History of Densities of the densest known lattice sphere packings

 Display properties:   number-header: $\Delta_n$ or $\delta_n$-Numbers: []+Numbers:+- params:+    n: '1'+    expression: density+  number: '1'+  comment: $\Delta_{1}=1$, the density of $\Lambda_1=\mathbb{Z}$; the densest packing+    of any kind in dimension 1+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Z,1,density}+- params:+    n: '1'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Z,1,centre}+- params:+    n: '2'+    expression: density+  number: '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 CITE{ThueFejesToth}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,2,density}+- params:+    n: '2'+    expression: centre+  number: '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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,2,centre}+- params:+    n: '3'+    expression: density+  number: '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+    CITE{Hales}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,3,density}+- params:+    n: '3'+    expression: centre+  number: '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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,3,centre}+- params:+    n: '4'+    expression: density+  number: '0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145'+  comment: $\Delta_{4}=\frac{\pi^{2}}{16}$, the density of $\Lambda_4=D_4$; the densest+    lattice packing in dimension 4 CITE{KZ}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,4,density}+- params:+    n: '4'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,4,centre}+- params:+    n: '5'+    expression: density+  number: '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 CITE{KZ}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,5,density}+- params:+    n: '5'+    expression: centre+  number: '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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,5,centre}+- params:+    n: '6'+    expression: density+  number: '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 CITE{Blichfeldt}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,6,density}+- params:+    n: '6'+    expression: centre+  number: '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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,6,centre}+- params:+    n: '7'+    expression: density+  number: '0.2952978731457125730997744292104894781164313196750962637537575057505370944520543432149209622152655828'+  comment: $\Delta_{7}=\frac{\pi^{3}}{105}$, the density of $\Lambda_7=E_7$; the densest+    lattice packing in dimension 7 CITE{Blichfeldt}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,7,density}+- params:+    n: '7'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,7,centre}+- params:+    n: '8'+    expression: density+  number: '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 CITE{Viazovska}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,8,density}+- params:+    n: '8'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,8,centre}+- params:+    n: '9'+    expression: density+  number: '0.1457748758081711105332137544576519995425739925107256670963211484639631773090752900724100031467048256'+  comment: $\Delta_{9}=\frac{\sqrt{2}\,\pi^{4}}{945}$, the density of $\Lambda_{9}$;+    the densest lattice packing in dimension 9 CITE{DSvW}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,9,density}+- params:+    n: '9'+    expression: centre+  number: '0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915'+  comment: $\delta_{9}=\frac{\sqrt{2}}{32}$, the centre density of $\Lambda_{9}$;+    $\det\Lambda_{9}=512$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,9,centre}+- params:+    n: '10'+    expression: density+  number: '0.09202111843130555779386059577030574946623950376510629227349760612731057277995423712397819980248230119'+  comment: $\Delta_{10}=\frac{\sqrt{3}\,\pi^{5}}{5760}$, the density of $\Lambda_{10}$;+    the nonlattice packing $P_{10c}$ CITE{Catalogue-density} is denser, with centre+    density $\frac{5}{128}$ and density $0.09961578\ldots$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,10,density}+- params:+    n: '10'+    