualpackings. There is a large class of packings
P
ofcompactobjectsinhyperbolicspaceforwhichsuchdensityiswell-defined. Using
ergodicmethods,theyshowthatifthereisonlyoneoptimallydensepackingofEn or
Hn,uptocongruence,bycongruentcopiesofbodiesfromsomefixedfinitecollection,
then that packing must have a symmetry group with compact fundamental domain.
Moreover,for almost any radius r [0, ) the optimal ball packing in Hn has low
∈ ∞
symmetry.
ACoxetersimplexisann-dimensionalsimplexinX suchthatitsdihedralangles
areeithersubmultiplesofπ,orzero.Thegroupgeneratedbyreflectionsonthesidesof
2
aCoxetersimplexiscalledaCoxetersimplexreflectiongroup.Suchreflectionsgivea
discretegroupofisometriesofX withtheCoxetersimplexasitsfundamentaldomain;
hencethegroupsgiveregulartessellationsofX. TheCoxetergroupsarefiniteforSn,
andinfiniteforEnorHn.
InHn weallowunboundedsimpliceswithidealverticesatinfinity∂Hn. Coxeter
simplicesexistonlyfordimensionsn=2,3,...,9;furthermore,onlyafinitenumber
exist in dimensions n 3. Johnson et al. [10] computed the volumes of all Cox-
≥
etersimplicesinhyperbolicn-space,seealsoKellerhals[13]. Suchsimplicesarethe
mostelementarybuildingblocksof hyperbolicmanifolds,the volumeof whichis an
importanttopologicalinvariant.
Inthen-dimensionalspaceX ofconstantcurvature(n 2),definethesimplicial
≥
densityfunctiond (r)tobethedensityofn+1spheresofradiusrmutuallytouching
n
oneanotherwithrespecttothesimplexspannedbythecentersofthespheres.L.Fejes
To´thandH.S.M.Coxeterconjecturedthatthepackingdensityofballsofradiusrin
X cannot exceed d (r). Rogers [20] proved this conjecture in Euclidean space En.
n
The2-dimensionalsphericalcasewassettledbyL.FejesTo´th[8],andBo¨ro¨czky[3],
whoprovedthefollowingextension:
Theorem1.1(K. Bo¨ro¨czky). In an n-dimensionalspaceof constantcurvature, con-
sider a packing of spheres of radius r. In the case of spherical space, assume that
r < π. ThenthedensityofeachsphereinitsDirichlet–Voronoicellcannotexceedthe
4
densityofn+1spheresofradiusrmutuallytouchingoneanotherwithrespecttothe
simplexspannedbytheircenters.
Inhyperbolicspace,themonotonicityofd (r) wasprovedbyBo¨ro¨czkyandFlo-
3
rian [4]; Marshall [16] showed that for sufficiently large n, function d (r) is strictly
n
increasingin variabler. Kellerhals[14] showedd (r) < d (r), andthat in cases
n n 1
−
considered by Marshall the local density of each ball in its Dirichlet–Voronoicell is
boundedabovebythesimplicialhoroballdensityd ( ).
n
Thisupperboundfordensityinhyperbolicspace∞H3 is0.85327613..., whichis
notrealizedbypackingregularballs. However,itisattainedbyahoroballpackingof
H3 wherethe idealcentersofhoroballslie onan absolutefigureofH3; forexample,
they may lie at the vertices of the ideal regular simplex tiling with Coxeter-Schla¨fli
symbol(3,3,6).
In[15]weprovedthattheoptimalballpackingarrangementinH3mentionedabove
isnotunique. Wegaveseveralnewexamplesofhoroballpackingarrangementsbased
ontotallyasymptoticCoxetertilingsthatyieldtheBo¨ro¨czky–Florianupperbound[4].
Furthermore, in [25], [26] we foundthat by allowing horoballsof differenttypes
ateachvertexofa totally asymptoticsimplexandgeneralizingthesimplicialdensity
functiontoHn for(n 2),theBo¨ro¨czky-typedensityupperboundisnolongervalid
≥
forthefullyasymptoticsimplicesforn 3. Forexample,inH4 the locallyoptimal
≥
packingdensityis0.77038...,higherthantheBo¨ro¨czky-typedensityupperboundof
0.73046.... Howeverthese ballpackingconfigurationsare onlylocallyoptimaland
cannotbeextendedtotheentiretyofthehyperbolicspacesHn. Furtheropenproblems
andconjectureson4-dimensionalhyperbolicpackingsare discussed in [7]. A recent
result of Jacquemet [9] gives a formula for the inradius of a hyperbolictruncated n-
3
simplexbasedonitsGrammatrix.
