Covering radii and covering densities of the classical lattices

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Numbers

family

$n$

$R/\rho$ or $\Theta(L)$

$\mathbb{Z}^n$

$R/\rho$:

comment: $R/\rho=1$; $R^2=\frac{1}{4}$ with $\mu=1$
equals: One

$\mathbb{Z}^n$

$\Theta(L)$:

comment: $\Theta(\mathbb{Z})=1$; the thinnest covering of $\mathbb{R}$ by equal intervals
equals: One

$\mathbb{Z}^n$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=\frac{1}{2}$ with $\mu=1$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$\mathbb{Z}^n$

$\Theta(L)$:

1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534

comment: $\Theta(\mathbb{Z}^{2})=\frac{\pi}{2}$
equals: Rational_multiples_of_pi#1/2

$\mathbb{Z}^n$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{4}$ with $\mu=1$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$\mathbb{Z}^n$

$\Theta(L)$:

2.720699046351326775891117386463233598426099372139110863354827403082184771689530825526187482318090253

comment: $\Theta(\mathbb{Z}^{3})=\frac{\sqrt{3}\,\pi}{2}$

$\mathbb{Z}^n$

$R/\rho$:

comment: $R/\rho=2$; $R^2=1$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116

comment: $\Theta(\mathbb{Z}^{4})=\frac{\pi^{2}}{2}$

$\mathbb{Z}^n$

$R/\rho$:

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{4}$ with $\mu=1$
equals: Algebraic_numbers_of_degree_2#1,0,-5,2

$\mathbb{Z}^n$

$\Theta(L)$:

9.195460979944543879309937184408385776826125935585245624246320340698927141124778745877273000137677141

comment: $\Theta(\mathbb{Z}^{5})=\frac{5\sqrt{5}\,\pi^{2}}{12}$

$\mathbb{Z}^n$

$R/\rho$:

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $R/\rho=\sqrt{6}$; $R^2=\frac{3}{2}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

17.44103063266864884870542722524453480125172481831037307795630268339109714107445964613126933083912348

comment: $\Theta(\mathbb{Z}^{6})=\frac{9\pi^{3}}{16}$

$\mathbb{Z}^n$

$R/\rho$:

2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611

comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

33.49758301440499875246683298329190753168095765962112073162977089625749335692405662733881463813925684

comment: $\Theta(\mathbb{Z}^{7})=\frac{49\sqrt{7}\,\pi^{3}}{120}$

$\mathbb{Z}^n$

$R/\rho$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $R/\rho=2\sqrt{2}$; $R^2=2$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

64.93939402266829149096022179247007416648505711512361446097857292664723697121813079341457815650199503

comment: $\Theta(\mathbb{Z}^{8})=\frac{2\pi^{4}}{3}$

$\mathbb{Z}^n$

$R/\rho$:

comment: $R/\rho=3$; $R^2=\frac{9}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

126.8057631496210299024375045179750466090203749203351293090715526701584886214768321832122164538123778

comment: $\Theta(\mathbb{Z}^{9})=\frac{729\pi^{4}}{560}$

$\mathbb{Z}^n$

$R/\rho$:

3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295

comment: $R/\rho=\sqrt{10}$; $R^2=\frac{5}{2}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

249.0394570192720160015798421577438203778488823470662358019958037811157162545687048224149580676311189

comment: $\Theta(\mathbb{Z}^{10})=\frac{625\pi^{5}}{768}$

$\mathbb{Z}^n$

$R/\rho$:

3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651

comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

491.3994429783274077305032733510548573679085060649560551376626397784182814447335498269596449321539448

comment: $\Theta(\mathbb{Z}^{11})=\frac{14641\sqrt{11}\,\pi^{5}}{30240}$

$\mathbb{Z}^n$

$R/\rho$:

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $R/\rho=2\sqrt{3}$; $R^2=3$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