expression: centre+  number: '0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533'+  comment: $\delta_{10}=\frac{\sqrt{3}}{48}$, the centre density of $\Lambda_{10}$;+    $\det\Lambda_{10}=768$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,10,centre}+- params:+    n: '11'+    expression: density+  number: '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 $P_{11a}$+    CITE{Catalogue-density} is denser, with centre density $\frac{9}{256}$ and density+    $0.06623802\ldots$+- params:+    n: '11'+    expression: centre+  number: '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 $4$+- params:+    n: '12'+    expression: density+  number: '0.04945417662424405540278906603150308121744946132104149258956153227798837000217850892485706837201212635'+  comment: $\Delta_{12}=\frac{\pi^{6}}{19440}$, the density of $K_{12}$, the Coxeter–Todd+    lattice+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#K,12,density}+- params:+    n: '12'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#K,12,centre}+- params:+    n: '13'+    expression: density+  number: '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 $P_{13a}$+    CITE{Catalogue-density} is denser, with centre density $\frac{9}{256}$ and density+    $0.03201429\ldots$+- params:+    n: '13'+    expression: centre+  number: '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 $4$+- params:+    n: '14'+    expression: density+  number: '0.02162409608244710546249221367868409564177065936712948464424487710130327456120990167726026005937182741'+  comment: $\Delta_{14}=\frac{\sqrt{3}\,\pi^{7}}{241920}$, the density of $\Lambda_{14}$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,14,density}+- params:+    n: '14'+    expression: centre+  number: '0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533'+  comment: $\delta_{14}=\frac{\sqrt{3}}{48}$, the centre density of $\Lambda_{14}$;+    $\det\Lambda_{14}=768$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,14,centre}+- params:+    n: '15'+    expression: density+  number: '0.01685757065676269764781550358540912002543295423644778555552670648966997295106061582291034513965069853'+  comment: $\Delta_{15}=\frac{8\sqrt{2}\,\pi^{7}}{2027025}$, the density of $\Lambda_{15}$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,15,density}+- params:+    n: '15'+    expression: centre+  number: '0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915'+  comment: $\delta_{15}=\frac{\sqrt{2}}{32}$, the centre density of $\Lambda_{15}$;+    $\det\Lambda_{15}=512$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,15,centre}+- params:+    n: '16'+    expression: density+  number: '0.01470816439743082528386745954846658884656680845678491240716470023416869559435667208586846528865779255'+  comment: $\Delta_{16}=\frac{\pi^{8}}{645120}$, the density of $\Lambda_{16}=BW_{16}$,+    the Barnes–Wall lattice+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,16,density}+- params:+    n: '16'+    expression: centre+  number: 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: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,16,centre}+- params:+    n: '17'+    expression: density+  number: '0.008811319182321189869770444983484678153924789606690031715489243517038983815272716868407840771127355841'+  comment: $\Delta_{17}=\frac{32\pi^{8}}{34459425}$, the density of $\Lambda_{17}$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,17,density}+- params:+    n: '17'+    expression: centre+  number: 1/16+  comment: $\delta_{17}=\frac{1}{16}$, the centre density of $\Lambda_{17}$; $\det\Lambda_{17}=256$+    at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,17,centre}+- params:+    n: '18'+    expression: density+  number: '0.005928368718469419730049891486523909806100960042824357061950447334418422452973803955425827012554145075'+  comment: $\Delta_{18}=\frac{\sqrt{3}\,\pi^{9}}{8709120}$, the density of $\Lambda_{18}$;+    the nonlattice packing $B_{18}$ CITE{BierbrauerEdel} is denser, with centre density+    $\frac{3^{9}}{2^{18}}$ and density $0.006167898\ldots$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,18,density}+- params:+    n: '18'+    expression: centre+  number: '0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065'+  comment: $\delta_{18}=\frac{\sqrt{3}}{24}$, the centre density of $\Lambda_{18}$;+    $\det\Lambda_{18}=192$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,18,centre}+- params:+    n: '19'+    expression: density+  number: '0.004120806279768667500886821058357660528632336189215187578663362443680168202332562149057259835876284349'+  comment: $\Delta_{19}=\frac{64\sqrt{2}\,\pi^{9}}{654729075}$, the density of $\Lambda_{19}$;+    the antipode packing of dimension 19 CITE{Antipode2025} is denser, with centre+    density $\frac{13^{19/2}}{3^{9}\cdot 5^{21/2}}$ and density $0.004147369\ldots$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,19,density}+- params:+    n: '19'+    expression: centre+  number: '0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830'+  comment: $\delta_{19}=\frac{\sqrt{2}}{16}$, the centre density of $\Lambda_{19}$;+    $\det\Lambda_{19}=128$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,19,centre}+- params:+    n: '20'+    expression: density+  number: '0.003225861423751757501574786781612356207148305131018335748975296564745861166993469481943592281654563139'+  comment: $\Delta_{20}=\frac{\pi^{10}}{29030400}$, the density of $\Lambda_{20}$;+    the antipode packing of dimension 20 CITE{Antipode2025} is denser, with centre+    density $\frac{3^{20}}{2^{10}\cdot 5^{21/2}}$ and density $0.004024165\ldots$,+    as is $B_{20}$ CITE{Vardy20} with centre density $\frac{7^{10}}{2^{31}}$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,20,density}+- params:+    n: '20'+    expression: centre+  number: 1/8+  comment: $\delta_{20}=\frac{1}{8}$, the centre density of $\Lambda_{20}$; $\det\Lambda_{20}=64$+    at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,20,centre}+- params:+    n: '21'+    expression: density+  number: '0.002465884711502463245944355824165164696567691388297833979591321081883716334406532970306523662963463407'+  comment: $\Delta_{21}=\frac{256\sqrt{2}\,\pi^{10}}{13749310575}$, the density of+    $\Lambda_{21}$; the antipode packing of dimension 21 CITE{Antipode2025} is denser,+    with centre density $\frac{43^{21/2}}{2^{41}\cdot 3^{23/2}}$ and density $0.002929829\ldots$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,21,density}+- params:+    n: '21'+    expression: centre+  number: '0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966'+  comment: $\delta_{21}=\frac{\sqrt{2}}{8}$, the centre density of $\Lambda_{21}$;+    $\det\Lambda_{21}=32$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,21,centre}+- params:+    n: '22'+    expression: density+  number: '0.002127660145275864210626933444611565330655697515338490683084571822978614708729444826368239332278713761'+  comment: $\Delta_{22}=\frac{\sqrt{3}\,\pi^{11}}{239500800}$, the density of $\Lambda_{22}$;+    the nonlattice packing $R_{22}$ CITE{Antipode} is denser, with centre density+    $0.33254$ CITE{Catalogue-density}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,22,density}+- params:+    n: '22'+    expression: centre+  number: '0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626'+  comment: $\delta_{22}=\frac{\sqrt{3}}{6}$, the centre density of $\Lambda_{22}$;+    $\det\Lambda_{22}=12$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,22,centre}+- params:+    n: '23'+    expression: density+  number: '0.001905328193426062470906566906903334103361515823538275509259618028319837706257289936344921562809463883'+  comment: $\Delta_{23}=\frac{2048\pi^{11}}{316234143225}$, the density of $\Lambda_{23}$;+    the antipode packing of dimension 23 CITE{Antipode2025} is denser, with centre+    density $\frac{23^{23/2}}{2^{34}\cdot 3^{12}}$ and density $0.001907194\ldots$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,23,density}+- params:+    n: '23'+    expression: centre+  number: 1/2+  comment: $\delta_{23}=\frac{1}{2}$, the centre density of $\Lambda_{23}$; $\det\Lambda_{23}=4$+    at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,23,centre}+- params:+    n: '24'+    expression: density+  number: '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 CITE{CKMRV}+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,24,density}+- params:+    n: '24'+    expression: centre+  number: '1'+  comment: $\delta_{24}=1$, the centre density of $\Lambda_{24}$, the Leech lattice;+    $\det\Lambda_{24}=1$ at minimal norm $4$+  equals: HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,24,centre}+- params:+    n: '25'+    expression: density+  number: '0.0006772120097731805113471473269154748776650137636745243615753164714289538439340188299756603053478865394'+  comment: $\Delta_{25}=\frac{2^{13}\pi^{12}}{25!!