The second-named author has several additional results on globally and locally
optimal ball packings in Hn, Sn, and the eight Thurston geomerties arising from
Thurston’sgeometrizationconjecture[21],[22],[23],[24],[27],[28]. Thesepacking
densities are global or local, depending on whether the density obtained in a funda-
mentaldomaincanorcannotbeextendedtotheentirespace.
Inthispaperwecontinueourinvestigationsonballpackingsinhyperbolic4-space.
Using horoball packings, allowing horoballs of different types, we find seven coun-
terexamples(realizedbyallowinguptothreehoroballtypes)tooneofL.FejesTo´th’s
conjecturesstatedintheconcludingsectionofhisbookRegularFigures:
Finallywedrawattentiontothetessellations 5,3,3,3 of4-dimensional
{ }
hyperbolicspace, the cell-inspheresand cell-circumspheresof whichare
also expectedtoforma closestpackingandloosestcovering. Thecorre-
spondingdensitiesare(5 √5)/4 = 0.690... and(4+6√5)/√125 =
−
1.557... [8]
2 Higher Dimensional Hyperbolic Geometry
In this paper we use the Cayley–Klein ball model, and a projective interpretation of
hyperbolicgeometry.Thishastheadvantageofgreatlysimplifyingourcalculationsin
higher dimensions as comparedto other models such as the Poincare´ model. In this
section we give a brief review of the concepts used in this paper. For a generaldis-
cussionandbackgroundinhyperbolicgeometryandtheprojectivemodelsofThurston
geometriessee[17]and[18].
2.1 The ProjectiveModel
We use the projective model in Lorentzian (n + 1)-space E1,n of signature (1,n),
i.e. E1,n is the real vector space Vn+1 equipped with the bilinear form of signature
(1,n)
x, y = x0y0+x1y1+ +xnyn (1)
h i − ···
wherethenon-zerorealvectorsx=(x0,x1,...,xn) Vn+1andy=(y0,y1,...,yn)
Vn+1 representpointsinprojectivespace n(R).∈Hn isrepresentedastheinterior
∈ P
oftheabsolutequadraticform
Q= [x] n x, x =0 =∂Hn (2)
{ ∈P |h i }
in realprojectivespace n(Vn+1,V ). Allproperinteriorpointsx Hn satisfy
n+1
P ∈
x, x <0.
h Thie boundary points ∂Hn in n represent the absolute points at infinity of Hn.
Pointsysatisfying y, y > 0liePoutside∂Hn andarereferredtoasouterpointsof
Hn. Take P([x]) h n, ipoint [y] n is said to be conjugate to [x] relative to Q
∈ P ∈ P
when x, y = 0. ThesetofallpointsconjugatetoP([x])formaprojective(polar)
h i
hyperplane
pol(P)= [y] n x, y =0 . (3)
{ ∈P |h i }
4
HencethebilinearformQin(1)inducesabijectionorlinearpolarityVn+1 V
n+1
→
between the points of n and its hyperplanes. Point X[x] and hyperplane α[a] are
P
incidentif the value of the linear form a evaluatedon vectorx is zero, i.e. xa = 0
wherex Vn+1 0 ,and a V 0 .Similarly,linesin narecharacterized
n+1
∈ \{ } ∈ \{ } P
by2-subspacesofVn+1 or(n 1)-spacesof V [17].
n+1
LetP Hn denoteapoly−hedronboundedbyafinitesetofhyperplanesHi with
unitnorma⊂lvectorsbi V directedtowardstheinteriorofP:
n+1
∈
Hi = x Hd x, bi =0 with bi,bi =1. (4)
{ ∈ |h i } h i
InthispaperP isassumedtobeanacute-angledpolyhedronwithproperoridealver-
tices. TheGrammianmatrixG(P)=( bi,bj ) i,j 0,1,2...n isanindecom-
i,j
posablesymmetricmatrixofsignature(h1,n)wiithentri∈es{bi,bi =1}and bi,bj 0
h i h i≤
fori=jwhere
6
0 ifHi Hj,
⊥
bi,bj = −cosαij ifHi,Hj intersect alonganedgeofP at angleαij,
h i  −1 if Hi,Hj areparallelinthehyperbolicsense,
coshlij ifHi,Hj admitacommonperpendicularoflengthlij.
