Densities of the densest known lattice sphere packings

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Numbers

$n$

$\Delta_n$ or $\delta_n$

$\Delta_n$:

comment: $\Delta_{1}=1$, the density of $\Lambda_1=\mathbb{Z}$; the densest packing of any kind in dimension 1
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Z,1,density

$\delta_n$:

1/2

comment: $\delta_{1}=\frac{1}{2}$, the centre density of $\Lambda_1=\mathbb{Z}$; $\det\Lambda_{1}=4$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Z,1,centre

$\Delta_n$:

0.9068996821171089252970391288210778661420331240463702877849424676940615905631769418420624941060300844

comment: $\Delta_{2}=\frac{\sqrt{3}\,\pi}{6}$, the density of $\Lambda_2=A_2$, the hexagonal lattice; the densest packing of any kind in dimension 2 [18]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,2,density

$\delta_n$:

0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626

comment: $\delta_{2}=\frac{\sqrt{3}}{6}$, the centre density of $\Lambda_2=A_2$, the hexagonal lattice; $\det\Lambda_{2}=12$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,2,centre

$\Delta_n$:

0.7404804896930610411693134983434489497691036148959483705142326011594057988499123184292211557941275396

comment: $\Delta_{3}=\frac{\sqrt{2}\,\pi}{6}$, the density of $\Lambda_3=A_3=D_3$, the face-centred cubic lattice; the densest packing of any kind in dimension 3 [19]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,3,density

$\delta_n$:

0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966

comment: $\delta_{3}=\frac{\sqrt{2}}{8}$, the centre density of $\Lambda_3=A_3=D_3$, the face-centred cubic lattice; $\det\Lambda_{3}=32$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,3,centre

$\Delta_n$:

0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145

comment: $\Delta_{4}=\frac{\pi^{2}}{16}$, the density of $\Lambda_4=D_4$; the densest lattice packing in dimension 4 [16]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,4,density

$\delta_n$:

1/8

comment: $\delta_{4}=\frac{1}{8}$, the centre density of $\Lambda_4=D_4$; $\det\Lambda_{4}=64$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,4,centre

$\Delta_n$:

0.4652576133092586356105040624112936859946577513965361577435664445013271841888718143111600891540540958

comment: $\Delta_{5}=\frac{\sqrt{2}\,\pi^{2}}{30}$, the density of $\Lambda_5=D_5$; the densest lattice packing in dimension 5 [16]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,5,density

$\delta_n$:

0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830

comment: $\delta_{5}=\frac{\sqrt{2}}{16}$, the centre density of $\Lambda_5=D_5$; $\det\Lambda_{5}=128$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,5,centre

$\Delta_n$:

0.3729475455820649395634775586799581063936647972683873631114040655972831720296832195225267216353405428

comment: $\Delta_{6}=\frac{\sqrt{3}\,\pi^{3}}{144}$, the density of $\Lambda_6=E_6$; the densest lattice packing in dimension 6 [17]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,6,density

$\delta_n$:

0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065

comment: $\delta_{6}=\frac{\sqrt{3}}{24}$, the centre density of $\Lambda_6=E_6$; $\det\Lambda_{6}=192$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,6,centre

$\Delta_n$:

0.2952978731457125730997744292104894781164313196750962637537575057505370944520543432149209622152655828

comment: $\Delta_{7}=\frac{\pi^{3}}{105}$, the density of $\Lambda_7=E_7$; the densest lattice packing in dimension 7 [17]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,7,density

$\delta_n$:

1/16

comment: $\delta_{7}=\frac{1}{16}$, the centre density of $\Lambda_7=E_7$; $\det\Lambda_{7}=256$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,7,centre

$\Delta_n$:

0.2536695079010480136365633663768362272128322543559516189881975504947157694188208234117756959238359181

comment: $\Delta_{8}=\frac{\pi^{4}}{384}$, the density of $\Lambda_8=E_8$; the densest packing of any kind in dimension 8 [20]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,8,density

$\delta_n$:

1/16

comment: $\delta_{8}=\frac{1}{16}$, the centre density of $\Lambda_8=E_8$; $\det\Lambda_{8}=256$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,8,centre

$\Delta_n$:

0.1457748758081711105332137544576519995425739925107256670963211484639631773090752900724100031467048256

comment: $\Delta_{9}=\frac{\sqrt{2}\,\pi^{4}}{945}$, the density of $\Lambda_{9}$; the densest lattice packing in dimension 9 [13]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,9,density

$\delta_n$:

0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915

comment: $\delta_{9}=\frac{\sqrt{2}}{32}$, the centre density of $\Lambda_{9}$; $\det\Lambda_{9}=512$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,9,centre

$\Delta_n$:

0.09202111843130555779386059577030574946623950376510629227349760612731057277995423712397819980248230119

comment: $\Delta_{10}=\frac{\sqrt{3}\,\pi^{5}}{5760}$, the density of $\Lambda_{10}$; the nonlattice packing $P_{10c}$ [24] is denser, with centre density $\frac{5}{128}$ and density $0.09961578\ldots$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,10,density

$\delta_n$:

0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533

comment: $\delta_{10}=\frac{\sqrt{3}}{48}$, the centre density of $\Lambda_{10}$; $\det\Lambda_{10}=768$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,10,centre

$\Delta_n$:

0.06043266010816539437608674046419309761323024008688077170951518273167614335524972827638523430991943333

comment: $\Delta_{11}=\frac{32\sqrt{3}\,\pi^{5}}{280665}$, the density of $K_{11}$, a lamination of the Coxeter–Todd lattice $K_{12}$; the nonlattice packing $P_{11a}$ [24] is denser, with centre density $\frac{9}{256}$ and density $0.06623802\ldots$

$\delta_n$:

0.03207501495497920913939715447233096975820009729278482644547790702688764846127407476076196642143052918

comment: $\delta_{11}=\frac{\sqrt{3}}{54}$, the centre density of $K_{11}$, a lamination of the Coxeter–Todd lattice $K_{12}$; $\det K_{11}=972$ at minimal norm $4$

$\Delta_n$:

0.04945417662424405540278906603150308121744946132104149258956153227798837000217850892485706837201212635

comment: $\Delta_{12}=\frac{\pi^{6}}{19440}$, the density of $K_{12}$, the Coxeter–Todd lattice
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#K,12,density

$\delta_n$:

1/27

comment: $\delta_{12}=\frac{1}{27}$, the centre density of $K_{12}$, the Coxeter–Todd lattice; $\det K_{12}=729$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#K,12,centre

$\Delta_n$:

0.02920843092810790193767210634687198772921529582092394570064333230603993691845574246214915260239615754

comment: $\Delta_{13}=\frac{64\sqrt{3}\,\pi^{6}}{3648645}$, the density of $K_{13}$, a lamination of the Coxeter–Todd lattice $K_{12}$; the nonlattice packing $P_{13a}$ [24] is denser, with centre density $\frac{9}{256}$ and density $0.03201429\ldots$

$\delta_n$:

0.03207501495497920913939715447233096975820009729278482644547790702688764846127407476076196642143052918

comment: $\delta_{13}=\frac{\sqrt{3}}{54}$, the centre density of $K_{13}$, a lamination of the Coxeter–Todd lattice $K_{12}$; $\det K_{13}=972$ at minimal norm $4$

$\Delta_n$:

0.02162409608244710546249221367868409564177065936712948464424487710130327456120990167726026005937182741

comment: $\Delta_{14}=\frac{\sqrt{3}\,\pi^{7}}{241920}$, the density of $\Lambda_{14}$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,14,density

$\delta_n$:

0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533

comment: $\delta_{14}=\frac{\sqrt{3}}{48}$, the centre density of $\Lambda_{14}$; $\det\Lambda_{14}=768$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,14,centre

$\Delta_n$:

0.01685757065676269764781550358540912002543295423644778555552670648966997295106061582291034513965069853

comment: $\Delta_{15}=\frac{8\sqrt{2}\,\pi^{7}}{2027025}$, the density of $\Lambda_{15}$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,15,density

$\delta_n$:

0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915

comment: $\delta_{15}=\frac{\sqrt{2}}{32}$, the centre density of $\Lambda_{15}$; $\det\Lambda_{15}=512$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,15,centre

$\Delta_n$:

0.01470816439743082528386745954846658884656680845678491240716470023416869559435667208586846528865779255

comment: $\Delta_{16}=\frac{\pi^{8}}{645120}$, the density of $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,16,density

$\delta_n$:

1/16

comment: $\delta_{16}=\frac{1}{16}$, the centre density of $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice; $\det\Lambda_{16}=256$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,16,centre

$\Delta_n$:

0.008811319182321189869770444983484678153924789606690031715489243517038983815272716868407840771127355841

comment: $\Delta_{17}=\frac{32\pi^{8}}{34459425}$, the density of $\Lambda_{17}$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,17,density

$\delta_n$:

1/16

comment: $\delta_{17}=\frac{1}{16}$, the centre density of $\Lambda_{17}$; $\det\Lambda_{17}=256$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,17,centre

$\Delta_n$:

0.005928368718469419730049891486523909806100960042824357061950447334418422452973803955425827012554145075

comment: $\Delta_{18}=\frac{\sqrt{3}\,\pi^{9}}{8709120}$, the density of $\Lambda_{18}$; the nonlattice packing $B_{18}$ [8] is denser, with centre density $\frac{3^{9}}{2^{18}}$ and density $0.006167898\ldots$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,18,density

$\delta_n$:

0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065

comment: $\delta_{18}=\frac{\sqrt{3}}{24}$, the centre density of $\Lambda_{18}$; $\det\Lambda_{18}=192$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,18,centre

$\Delta_n$:

0.004120806279768667500886821058357660528632336189215187578663362443680168202332562149057259835876284349

comment: $\Delta_{19}=\frac{64\sqrt{2}\,\pi^{9}}{654729075}$, the density of $\Lambda_{19}$; the antipode packing of dimension 19 [12] is denser, with centre density $\frac{13^{19/2}}{3^{9}\cdot 5^{21/2}}$ and density $0.004147369\ldots$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,19,density

$\delta_n$:

0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830

comment: $\delta_{19}=\frac{\sqrt{2}}{16}$, the centre density of $\Lambda_{19}$; $\det\Lambda_{19}=128$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,19,centre