}\cdot \frac{\sqrt{2}}{2}$, the+    density of $\Lambda_{25}$+- params:+    n: '25'+    expression: centre+  number: '0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864'+  comment: $\delta_{25}=\frac{\sqrt{2}}{2}$, the centre density of $\Lambda_{25}$+  equals: HREF{Algebraic_numbers_of_degree_2#2,0,-1,2}+- params:+    n: '26'+    expression: density+  number: '0.0002692200504338088915924745768040714297011094434231232693144368460479860964520299557149709280683257129'+  comment: $\Delta_{26}=\frac{\pi^{13}}{13!}\cdot \frac{\sqrt{3}}{3}$, the density+    of $\Lambda_{26}$ and $T_{26}$+- params:+    n: '26'+    expression: centre+  number: '0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252'+  comment: $\delta_{26}=\frac{\sqrt{3}}{3}$, the centre density of $\Lambda_{26}$+    and $T_{26}$+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-1,2}+- params:+    n: '27'+    expression: density+  number: '0.0001286752812026914537542358296864780893844252168343427048415149520435305115637214720660845547777589254'+  comment: $\Delta_{27}=\frac{2^{14}\pi^{13}}{27!!}\cdot \frac{\sqrt{3}}{3}$, the+    density of Bacher's lattice $B_{27}$ CITE{Bacher}; the nonlattice packing $B_{27}^{*}$+    CITE{VardyDoubling} is denser, with centre density $\frac{\sqrt{2}}{2}$ and density+    $0.0001575943\ldots$+- params:+    n: '27'+    expression: centre+  number: '0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252'+  comment: $\delta_{27}=\frac{\sqrt{3}}{3}$, the centre density of Bacher's lattice+    $B_{27}$ CITE{Bacher}+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-1,2}+- params:+    n: '28'+    expression: density+  number: '0.00006975873661656380474534448556815955174019822099394214212351808051429010470306762143391879066489944667'+  comment: $\Delta_{28}=\frac{\pi^{14}}{14!}\cdot \frac{2}{3}$, the density of Bacher's+    lattice $B_{28}$ CITE{Bacher}; the nonlattice packing $B_{28}^{*}$ CITE{VardyDoubling}+    is denser, with centre density $1$ and density $0.0001046381\ldots$+- params:+    n: '28'+    expression: centre+  number: 2/3+  comment: $\delta_{28}=\frac{2}{3}$, the centre density of Bacher's lattice $B_{28}$+    CITE{Bacher}+- params:+    n: '29'+    expression: density+  number: '0.00002787898745689491625867746513359251450710853680071583775002735955969236331285330192390674633854318115'+  comment: $\Delta_{29}=\frac{2^{15}\pi^{14}}{29!!}\cdot \frac{\sqrt{3}}{3}$, the+    density of Bacher's lattice $B_{29}$ CITE{Bacher}; the nonlattice packing $B_{29}^{*}$+    CITE{VardyDoubling} is denser, with centre density $\frac{\sqrt{2}}{2}$ and density+    $3.414464\ldots\cdot 10^{-5}$+- params:+    n: '29'+    expression: centre+  number: '0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252'+  comment: $\delta_{29}=\frac{\sqrt{3}}{3}$, the centre density of Bacher's lattice+    $B_{29}$ CITE{Bacher}+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-1,2}+- params:+    n: '30'+    expression: density+  number: '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 $Q_{32}$ CITE{Quebbemann}; the+    nonlattice packing $T_{30}$ CITE{VardyDoubling} is denser, with centre density+    $1$ and density $2.191535\ldots\cdot 10^{-5}$+- params:+    n: '30'+    expression: centre+  number: '0.6583806132496917613145253261396090864375481845547503171119256856204853542607042316244421558164064457'+  comment: $\delta_{30}=\frac{3^{27/2}}{2^{22}}$, the centre density of $Q_{30}$,+    a section of Quebbemann's lattice $Q_{32}$ CITE{Quebbemann}+- params:+    n: '31'+    expression: density+  number: '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 $Q_{32}$ CITE{Quebbemann}+- params:+    n: '31'+    expression: centre+  number: '1.209522419251813897932523068585014714448148891943658461918912656856381129940287883550500729324021128'+  comment: $\delta_{31}=\frac{3^{15}}{2^{47/2}}$, the centre density of $Q_{31}$,+    a section of Quebbemann's lattice $Q_{32}$ CITE{Quebbemann}+- params:+    n: '32'+    expression: density+  number: '0.00001104074930885985419934596904309777447951275366061902008487075156881710592894952916757719442476445007'+  comment: $\Delta_{32}=\frac{\pi^{16}}{16!