This is visualize−d using the weighted graph or scheme of the polytope (P). The
graph nodes correspond to the hyperplanesHi and are connected if Hi and Hj not
P
perpendicular (i = j). If they are connected we write the positive weight k where
6
α = π/k onthe edge, andunlabelededgesdenotean angleofπ/3. Forexamples,
ij
seetheCoxeterdiagramsinTable1.
In this paperwe set the sectionalcurvatureof Hn, K = k2, to be k = 1. The
−
distancedoftwoproperpoints[x]and[y]iscalculatedbytheformula
x, y
coshd= −h i . (5)
x, x y, y
h ih i
TheperpendicularfootY[y]ofpointXp[x]droppedontoplane[u]isgivenby
x,u
y=x h iu, (6)
− u,u
h i
whereuisthepoleoftheplane[u].
2.2 Horospheres andHoroballsinHn
AhorosphereinHn(n 2)isahyperbolicn-spherewithinfiniteradiuscenteredatan
idealpointon∂Hn. Equ≥ivalently,ahorosphereisan(n 1)-surfaceorthogonaltothe
−
setofparallelstraightlinespassingthroughapointoftheabsolutequadraticsurface.
Ahoroballisahorospheretogetherwithitsinterior.
Inordertoderivetheequationofahorosphere,weintroduceaprojectivecoordinate
systemfor n withavectorbasisa (i = 0,1,2,...,n)sothattheCayley-Kleinball
i
P
5
model of Hn is centered at (1,0,0,...,0), and set an arbitrary point at infinity to
lie at A = (1,0,...,0,1). The equation of a horosphere with center A passing
0 0
through point S = (1,0,...,0,s) is derived from the equation of the the absolute
sphere x0x0+x1x1+x2x2+ +xnxn =0,andtheplanex0 xn =0tangent
− ··· −
totheabsolutesphereatA . Thegeneralequationofthehorosphereis
0
0=λ( x0x0+x1x1+x2x2+ +xnxn)+µ(x0 xn)2. (7)
− ··· −
PlugginginforS weobtain
λ 1 s
λ( 1+s2)+µ( 1+s)2 =0 and = − .
− − µ 1+s
Ifs= 1,theequationofahorosphereinprojectivecoordinatesis
6 ±
n
(s 1) x0x0+ (xi)2 (1+s)(x0 xn)2 =0, (8)
− − !− −
i=1
X
andincartesiancoordinatessettingh = xi itbecomes
i x0
2 n h2 4 h s+1 2
i=1 i + d− 2 =1. (9)
1 s (1 s)2
(cid:0)P− (cid:1) (cid:0) − (cid:1)
Inann-dimensionalhyperbolicspaceanytwohoroballsarecongruentintheclas-
sical sense: each have an infinite radius. However, it is often useful to distinguish
between certain horoballsof a packing. We use the notion of horoball type with re-
specttothepackingasintroducedin[26].
Twohoroballsofahoroballpackingaresaidtobeofthesametypeorequipacked
if and only if their local packing densities with respect to a given cell (in our case a
Coxetersimplex)areequal.Ifthisisnotthecase,thenwesaythetwohoroballsareof
differenttype. Forexample,intheabovediscussionhoroballscenteredatA passing
0
throughS withdifferentvaluesforthefinalcoordinatesareofdifferenttyperelative
toanappropriatecell.
In order to compute volumes of horoball pieces, we use Ja´nos Bolyai’s classical
formulasfromthemid19-thcentury:
1. ThehyperboliclengthL(x)ofahorosphericarcthatbelongstoachordsegment
oflengthxis
x
L(x)=2sinh . (10)
2
(cid:16) (cid:17)
2. TheintrinsicgeometryofahorosphereisEuclidean,sothe(n 1)-dimensional
−
volume ofapolyhedronAonthesurfaceofthehorospherecanbecalculated
as in EnA1. The volumeof the horoballpiece (A) determinedby A and the
−
H
aggregateofaxesdrawnfromAtothecenterofthehoroballis
1
vol( (A))= . (11)
H n 1A
−
6
3 Horoball packings of Coxeter Simplices with Ideal
Verticies
Let beaCoxetertiling.Arigidmotionmappingonecellof ontoanothermapsthe
T T
entiretilingontoitself. Thesymmetrygroupofa CoxetertilingisitsCoxetergroup,
denotedbyΓ . Anysimplexcellof actsasafundamentaldomain ofΓ ,where
T T FT T
the Coxeter group is generated by reflections on the (n 1)-dimensional facets of
−
. In this paperwe consider onlyasymptotic Coxter simplices, i.e. ones thathave
aFtTleastoneidealvertex. InTable1welistthenineasymptoticCoxetersimplicesthat
existinhyperbolic4-space,togetherwiththeirvolumes. Foracompletediscussionof
hyperbolicCoxetersimplicesandtheirvolumesfordimensionsn 3,seeJohnsonet
≥
al. [10].