$\Delta_n$:

0.003225861423751757501574786781612356207148305131018335748975296564745861166993469481943592281654563139

comment: $\Delta_{20}=\frac{\pi^{10}}{29030400}$, the density of $\Lambda_{20}$; the antipode packing of dimension 20 [12] is denser, with centre density $\frac{3^{20}}{2^{10}\cdot 5^{21/2}}$ and density $0.004024165\ldots$, as is $B_{20}$ [9] with centre density $\frac{7^{10}}{2^{31}}$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,20,density

$\delta_n$:

1/8

comment: $\delta_{20}=\frac{1}{8}$, the centre density of $\Lambda_{20}$; $\det\Lambda_{20}=64$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,20,centre

$\Delta_n$:

0.002465884711502463245944355824165164696567691388297833979591321081883716334406532970306523662963463407

comment: $\Delta_{21}=\frac{256\sqrt{2}\,\pi^{10}}{13749310575}$, the density of $\Lambda_{21}$; the antipode packing of dimension 21 [12] is denser, with centre density $\frac{43^{21/2}}{2^{41}\cdot 3^{23/2}}$ and density $0.002929829\ldots$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,21,density

$\delta_n$:

0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966

comment: $\delta_{21}=\frac{\sqrt{2}}{8}$, the centre density of $\Lambda_{21}$; $\det\Lambda_{21}=32$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,21,centre

$\Delta_n$:

0.002127660145275864210626933444611565330655697515338490683084571822978614708729444826368239332278713761

comment: $\Delta_{22}=\frac{\sqrt{3}\,\pi^{11}}{239500800}$, the density of $\Lambda_{22}$; the nonlattice packing $R_{22}$ [11] is denser, with centre density $0.33254$ [24]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,22,density

$\delta_n$:

0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626

comment: $\delta_{22}=\frac{\sqrt{3}}{6}$, the centre density of $\Lambda_{22}$; $\det\Lambda_{22}=12$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,22,centre

$\Delta_n$:

0.001905328193426062470906566906903334103361515823538275509259618028319837706257289936344921562809463883

comment: $\Delta_{23}=\frac{2048\pi^{11}}{316234143225}$, the density of $\Lambda_{23}$; the antipode packing of dimension 23 [12] is denser, with centre density $\frac{23^{23/2}}{2^{34}\cdot 3^{12}}$ and density $0.001907194\ldots$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,23,density

$\delta_n$:

1/2

comment: $\delta_{23}=\frac{1}{2}$, the centre density of $\Lambda_{23}$; $\det\Lambda_{23}=4$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,23,centre

$\Delta_n$:

0.001929574309403923047903345563685957640168471815000303352234647617331495634250985531487347698186143913

comment: $\Delta_{24}=\frac{\pi^{12}}{479001600}$, the density of $\Lambda_{24}$, the Leech lattice; the densest packing of any kind in dimension 24 [21]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,24,density

$\delta_n$:

comment: $\delta_{24}=1$, the centre density of $\Lambda_{24}$, the Leech lattice; $\det\Lambda_{24}=1$ at minimal norm $4$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,24,centre

$\Delta_n$:

0.0006772120097731805113471473269154748776650137636745243615753164714289538439340188299756603053478865394

comment: $\Delta_{25}=\frac{2^{13}\pi^{12}}{25!!}\cdot \frac{\sqrt{2}}{2}$, the density of $\Lambda_{25}$

$\delta_n$:

0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864

comment: $\delta_{25}=\frac{\sqrt{2}}{2}$, the centre density of $\Lambda_{25}$
equals: Algebraic_numbers_of_degree_2#2,0,-1,2

$\Delta_n$:

0.0002692200504338088915924745768040714297011094434231232693144368460479860964520299557149709280683257129

comment: $\Delta_{26}=\frac{\pi^{13}}{13!}\cdot \frac{\sqrt{3}}{3}$, the density of $\Lambda_{26}$ and of the lattice $T_{26}$

$\delta_n$:

0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252

comment: $\delta_{26}=\frac{\sqrt{3}}{3}$, the centre density of $\Lambda_{26}$ and of the lattice $T_{26}$
equals: Algebraic_numbers_of_degree_2#3,0,-1,2

$\Delta_n$:

0.0001286752812026914537542358296864780893844252168343427048415149520435305115637214720660845547777589254

comment: $\Delta_{27}=\frac{2^{14}\pi^{13}}{27!!}\cdot \frac{\sqrt{3}}{3}$, the density of Bacher's lattice $B_{27}$ [5]; the nonlattice packing $B_{27}^{*}$ [10] is denser, with centre density $\frac{\sqrt{2}}{2}$ and density $0.0001575943\ldots$

$\delta_n$:

0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252

comment: $\delta_{27}=\frac{\sqrt{3}}{3}$, the centre density of Bacher's lattice $B_{27}$ [5]
equals: Algebraic_numbers_of_degree_2#3,0,-1,2

$\Delta_n$:

0.00006975873661656380474534448556815955174019822099394214212351808051429010470306762143391879066489944667

comment: $\Delta_{28}=\frac{\pi^{14}}{14!}\cdot \frac{2}{3}$, the density of Bacher's lattice $B_{28}$ [5]; the nonlattice packing $B_{28}^{*}$ [10] is denser, with centre density $1$ and density $0.0001046381\ldots$

$\delta_n$:

2/3

comment: $\delta_{28}=\frac{2}{3}$, the centre density of Bacher's lattice $B_{28}$ [5]

$\Delta_n$:

0.00002787898745689491625867746513359251450710853680071583775002735955969236331285330192390674633854318115

comment: $\Delta_{29}=\frac{2^{15}\pi^{14}}{29!!}\cdot \frac{\sqrt{3}}{3}$, the density of Bacher's lattice $B_{29}$ [5]; the nonlattice packing $B_{29}^{*}$ [10] is denser, with centre density $\frac{\sqrt{2}}{2}$ and density $3.414464\ldots\cdot 10^{-5}$

$\delta_n$:

0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252

comment: $\delta_{29}=\frac{\sqrt{3}}{3}$, the centre density of Bacher's lattice $B_{29}$ [5]
equals: Algebraic_numbers_of_degree_2#3,0,-1,2

$\Delta_n$:

0.00001442864384256620421857489131773473395596412180139265575900403752200627712423813706538622689394868806

comment: $\Delta_{30}=\frac{\pi^{15}}{15!}\cdot \frac{3^{27/2}}{2^{22}}$, the density of $Q_{30}$, a section of Quebbemann's lattice $Q_{32}$ [6]; the nonlattice packing $T_{30}$ [10] is denser, with centre density $1$ and density $2.191535\ldots\cdot 10^{-5}$

$\delta_n$:

0.6583806132496917613145253261396090864375481845547503171119256856204853542607042316244421558164064457

comment: $\delta_{30}=\frac{3^{27/2}}{2^{22}}$, the centre density of $Q_{30}$, a section of Quebbemann's lattice $Q_{32}$ [6]

$\Delta_n$:

0.00001183776518593384998071759273835507885728390453732382817598636726702797292586777705496109319991088297

comment: $\Delta_{31}=\frac{2^{16}\pi^{15}}{31!!}\cdot \frac{3^{15}}{2^{47/2}}$, the density of $Q_{31}$, a section of Quebbemann's lattice $Q_{32}$ [6]

$\delta_n$:

1.209522419251813897932523068585014714448148891943658461918912656856381129940287883550500729324021128

comment: $\delta_{31}=\frac{3^{15}}{2^{47/2}}$, the centre density of $Q_{31}$, a section of Quebbemann's lattice $Q_{32}$ [6]

$\Delta_n$:

0.00001104074930885985419934596904309777447951275366061902008487075156881710592894952916757719442476445007

comment: $\Delta_{32}=\frac{\pi^{16}}{16!}\cdot \frac{3^{16}}{2^{24}}$, the density of Quebbemann's lattice $Q_{32}$ [6], one of several lattices of this density [24]

$\delta_n$:

43046721/16777216

comment: $\delta_{32}=\frac{3^{16}}{2^{24}}$, the centre density of Quebbemann's lattice $Q_{32}$ [6], one of several lattices of this density [24]

$\Delta_n$:

0.000004140688289649678228845104330902760244396139391856860344706085015540230528228227452679860288336994649

comment: $\Delta_{33}=\frac{2^{17}\pi^{16}}{33!!}\cdot \frac{3^{33/2}}{2^{25}}$, the density of $Q_{33}$, due to Elkies [24]

$\delta_n$:

2.222034569717709694436522975721180666726725122872282320252749188969138070629876781732492275880371754

comment: $\delta_{33}=\frac{3^{33/2}}{2^{25}}$, the centre density of $Q_{33}$, due to Elkies [24]

$\Delta_n$:

0.000001766973889154063266233146292413824721789458545713814925373042404669197424829268689185553018927793592

comment: $\Delta_{34}=\frac{\pi^{17}}{17!}\cdot \frac{3^{33/2}}{2^{25}}$, the density of $Q_{34}$, due to Elkies [24]

$\delta_n$:

2.222034569717709694436522975721180666726725122872282320252749188969138070629876781732492275880371754

comment: $\delta_{34}=\frac{3^{33/2}}{2^{25}}$, the centre density of $Q_{34}$, due to Elkies [24]

$\Delta_n$:

9.461904151153729608089621070158634848599355754381467737703194297509752400982400821965740121725247960e-7

comment: $\Delta_{35}=\frac{2^{18}\pi^{17}}{35!!}\cdot 2\sqrt{2}$, the density of $B_{35}$ [24]

$\delta_n$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\delta_{35}=2\sqrt{2}$, the centre density of $B_{35}$ [24]

$\Delta_n$:

6.161466094691810322232590442931743951888902608759339696407596696344044477025154721431238893638150756e-7

comment: $\Delta_{36}=\frac{\pi^{18}}{18!}\cdot \frac{2^{18}}{3^{10}}$, the density of the Kschischang–Pasupathy lattice $KP_{36}$ [7]

$\delta_n$:

262144/59049

comment: $\delta_{36}=\frac{2^{18}}{3^{10}}$, the centre density of the Kschischang–Pasupathy lattice $KP_{36}$ [7]