}\cdot \frac{3^{16}}{2^{24}}$, the density+    of Quebbemann's lattice $Q_{32}$ CITE{Quebbemann}, and others+- params:+    n: '32'+    expression: centre+  number: 43046721/16777216+  comment: $\delta_{32}=\frac{3^{16}}{2^{24}}$, the centre density of Quebbemann's+    lattice $Q_{32}$ CITE{Quebbemann}, and others+- params:+    n: '33'+    expression: density+  number: '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 CITE{Catalogue-density}+- params:+    n: '33'+    expression: centre+  number: '2.222034569717709694436522975721180666726725122872282320252749188969138070629876781732492275880371754'+  comment: $\delta_{33}=\frac{3^{33/2}}{2^{25}}$, the centre density of $Q_{33}$,+    due to Elkies CITE{Catalogue-density}+- params:+    n: '34'+    expression: density+  number: '0.000001766973889154063266233146292413824721789458545713814925373042404669197424829268689185553018927793592'+  comment: $\Delta_{34}=\frac{\pi^{17}}{17!}\cdot \frac{3^{33/2}}{2^{25}}$, the density+    of $Q_{34}$, due to Elkies CITE{Catalogue-density}+- params:+    n: '34'+    expression: centre+  number: '2.222034569717709694436522975721180666726725122872282320252749188969138070629876781732492275880371754'+  comment: $\delta_{34}=\frac{3^{33/2}}{2^{25}}$, the centre density of $Q_{34}$,+    due to Elkies CITE{Catalogue-density}+- params:+    n: '35'+    expression: density+  number: '9.461904151153729608089621070158634848599355754381467737703194297509752400982400821965740121725247960e-7'+  comment: $\Delta_{35}=\frac{2^{18}\pi^{17}}{35!!}\cdot 2\sqrt{2}$, the density of+    $B_{35}$ CITE{Catalogue-density}+- params:+    n: '35'+    expression: centre+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $\delta_{35}=2\sqrt{2}$, the centre density of $B_{35}$ CITE{Catalogue-density}+- params:+    n: '36'+    expression: density+  number: '6.161466094691810322232590442931743951888902608759339696407596696344044477025154721431238893638150756e-7'+  comment: $\Delta_{36}=\frac{\pi^{18}}{18!}\cdot \frac{2^{18}}{3^{10}}$, the density+    of the Kschischang–Pasupathy lattice $KP_{36}$ CITE{KP}+- params:+    n: '36'+    expression: centre+  number: 262144/59049+  comment: $\delta_{36}=\frac{2^{18}}{3^{10}}$, the centre density of the Kschischang–Pasupathy+    lattice $KP_{36}$ CITE{KP}+- params:+    n: '37'+    expression: density+  number: '3.213562007593008162341082150232664617214040385929130458362157889703619240459966874214828145318934216e-7'+  comment: $\Delta_{37}=\frac{2^{19}\pi^{18}}{37!!}\cdot 4\sqrt{2}$, the density of+    the lattice the catalogue calls $D_{37}$ CITE{Catalogue-density}, which is not+    the root lattice+- params:+    n: '37'+    expression: centre+  number: '5.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291'+  comment: $\delta_{37}=4\sqrt{2}$, the centre density of the lattice the catalogue+    calls $D_{37}$ CITE{Catalogue-density}, which is not the root lattice+- params:+    n: '38'+    expression: density+  number: '1.835874319781589848816309790845727028485982846487039096526555894380331837526438410356589312440255285e-7'+  comment: $\Delta_{38}=\frac{\pi^{19}}{19!}\cdot 8$, the density of the lattice the+    catalogue calls $D_{38}$ CITE{Catalogue-density}, which is not the root lattice+- params:+    n: '38'+    expression: centre+  number: '8'+  comment: $\delta_{38}=8$, the centre density of the lattice the catalogue calls+    $D_{38}$ CITE{Catalogue-density}, which is not the root lattice+- params:+    n: '39'+    expression: density+  number: '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 $P_{48p}$+- params:+    n: '39'+    expression: centre+  number: '10.97175609084857837783112211323698462591417821720153867127005116508621291322273490778575148095210608'+  comment: $\delta_{39}=\frac{3^{16}}{2^{41/2}\,\sqrt{7}}$, the centre density of+    a section of $P_{48p}$+- params:+    n: '40'+    expression: density+  number: '7.848004886817056107952147964588977935929629771229549931566154754727770719399413185430635961271491419e-8'+  comment: $\Delta_{40}=\frac{\pi^{20}}{20!