Wedefinethedensityofahoroballpacking ofaCoxetersimplextiling as
BT T
n
vol( )
δ( )= i=1 Bi∩FT . (12)
BT vol( )
P FT
Here denotesthe simplicial fundamentaldomain of tiling , n is the number of
FT T
ideal vertices of , and are the horoballs centered at ideal vertices. We allow
i
horoballsofdiffeFreTnttypesBattheasymptoticverticesofthetiling. Ahoroballtypeis
allowed if it yields a packing: no two horoballs may have an interior point in com-
mon. In addition we require that no horoball extend beyond the facet opposite the
vertexwhereit is centeredso thatthe packingremainsinvariantunderthe actionsof
the Coxeter group of the tiling. If these conditions are satisfied, we can extend the
packing density from the simplicial fundamental domain to the entire H4 using
FT
the Coxetergroup associated with a tiling. In the case of Coxeter simplex tilings,
T
Dirichlet–Voronoicells coincide with the Coxeter simplices. We denote the optimal
horoballpackingdensityas
δ ( )= sup δ( ). (13)
opt
T T packing BT
B
TheasymptoticCoxetersimplextilingsarerelatedthroughthesubgroupstructure
oftheirCoxetersymmetrygroupsasshownin Figure1 [11], [12]. LetΓ andΓ be
1 2
thetwoCoxetersymmetrygroupsofCoxetertilings and ,respectively.Whenthe
1 2
T T
indexofCoxetergroupΓ inΓ istwo,i.e. Γ :Γ =2,thenthetwoCoxetergroups
1 2 1 2
| |
differ by one reflection, and the fundamental domain of Γ is obtained from that of
2
Γ bydomaindoubling,that is by merginga certain pair of neighboringdomainsby
1
removingacommonfacet.InthecaseofasymptoticCoxetersimplicesif Γ :Γ =2
1 2
| |
and the numberof asymptoticverticesof the fundamentaldomainsof Γ and Γ are
1 2
equal, then the new fundamentaldomainis obtainedby removinga facet adjacentto
anasymptoticvertex. Ifthenumberofasymptoticverticesincreasesbyone,thenthe
cellsofΓ areobtainedbyremovingafacetoppositetotheasymptoticverticesofΓ
2 1
andmergingthecells. TherelationshipbetweenthevolumesofthecellsofΓ andΓ
1 2
when Γ :Γ =misgivenbyvol( )=m vol( ). Iftheindexofthegroupsis
| 1 2| FT1 · FT2
two,thenapackingdensityδ( )forthebiggergroupcanbeextendedtothesmaller
groupΓ . BT1
2
7
Coxeter Witt Simplex Packing
Diagram Notation Symbol Volume Density
SimplyAsymptotic
4 [4,32,1] S π2/1440 0.71644896
4
[3,3[4]] P π2/720 0.71644896
4
4 4 [3,4,3,4] R π2/864 0.60792710
4
4
[3,4,31,1] O π2/432 0.60792710
4
4 4
[(32,4,3,4)] FR π2/108 0.71644896
4
DoublyAsymptotic
4 d
4 [4, 3,4] N π2/288 0.71644896
4 4
4
[4,3[4]] BP π2/144 0.71644896
4
TriplyAsymptotic
4 [4,31,1,1] M π2/144 0.71644896
4
[3[3] []] DP π2/72 0.71644896
× 4
Table1: NotationandvolumesforthenineasymptoticCoxeterSimplicesinH4.
Figure 1: Lattice of Subgroupsof cocompactCoxeter groupsin H4. The numberof
starsinthesuperscript and indicatesthatthefundamentalsimplexofthegroup
∗∗ ∗∗∗
hastwoorthreeidealvertices.
8
3.1 Simply AsymptoticCases
We compute the optimal horoballpacking density for the Coxeter simplex tiling S ;
4
the other simply asymptotic cases can be obtained using the same method. Case R
4
wascomputedbythesecond-namedauthorin[21].