$\Delta_n$:

3.213562007593008162341082150232664617214040385929130458362157889703619240459966874214828145318934216e-7

comment: $\Delta_{37}=\frac{2^{19}\pi^{18}}{37!!}\cdot 4\sqrt{2}$, the density of the lattice the catalogue calls $D_{37}$ [24], which is not the root lattice

$\delta_n$:

5.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291

comment: $\delta_{37}=4\sqrt{2}$, the centre density of the lattice the catalogue calls $D_{37}$ [24], which is not the root lattice

$\Delta_n$:

1.835874319781589848816309790845727028485982846487039096526555894380331837526438410356589312440255285e-7

comment: $\Delta_{38}=\frac{\pi^{19}}{19!}\cdot 8$, the density of the lattice the catalogue calls $D_{38}$ [24], which is not the root lattice

$\delta_n$:

comment: $\delta_{38}=8$, the centre density of the lattice the catalogue calls $D_{38}$ [24], which is not the root lattice

$\Delta_n$:

1.004160423807392571953915123402160513103627638984656001369871883369923077631991214873236331735606926e-7

comment: $\Delta_{39}=\frac{2^{20}\pi^{19}}{39!!}\cdot \frac{3^{16}}{2^{41/2}\,\sqrt{7}}$, the density of a section of $P_{48p}$

$\delta_n$:

10.97175609084857837783112211323698462591417821720153867127005116508621291322273490778575148095210608

comment: $\delta_{39}=\frac{3^{16}}{2^{41/2}\,\sqrt{7}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

7.848004886817056107952147964588977935929629771229549931566154754727770719399413185430635961271491419e-8

comment: $\Delta_{40}=\frac{\pi^{20}}{20!}\cdot \frac{3^{17}}{2^{45/2}}$, the density of a section of $P_{48p}$

$\delta_n$:

21.77140354653265016278541523453026486006668005498585231454042782341486033892518190390901312783238031

comment: $\delta_{40}=\frac{3^{17}}{2^{45/2}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

6.107161314064865911521112990256269995951628211028227498660539415897274256760684592677984090841666792e-8

comment: $\Delta_{41}=\frac{2^{21}\pi^{20}}{41!!}\cdot \frac{3^{17}}{2^{43/2}}$, the density of a section of $P_{48p}$

$\delta_n$:

43.54280709306530032557083046906052972013336010997170462908085564682972067785036380781802625566476062

comment: $\delta_{41}=\frac{3^{17}}{2^{43/2}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

4.981109572888996969985256478761774860906747743037860734407016630223122056146991080277434290058270008e-8

comment: $\Delta_{42}=\frac{\pi^{21}}{21!}\cdot \frac{3^{18}}{2^{22}}$, the density of a section of $P_{48p}$

$\delta_n$:

387420489/4194304

comment: $\delta_{42}=\frac{3^{18}}{2^{22}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

4.015719024813621787458341847586705407343066262469126089074091267698266080902469583097897564655669231e-8

comment: $\Delta_{43}=\frac{2^{22}\pi^{21}}{43!!}\cdot \frac{3^{19}}{2^{45/2}}$, the density of a section of $P_{48p}$

$\delta_n$:

195.9426319187938514650687371107723837406001204948726708308638504107337430503266371351811181504914228

comment: $\delta_{43}=\frac{3^{19}}{2^{45/2}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

3.200853526550563420549722108024351982826479394721109384666999674588104498216971555352083913903218733e-8

comment: $\Delta_{44}=\frac{\pi^{22}}{22!}\cdot \frac{3^{20}}{2^{23}}$, the density of a section of $P_{48p}$; the antipode packing of dimension 44 [12] is denser, with centre density $\frac{157^{22}}{2^{22}\cdot 5^{43/2}\cdot 11^{23}}$ and density $3.924426\ldots\cdot 10^{-8}$, as is $T_{44}$ [11] with centre density $\frac{17^{22}}{2^{43}\cdot 3^{24}}$

$\delta_n$:

3486784401/8388608

comment: $\delta_{44}=\frac{3^{20}}{2^{23}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

2.523150677447048566040145693682576538875300138562733003349171752775916756915197431774818524452253809e-8

comment: $\Delta_{45}=\frac{2^{23}\pi^{22}}{45!!}\cdot \frac{3^{21}}{2^{47/2}}$, the density of a section of $P_{48p}$; the antipode packing of dimension 45 [12] is denser, with centre density $\frac{23^{45/2}}{2^{183/2}}$ and density $3.558225\ldots\cdot 10^{-8}$, as is $T_{45}$ [11] with centre density $\frac{17^{45/2}}{2^{44}\cdot 3^{24}}$

$\delta_n$:

881.7418436345723315928093169984757268327005422269270187388873268483018437264698671083150316772114025

comment: $\delta_{45}=\frac{3^{21}}{2^{47/2}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

2.271798035790364987469960422140790056284082475950683765178309964525128247789243100146897370589250298e-8

comment: $\Delta_{46}=\frac{\pi^{23}}{23!}\cdot \frac{3^{43/2}}{2^{23}}$, the density of a section of $P_{48p}$; the nonlattice packing $T_{46}$ [11] is denser, with centre density $\frac{13^{23}}{3^{93/2}}$ and density $2.860957\ldots\cdot 10^{-8}$