}\cdot \frac{3^{17}}{2^{45/2}}$, the density+    of a section of $P_{48p}$+- params:+    n: '40'+    expression: centre+  number: '21.77140354653265016278541523453026486006668005498585231454042782341486033892518190390901312783238031'+  comment: $\delta_{40}=\frac{3^{17}}{2^{45/2}}$, the centre density of a section+    of $P_{48p}$+- params:+    n: '41'+    expression: density+  number: '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 $P_{48p}$+- params:+    n: '41'+    expression: centre+  number: '43.54280709306530032557083046906052972013336010997170462908085564682972067785036380781802625566476062'+  comment: $\delta_{41}=\frac{3^{17}}{2^{43/2}}$, the centre density of a section+    of $P_{48p}$+- params:+    n: '42'+    expression: density+  number: '4.981109572888996969985256478761774860906747743037860734407016630223122056146991080277434290058270008e-8'+  comment: $\Delta_{42}=\frac{\pi^{21}}{21!}\cdot \frac{3^{18}}{2^{22}}$, the density+    of a section of $P_{48p}$+- params:+    n: '42'+    expression: centre+  number: 387420489/4194304+  comment: $\delta_{42}=\frac{3^{18}}{2^{22}}$, the centre density of a section of+    $P_{48p}$+- params:+    n: '43'+    expression: density+  number: '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 $P_{48p}$+- params:+    n: '43'+    expression: centre+  number: '195.9426319187938514650687371107723837406001204948726708308638504107337430503266371351811181504914228'+  comment: $\delta_{43}=\frac{3^{19}}{2^{45/2}}$, the centre density of a section+    of $P_{48p}$+- params:+    n: '44'+    expression: density+  number: '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 CITE{Antipode2025}+    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 $T_{44}$ CITE{Antipode} with+    centre density $\frac{17^{22}}{2^{43}\cdot 3^{24}}$+- params:+    n: '44'+    expression: centre+  number: 3486784401/8388608+  comment: $\delta_{44}=\frac{3^{20}}{2^{23}}$, the centre density of a section of+    $P_{48p}$+- params:+    n: '45'+    expression: density+  number: '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 CITE{Antipode2025}+    is denser, with centre density $\frac{23^{45/2}}{2^{183/2}}$ and density $3.558225\ldots\cdot+    10^{-8}$, as is $T_{45}$ CITE{Antipode} with centre density $\frac{17^{45/2}}{2^{44}\cdot+    3^{24}}$+- params:+    n: '45'+    expression: centre+  number: '881.7418436345723315928093169984757268327005422269270187388873268483018437264698671083150316772114025'+  comment: $\delta_{45}=\frac{3^{21}}{2^{47/2}}$, the centre density of a section+    of $P_{48p}$+- params:+    n: '46'+    expression: density+  number: '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 $T_{46}$ CITE{Antipode} is denser,+    with centre density $\frac{13^{23}}{3^{93/2}}$ and density $2.860957\ldots\cdot+    10^{-8}$+- params:+    n: '46'+    expression: centre+  number: '2159.817601765613822992300332400987608058376819431858415285672211678002204652240231843982492155721345'+  comment: $\delta_{46}=\frac{3^{43/2}}{2^{23}}$, the centre density of a section+    of $P_{48p}$+- params:+    n: '47'+    expression: density+  number: '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 CITE{Antipode2025}+    is denser, with centre density $\frac{47^{47/2}}{2^{118}}$ and density $2.266960\ldots\cdot+    10^{-8}$, as is $T_{47}$ CITE{Antipode} with centre density $\frac{5^{47/2}\cdot+    7^{47/2}}{2^{70}\cdot 3^{24}}$+- params:+    n: '47'+    expression: centre+  number: 94143178827/16777216+  comment: $\delta_{47}=\frac{3^{23}}{2^{24}}$, the centre density of a section of+    $P_{48p}$+- params:+    n: '48'+    expression: density+  number: '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+    $6$+- params:+    n: '48'+    expression: centre+  number: 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 $6$