Proposition3.1. TheoptimalhoroballpackingdensityforsimplyasymptoticCoxeter
simplextiling isδ (S ) 0.71644896.
TS4 opt 4 ≈
Proof. Let be the simplicial fundamentaldomain of Coxeter tiling . We set
FS4 TS4
coordinates for its vertices A ,A ,...,A that satisfy the angle requirements. Our
0 1 4
choice of vertices, as well as forms for hyperplanes[u ] opposite to vertices A , are
i i
givenin Table2. Inordertomaximizethe packingdensity,we determinethe largest
horoballtype (s)centeredatidealvertexA thatisadmissibleincell . Thisis
B0 0 FS4
thehoroballwith type-parameters (intuitivelythe “radius”ofthe horoball)suchthat
the horoball (s) is tangent to the plane of the hyperface[u ] boundingthe funda-
0 0
B
mentalsimplexoppositeofA . TheperpendicularfootF [f ]ofvertexA onplane
0 0 0 0
[u ],
0
a ,u 2 1
f =a h 0 0iu = 1,0, , ,0 , (14)
0 0− u ,u 0 −5 5
h 0 0i (cid:18) (cid:19)
isthepointoftangencyofhoroball (s)andhyperfaceu ofthethesimplexcell.
0 0
B
Plugging in for F and solving equation(9), we find that the horoballwith type-
0
parameters = 1 istheoptimaltype. Theequationofhorosphere∂ = ∂ ( 1)
−9 B0 B0 −9
centeredatA passingthroughF is
0 0
9 81 4 2
h2+h2+h2 + h =1. (15)
5 1 2 3 25 4− 9
(cid:18) (cid:19)
(cid:0) (cid:1)
A
1
A
2
A
3 A=H
2 2
A
1
A A =H
H 3 4 4
H 1 H H
3 1 2
H
3
A
0 A
0
(a) (b)
Figure2:Simplyasymptoticcase. (a)Horoball intersectingthesidesofthesimplex
0
B
atH ,H ,andH . (b)HorospherictetrahedrononhyperfaceoppositeA .
1 2 3 0
9
The intersections H [h ] of horosphere ∂ and simplex edges are found by pa-
i i 0
B
rameterizingthesimplexedgesash (λ) = λa +a (i = 1,2,3,4),andcomputing
i 0 i
their intersections with ∂ . See Figure (2), and Table 2 for the intersection points.
0
B
The volume of the horospherical tetrahedron determines the volume of the horoball
piecebyequation(11). Inordertodeterminethedataofthehorospherictetrahedron,
we compute the hyperbolic distances l by the formula (5) l = d(H ,H ) where
ij ij i j
d(hi,hj)=arccos(cid:18)√hh−i,hhhiii,hhhjji,hji(cid:19).Moreover,thehorosphericaldistancesLij can
becalculatedbyformula(10). TheintrinsicgeometryofthehorosphereisEuclidean,
so we use the Cayley-Menger determinant to find the volume of the horospheric
A
tetrahedronA,
0 1 1 1 1
1 0 L2 L2 L2
1 (cid:12) 12 13 14(cid:12)
= (cid:12)1 L2 0 L2 L2 (cid:12) 0.0147314. (16)
A 288(cid:12) 12 23 24(cid:12)≈
(cid:12)1 L2 L2 0 L2 (cid:12)
(cid:12) 13 23 34(cid:12)
(cid:12)1 L2 L2 L2 0 (cid:12)
(cid:12) 14 24 34 (cid:12)
(cid:12) (cid:12)
Thevolumeoftheoptim(cid:12) alhoroballpieceinthefu(cid:12)ndamentalsimplexis
(cid:12) (cid:12)
1 1
vol( )= 0.0147314 0.00491046. (17)
B0∩FS4 n 1A≈ 3 · ≈
−
HencebytheCoxetergroupΓ theoptimalhoroballpackingdensityoftheCox-
S4
eterSimplextiling becomes
TS4
δ (S )= vol(B0∩FS4) 0.00491046 0.71644896. (18)
opt 4 vol( ) ≈ π2/1440 ≈
FS4
The same method is used to find the optimal packing density of the remaining
simply asymptotic Coxeter simplex tilings. Results of the computationsare given in
Table2. Wesummarizetheresults:
Corollary 3.2. The optimal horoball packing density for simply asymptotic Coxeter
simplextiling ,Γ S ,P ,FR isδ (Γ) 0.71644896.
Γ 4 4 4 opt
T ∈ ≈
n o
d
10