$\delta_n$:

2159.817601765613822992300332400987608058376819431858415285672211678002204652240231843982492155721345

comment: $\delta_{46}=\frac{3^{43/2}}{2^{23}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

2.146607826746594433694467372566267049167157488861185240643274998764633203947081498001668427708126404e-8

comment: $\Delta_{47}=\frac{2^{24}\pi^{23}}{47!!}\cdot \frac{3^{23}}{2^{24}}$, the density of a section of $P_{48p}$; the antipode packing of dimension 47 [12] is denser, with centre density $\frac{47^{47/2}}{2^{118}}$ and density $2.266960\ldots\cdot 10^{-8}$, as is $T_{47}$ [11] with centre density $\frac{5^{47/2}\cdot 7^{47/2}}{2^{70}\cdot 3^{24}}$

$\delta_n$:

94143178827/16777216

comment: $\delta_{47}=\frac{3^{23}}{2^{24}}$, the centre density of a section of $P_{48p}$

$\Delta_n$:

2.317829531054123758796747066071034647103782879570623868579861117106670677649424225296858591024593000e-8

comment: $\Delta_{48}=\frac{\pi^{24}}{24!}\cdot \frac{3^{24}}{2^{24}}$, the density of $P_{48n}$, $P_{48p}$ and $P_{48q}$, even unimodular lattices of minimal norm $6$

$\delta_n$:

282429536481/16777216

comment: $\delta_{48}=\frac{3^{24}}{2^{24}}$, the centre density of $P_{48n}$, $P_{48p}$ and $P_{48q}$, even unimodular lattices of minimal norm $6$

Definition

For each dimension $n\geq 1$, the packing density $\Delta_n=V_n\delta_n$ and the centre density $\delta_n=\rho^n/\sqrt{\det L}$ of the densest lattice sphere packing [23] known in $\mathbb{R}^n$, according to the table of densest packings of the Catalogue of Lattices [24], where $L$ is the lattice, $\rho$ its packing radius, half the length of a shortest nonzero vector, and $V_n$ the volume of the unit ball.

Parameters

$n$

— dimension ($n\geq 1$)

Formulas

(1)

$\Delta_n=V_n\,\delta_n$ with $V_n=\pi^{n/2}/\Gamma(n/2+1)$ the volume of the unit ball, and $\delta_n=(\gamma(L)/4)^{n/2}$ where $\gamma(L)=\mu/(\det L)^{1/n}$ is the Hermite number of the record lattice $L$ of minimal norm $\mu$.

(2)

$\delta_n=1/\sqrt{\det\Lambda_n}$ for $1\leq n\leq 24$, $n\neq 11,12,13$, with $\Lambda_n$ scaled to minimal norm $4$, where $\det\Lambda_n$ for $n=0,1,\ldots,24$ is $1$, $4$, $12$, $32$, $64$, $128$, $192$, $256$, $256$, $512$, $768$, $1024$, $1024$, $1024$, $768$, $512$, $256$, $256$, $192$, $128$, $64$, $32$, $12$, $4$, $1$ [27]; $\delta_{11}=\delta_{13}=\frac{1}{18\sqrt{3}}$ and $\delta_{12}=\frac{1}{27}$ from $\det K_{11}=\det K_{13}=972$ and $\det K_{12}=729$. Hence $\delta_{24-n}=\delta_n$ for $0\leq n\leq 24$.

(3)

$\Delta_1=1$, $\Delta_2=\frac{\pi}{2\sqrt{3}}$, $\Delta_3=\frac{\pi}{3\sqrt{2}}$, $\Delta_4=\frac{\pi^2}{16}$, $\Delta_8=\frac{\pi^4}{384}$ and $\Delta_{24}=\frac{\pi^{12}}{12!}$; $\delta_{48}=\left(\frac{3}{2}\right)^{24}$, since an even unimodular lattice of minimal norm $6$ has $\rho^2=\frac{3}{2}$ and $\det L=1$.

Comments

(4)

$\det L$ is the determinant of a Gram matrix of $L$, the square of the volume of a fundamental domain, and $V_n=\pi^{n/2}/\Gamma(n/2+1)$ is the volume of the unit ball. $\Delta_n$ is the fraction of $\mathbb{R}^n$ covered by balls of radius $\rho$ centred at the points of $L$, and $\delta_n$ is $\Delta_n$ with the ball's volume divided out, the quantity Conway and Sloane [1] tabulate; both are unchanged when $L$ is scaled. $\gamma(L)=4\,\delta_n^{2/n}$ is the Hermite number of the record lattice $L$, the best lower bound known for Hermite's constant $\gamma_n$ in dimension $n$, and equal to $\gamma_n$ where the record is proven.

(5)

"Known" means listed in the catalogue's table of densest packings [24] as revised in February 2012, which gives the centre density of the densest lattice known in every dimension $n\leq 48$ in closed form; Table 1 of Cohn's survey [2], taken from Conway and Sloane [1], lists the same record densities for $n\leq 36$. The record is proven for $n\leq 8$ (Lagrange for $n=2$ [14], Gauss for $n=3$ [15], Korkine and Zolotareff for $n=4,5$ [16], Blichfeldt for $n=6,7,8$ [17]), for $n=24$ [22], and for $n=9$ by the enumeration of the perfect lattices in dimension $9$ of Dutour Sikirić and van Woerden [13]; in every other dimension a denser lattice may exist. The densest packing of any kind is known for $n=1$, $2$ [18], $3$ [19], $8$ [20] and $24$ [21], and is the lattice packing in each case.

(6)

For $n\leq 24$ the record lattice is the laminated lattice $\Lambda_n$ [3] except in dimensions $11$, $12$ and $13$, where the Coxeter–Todd lattice $K_{12}$ [4] and its laminations $K_{11}$ and $K_{13}$ are denser; $\Lambda_n$ for $n\leq 8$ is $\mathbb{Z}$, $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$, $E_8$, $\Lambda_{16}$ is the Barnes–Wall lattice and $\Lambda_{24}$ the Leech lattice. The densities of $\Lambda_n$ for $n\leq 24$ and of $K_{12}$ are also in the table of the classical lattices, which the entries for $n\leq 24$ other than $n=11$ and $13$ link. Beyond $n=24$ the records are $\Lambda_{25}$ and $\Lambda_{26}$, Bacher's lattices $B_{27}$, $B_{28}$ and $B_{29}$ [5], Quebbemann's lattice $Q_{32}$ [6] and its sections $Q_{30}$ and $Q_{31}$, lattices $Q_{33}$ and $Q_{34}$ of Elkies, $B_{35}$, the Kschischang–Pasupathy lattice $KP_{36}$ [7], lattices the catalogue calls $D_{37}$ and $D_{38}$ (not the root lattices), sections of $P_{48p}$ for $39\leq n\leq 47$, and in dimension $48$ the even unimodular lattices $P_{48n}$, $P_{48p}$ and $P_{48q}$ of minimal norm $6$, all as named on the catalogue's page [24]. The entry comment names the lattice of each row.

(7)

In dimensions $10$, $11$, $13$, $18$, $19$, $20$, $21$, $22$, $23$, $27$, $28$, $29$, $30$, $44$, $45$, $46$ and $47$ a nonlattice packing denser than every lattice packing known exists, and the entry comment gives its centre density: the packings $P_{10c}$, $P_{11a}$, $P_{13a}$, $B_{18}$ [8], $R_{22}$ [11], $B_{27}^{*}$, $B_{28}^{*}$, $B_{29}^{*}$, $T_{30}$ [10] and $T_{46}$ [11] of the catalogue's table, and the antipode packings of Chen, Hu, Li, Wang and Wu [12] in dimensions $19$, $20$, $21$, $23$, $44$, $45$ and $47$, which are denser than the packings $B_{20}$ [9], $T_{44}$, $T_{45}$ and $T_{47}$ [11] that the table of 2012 names. Their densities are not entries, since a nonlattice packing has no Gram matrix and this table is about lattices.

(8)

The comment on each entry gives its value in closed form, for $\Delta_n$ always $\pi^{\lfloor n/2\rfloor}$ times an algebraic number of degree at most $2$ and for $\delta_n$ that algebraic number, and names the lattice; on a centre density for $n\leq 24$ it also gives the determinant of the lattice scaled to minimal norm $4$, which for $\Lambda_n$ is OEIS A028921 [27].

Programs

(P1)

Sage

n = 24; delta = 1                                     # Leech lattice: rho^2 = 1, det 1
Delta = pi^(n/2) / gamma(n/2 + 1) * delta             # 1/479001600*pi^12
n = 9; delta = sqrt(2)/32                             # Lambda_9: 1/sqrt(det) at minimal norm 4, det 512
Delta = pi^(n/2) / gamma(n/2 + 1) * delta             # 1/945*sqrt(2)*pi^4
n = 32; delta = 3^16 / 2^24                           # Q_32: rho^2 = 3/2, det 2^16
Delta = pi^(n/2) / gamma(n/2 + 1) * delta             # 59049/481517373489152000*pi^16

References

[1]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der mathematischen Wissenschaften 290, Springer, 1999; Table I.1 and Table 1.2.

[2]

H. Cohn, A conceptual breakthrough in sphere packing, Notices of the American Mathematical Society 64 (2017), 102–115;. (arXiv)

[3]

J. H. Conway and N. J. A. Sloane, Laminated lattices, Annals of Mathematics 116 (1982), 593–620.

[4]

H. S. M. Coxeter and J. A. Todd, An extreme duodenary form, Canadian Journal of Mathematics 5 (1953), 384–392.

[5]

R. Bacher, Dense lattices in dimensions 27–29, Inventiones Mathematicae 130 (1997), 153–158.

[6]

H.-G. Quebbemann, Lattices with theta functions for $G(\sqrt{2})$ and linear codes, Journal of Algebra 105 (1987), 443–450.

[7]

F. R. Kschischang and S. Pasupathy, Some ternary and quaternary codes and associated sphere packings, IEEE Transactions on Information Theory 38 (1992), 227–246.

[8]

J. Bierbrauer and Y. Edel, Dense sphere packings from new codes, Journal of Algebraic Combinatorics 11 (2000), 95–100.

[9]

A. Vardy, A new sphere packing in 20 dimensions, Inventiones Mathematicae 121 (1995), 119–133.

[10]

A. Vardy, Density doubling, double-circulants, and new sphere packings, Transactions of the American Mathematical Society 351 (1999), 271–283.

[11]

J. H. Conway and N. J. A. Sloane, The antipode construction for sphere packings, Inventiones Mathematicae 123 (1996), 309–313.

[12]

R. Chen, J. Hu, B. Li, L. Wang and T. Wu, New sphere packings from the antipode construction (2025), preprint. (arXiv)

[13]

M. Dutour Sikirić and W. van Woerden, The lattice packing problem in dimension 9 by Voronoi's algorithm (2025), preprint. (arXiv)

[14]

J. L. Lagrange, Recherches d'arithmétique, Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin (1773), 265–312.

[15]

C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen (1831); Werke II, 188–196.

[16]

A. Korkine and G. Zolotareff, Sur les formes quadratiques positives quaternaires, Mathematische Annalen 5 (1872), 581–583; Sur les formes quadratiques positives, Mathematische Annalen 11 (1877), 242–292.

[17]

H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Mathematische Zeitschrift 39 (1935), 1–15.

[18]

A. Thue, Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene, Norske Videnskabs-Selskabets Skrifter 1 (1910), 1–9; L. Fejes Tóth, Über einen geometrischen Satz, Mathematische Zeitschrift 46 (1940), 83–85.

[19]

T. C. Hales, A proof of the Kepler conjecture, Annals of Mathematics 162 (2005), 1065–1185.

[20]

M. S. Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics 185 (2017), 991–1015.

[21]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics 185 (2017), 1017–1033.

[22]

H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Annals of Mathematics 170 (2009), 1003–1050.

Links

[23]

Wikipedia: Sphere packing

[24]

G. Nebe and N. J. A. Sloane, A Catalogue of Lattices: Table of densest packings presently known

[25]

G. Nebe and N. J. A. Sloane, A Catalogue of Lattices

[26]

MathWorld: Hypersphere Packing

[27]

OEIS A028921: determinant of the laminated lattice $\Lambda_n$ with minimal norm $4$

Similar tables

Packing densities and Hermite numbers of the classical lattices — the same numbers for $n\leq 24$, $n\neq 11,13$, indexed by the lattice, beside every other classical lattice

Volume of the $d$-dimensional unit ball — $V_n$, the factor by which $\Delta_n$ and $\delta_n$ differ

Algebraic numbers of degree 2 — holds $\delta_{25}=1/\sqrt{2}$ and $\delta_{26}=\delta_{27}=\delta_{29}=1/\sqrt{3}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (every dimension $1\leq n\leq 48$ is here, the contiguous range of the catalogue's table; its six entries beyond, for $n=54$, $56$, $64$, $72$, $80$ and $128$, are not listed)

How they were obtained:

$\delta_n^2$ is a rational number for every $n\leq 48$, transcribed from the closed forms of the catalogue's table; $\delta_n$ is written exactly where that rational is a square and otherwise as a ball in arb, and $\Delta_n$ as the product of $\pi^{\lfloor n/2\rfloor}$, the exact rational $V_n/\pi^{\lfloor n/2\rfloor}$ and $\delta_n$, computed with 64 guard bits beyond the 100 digits written; the widest ball relative to its value, $\Delta_{46}$, has radius $6.7\cdot 10^{-119}$.

Before any entry was written the 48 rationals were compared with the same page parsed by a separate program and with the page's own decimals; for $n\leq 26$ and $n=30$, $31$, $32$, $36$ they were recomputed as $(\mu/4)^n/\det L$ from a Gram matrix of the record lattice (the Cartan matrices and the catalogue's GRAM blocks for $\Lambda_9$ to $\Lambda_{26}$, $K_{11}$, $K_{12}$, $K_{13}$, $Q_{30}$, $Q_{31}$, $Q_{32}$ and $KP_{36}$), with the minimal norm and the number of minimal vectors from PARI's qfminim, or from qfrep for $Q_{32}$ and $KP_{36}$, and the determinant exact; for $n=48$ the Gram matrices of $P_{48p}$ and $P_{48q}$ were checked to be even of determinant $1$, and the minimal norm $6$ is taken from the page; $\Delta_2$ to $\Delta_8$, $\Delta_{24}$, $\delta_5$ and $\delta_6$ agree with OEIS A093766, A093825, A222068 to A222072, A260646, A222066 and A222067 to every digit those entries give; the 36 record densities of Table 1 of Cohn's survey are $\Delta_n$, or the density of the nonlattice packing the comment names, rounded down as that table says; $\Delta_n=V_n\delta_n$ holds in balls against the stored digits of the table of unit-ball volumes; the entries for $n\leq 24$ agree with the stored digits of the table of the classical lattices they link; $\delta_{24-n}=\delta_n$; and the closed form and the nonlattice densities in every entry's comment, evaluated in balls, enclose the stored values.

Repeats values from: Packing_densities_and_Hermite_numbers_of_the_classical_lattices