973.4065584949957424930971866980751476030577471820596986403396398276450867528795911679616767663146829

comment: $\Theta(\mathbb{Z}^{12})=\frac{81\pi^{6}}{80}$

$\mathbb{Z}^n$

$R/\rho$:

3.605551275463989293119221267470495946251296573845246212710453056227166948293010445204619082018490718

comment: $R/\rho=\sqrt{13}$; $R^2=\frac{13}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

1934.564465080156808372399720669519650526715713110823667474564565810867384341276384000127792539696079

comment: $\Theta(\mathbb{Z}^{13})=\frac{371293\sqrt{13}\,\pi^{6}}{665280}$

$\mathbb{Z}^n$

$R/\rho$:

3.741657386773941385583748732316549301756019807778726946303745467320035156306939027976809895194379572

comment: $R/\rho=\sqrt{14}$; $R^2=\frac{7}{2}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

3855.625845862758354502809024559816075122113859440416458450227146206417214327540626958854303970448735

comment: $\Theta(\mathbb{Z}^{14})=\frac{117649\pi^{7}}{92160}$

$\mathbb{Z}^n$

$R/\rho$:

3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673518

comment: $R/\rho=\sqrt{15}$; $R^2=\frac{15}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

7703.081221506743320238474181423334429672389139837316704348374248648010622365430603166596024704184369

comment: $\Theta(\mathbb{Z}^{15})=\frac{84375\sqrt{15}\,\pi^{7}}{128128}$

$\mathbb{Z}^n$

$R/\rho$:

comment: $R/\rho=4$; $R^2=4$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

15422.62819120042505285660526349290186637763774438169631225513271274367415154814178911161185851963348

comment: $\Theta(\mathbb{Z}^{16})=\frac{512\pi^{8}}{315}$

$\mathbb{Z}^n$

$R/\rho$:

4.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297

comment: $R/\rho=\sqrt{17}$; $R^2=\frac{17}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

30936.18965664557911254280529800617630643392676764917432691134470204987726982785398091695548836627243

comment: $\Theta(\mathbb{Z}^{17})=\frac{410338673\sqrt{17}\,\pi^{8}}{518918400}$

$\mathbb{Z}^n$

$R/\rho$:

4.242640687119285146405066172629094235709015626130844219530039213972197435386321116551162602982924718

comment: $R/\rho=3\sqrt{2}$; $R^2=\frac{9}{2}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

62158.20226605830320885917004989834733191992206012469738132381874101551170378363722030136547961342968

comment: $\Theta(\mathbb{Z}^{18})=\frac{4782969\pi^{9}}{2293760}$

$\mathbb{Z}^n$

$R/\rho$:

4.358898943540673552236981983859615659137003925232444936890344138159557328203158085656159155851944527

comment: $R/\rho=\sqrt{19}$; $R^2=\frac{19}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

125076.7174480546452783241250492294740758993177905588649085842250941399772275662045993518655132078002

comment: $\Theta(\mathbb{Z}^{19})=\frac{16983563041\sqrt{19}\,\pi^{9}}{17643225600}$

$\mathbb{Z}^n$

$R/\rho$:

4.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

comment: $R/\rho=2\sqrt{5}$; $R^2=5$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

252020.4237306060548105302173134653286834613383608074803886950441207704036713648032768431470042627452

comment: $\Theta(\mathbb{Z}^{20})=\frac{390625\pi^{10}}{145152}$

$\mathbb{Z}^n$

$R/\rho$:

4.582575694955840006588047193728008488984456576767971902607242123906868425547770886604361559493445033

comment: $R/\rho=\sqrt{21}$; $R^2=\frac{21}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

508417.4439562997111121903708341768348796453237360756387586965398546909507526642147414598426410216698

comment: $\Theta(\mathbb{Z}^{21})=\frac{1400846643\sqrt{21}\,\pi^{10}}{1182438400}$

$\mathbb{Z}^n$

$R/\rho$:

4.690415759823429554565630113544466280588228353411737153605701891017024632753239721482115596061543135

comment: $R/\rho=\sqrt{22}$; $R^2=\frac{11}{2}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

1026791.975705456089383328761377033940244368664939119091653073681301466843464298285412317544519609581

comment: $\Theta(\mathbb{Z}^{22})=\frac{25937424601\pi^{11}}{7431782400}$

$\mathbb{Z}^n$

$R/\rho$:

4.795831523312719541597438064162693919996707041904129346485309114448257235907464082492191446436918861

comment: $R/\rho=\sqrt{23}$; $R^2=\frac{23}{4}$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

2075773.691736594681880477768612226317390481007737044003588831670484793890888867034679469930329289673

comment: $\Theta(\mathbb{Z}^{23})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{28158588057600}$

$\mathbb{Z}^n$

$R/\rho$:

4.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470

comment: $R/\rho=2\sqrt{6}$; $R^2=6$ with $\mu=1$

$\mathbb{Z}^n$

$\Theta(L)$:

4200263.272709858379779284458335555642362973511006520171785967060591687153098661895533230276901857311

comment: $\Theta(\mathbb{Z}^{24})=\frac{8748\pi^{12}}{1925}$

$A_n$

$R/\rho$:

comment: $R/\rho=1$; $R^2=\frac{1}{2}$ with $\mu=2$; $A_1=\sqrt{2}\,\mathbb{Z}$
equals: One

$A_n$

$\Theta(L)$:

comment: $\Theta(A_{1})=1$; the thinnest covering of $\mathbb{R}$ by equal intervals; $A_1=\sqrt{2}\,\mathbb{Z}$
equals: One

$A_n$

$R/\rho$:

1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050

comment: $R/\rho=\frac{2\sqrt{3}}{3}$; $R^2=\frac{2}{3}$ with $\mu=2$; $A_2$ is the hexagonal lattice, $A_2^{*}\cong A_2$
equals: Algebraic_numbers_of_degree_2#3,0,-4,2

$A_n$

$\Theta(L)$:

1.209199576156145233729385505094770488189377498728493717046589956925415454084235922456083325474706779

comment: $\Theta(A_{2})=\frac{2\sqrt{3}\,\pi}{9}$; the thinnest covering of the plane by equal discs [4]; $A_2$ is the hexagonal lattice, $A_2^{*}\cong A_2$

$A_n$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $A_3=D_3$, the face-centred cubic lattice
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$A_n$

$\Theta(L)$:

2.094395102393195492308428922186335256131446266250070547316629728205210937524139332418689883561411379

comment: $\Theta(A_{3})=\frac{2\pi}{3}$; $A_3=D_3$, the face-centred cubic lattice
equals: Rational_multiples_of_pi#2/3

$A_n$

$R/\rho$:

1.549193338482966754071706159912959844333168682116636330635029506445393236774791613407714950743469407

comment: $R/\rho=\frac{2\sqrt{15}}{5}$; $R^2=\frac{6}{5}$ with $\mu=2$

$A_n$

$\Theta(L)$:

3.177951314668834364689514290931538124471109123338260887739528309745549219972723534575185548847581220

comment: $\Theta(A_{4})=\frac{18\sqrt{5}\,\pi^{2}}{125}$

$A_n$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$A_n$

$\Theta(L)$:

5.921762640653615171300694599925690681188219644344474375848009625732026893451523145801064042231131339

comment: $\Theta(A_{5})=\frac{3\pi^{2}}{5}$

$A_n$

$R/\rho$:

1.851640199545102923133133553167999045058629802016670442774672685088462107955441248971531687601602924

comment: $R/\rho=\frac{2\sqrt{42}}{7}$; $R^2=\frac{12}{7}$ with $\mu=2$

$A_n$

$\Theta(L)$:

9.840087624865802147789218335069302113021733263491452817152078489189529621291259569064160629449402174

comment: $\Theta(A_{6})=\frac{288\sqrt{7}\,\pi^{3}}{2401}$

$A_n$

$R/\rho$:

comment: $R/\rho=2$; $R^2=2$ with $\mu=2$

$A_n$

$\Theta(L)$:

18.89906388132560467838556346947132659945160445920616088024048036803437404493147796575494158177699730

comment: $\Theta(A_{7})=\frac{64\pi^{3}}{105}$

$A_n$

$R/\rho$:

2.108185106778919554665929029621812355813036759550144551238336568528396292426158814229498738919533530

comment: $R/\rho=\frac{2\sqrt{10}}{3}$; $R^2=\frac{20}{9}$ with $\mu=2$

$A_n$

$\Theta(L)$:

32.99263020000421251382422486027032168190065392222913908498631962945040744358996636356987154219478486

comment: $\Theta(A_{8})=\frac{20000\pi^{4}}{59049}$

$A_n$

$R/\rho$:

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{2}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#1,0,-5,2

$A_n$

$\Theta(L)$:

64.42400200661536854261926765125999421278279475706707783827239377643575096351005039029224023462499507

comment: $\Theta(A_{9})=\frac{125\pi^{4}}{189}$

$A_n$

$R/\rho$:

2.335496832484568912748526995934534621721336820283684340052085678956094491777786074731242971705112228

comment: $R/\rho=\frac{2\sqrt{165}}{11}$; $R^2=\frac{30}{11}$ with $\mu=2$

$A_n$

$\Theta(L)$:

116.0151277666199205534033600290105967016136957189555050705394796936231945156708878183957860599676542

comment: $\Theta(A_{10})=\frac{202500\sqrt{11}\,\pi^{5}}{1771561}$

$A_n$

$R/\rho$:

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $R/\rho=\sqrt{6}$; $R^2=3$ with $\mu=2$

$A_n$

$\Theta(L)$:

228.9186213458728793238168760844401419904587104126898630633917500821705915181320494854692159386220110

comment: $\Theta(A_{11})=\frac{288\pi^{5}}{385}$

$A_n$

$R/\rho$:

2.541955637208970163356527125017880063513596897814620095448453248091796466218290498699055645452203161

comment: $R/\rho=\frac{2\sqrt{273}}{13}$; $R^2=\frac{42}{13}$ with $\mu=2$

$A_n$

$\Theta(L)$:

421.1440721554849571039071732338507912431303908887815319863499991010193924434978791470135655291125746

comment: $\Theta(A_{12})=\frac{38118276\sqrt{13}\,\pi^{6}}{313742585}$

$A_n$

$R/\rho$:

2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611

comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{2}$ with $\mu=2$

$A_n$

$\Theta(L)$:

836.9887685273318660019113281256783859239225586354908300503324259541655733302968162797935365768582117

comment: $\Theta(A_{13})=\frac{16807\pi^{6}}{19305}$

$A_n$

$R/\rho$:

2.732520204255892902192276981472280530862675376921288781211194266637426129521427752780595100900945797

comment: $R/\rho=\frac{4\sqrt{105}}{15}$; $R^2=\frac{56}{15}$ with $\mu=2$

$A_n$

$\Theta(L)$:

1564.048868967261339918402173297274792210321487414513307370756756514193122623198997939704705761844104

comment: $\Theta(A_{14})=\frac{15420489728\sqrt{15}\,\pi^{7}}{115330078125}$

$A_n$

$R/\rho$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $R/\rho=2\sqrt{2}$; $R^2=4$ with $\mu=2$

$A_n$

$\Theta(L)$:

3124.783356504510323242956142799925344724069683615497315592835990759993405993578146956173601913375561

comment: $\Theta(A_{15})=\frac{2097152\pi^{7}}{2027025}$

$A_n$

$R/\rho$:

2.910427500435995682226877545393466135398022982616673247810800169243497185650085830767903753513424209

comment: $R/\rho=\frac{12\sqrt{17}}{17}$; $R^2=\frac{72}{17}$ with $\mu=2$

$A_n$

$\Theta(L)$:

5909.117119637672521178938156835114526317382294948169260067935599042092472828903412118638899475368048

comment: $\Theta(A_{16})=\frac{626913312768\sqrt{17}\,\pi^{8}}{4150575677395}$

$A_n$

$R/\rho$:

comment: $R/\rho=3$; $R^2=\frac{9}{2}$ with $\mu=2$

$A_n$

$\Theta(L)$:

11853.07495260401227225108372655982963333735853696517277305677638059487132561125568582331362147325445

comment: $\Theta(A_{17})=\frac{531441\pi^{8}}{425425}$

$A_n$

$R/\rho$:

3.077935056255462286370025221020926410290289652738155062734236843911604770250605060057488030327810167

comment: $R/\rho=\frac{6\sqrt{95}}{19}$; $R^2=\frac{90}{19}$ with $\mu=2$

$A_n$

$\Theta(L)$:

22626.07369227618881414884369100518160995594098808618999002915841013185819264619700550038521647485653

comment: $\Theta(A_{18})=\frac{7473389062500\sqrt{19}\,\pi^{9}}{42917463804607}$

$A_n$

$R/\rho$:

3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295

comment: $R/\rho=\sqrt{10}$; $R^2=5$ with $\mu=2$

$A_n$

$\Theta(L)$:

45528.90725594584548808895099280516198836603945576825546760872965333731487930813552082556576637603755

comment: $\Theta(A_{19})=\frac{40000000\pi^{9}}{26189163}$

$A_n$

$R/\rho$:

3.236694374850748275461428822564184587358224216591466526958304990085313815593854740342786968592845840

comment: $R/\rho=\frac{2\sqrt{1155}}{21}$; $R^2=\frac{110}{21}$ with $\mu=2$

$A_n$

$\Theta(L)$:

87570.92778545918962448858509365594528786886967788841927199463253952507186427440208213583374808965141

comment: $\Theta(A_{20})=\frac{40527225939062500\sqrt{21}\,\pi^{10}}{198607342807439307}$

$A_n$

$R/\rho$:

3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651

comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{2}$ with $\mu=2$

$A_n$

$\Theta(L)$:

176662.6157138625626923172658354633373826425908419485943852861451933889245441946127689478956095974321

comment: $\Theta(A_{21})=\frac{2357947691\pi^{10}}{1249937325}$

$A_n$

$R/\rho$:

3.387958215439679443642581910197069263383989530041460190338454563911336627786030601895796540773809870

comment: $R/\rho=\frac{2\sqrt{1518}}{23}$; $R^2=\frac{132}{23}$ with $\mu=2$

$A_n$

$\Theta(L)$:

341929.0586685800579507269337962683253952744794417548676661526392367711712508086234450897711072557967

comment: $\Theta(A_{22})=\frac{929384818317508608\sqrt{23}\,\pi^{11}}{3835059275603556175}$

$A_n$

$R/\rho$:

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $R/\rho=2\sqrt{3}$; $R^2=6$ with $\mu=2$

$A_n$

$\Theta(L)$:

691247.4592887740181169881248915556892839576444771018580762901603589617462279375674443649435381925856

comment: $\Theta(A_{23})=\frac{3057647616\pi^{11}}{1301375075}$

$A_n$

$R/\rho$:

3.532704346531138741905617090783701585669855792372567130484081189022855975294177136148174215765908607

comment: $R/\rho=\frac{2\sqrt{78}}{5}$; $R^2=\frac{156}{25}$ with $\mu=2$

$A_n$

$\Theta(L)$:

1344951.365215002441106465210126379275553804007369829343396956807252656785760648054718974570468529901

comment: $\Theta(A_{24})=\frac{834812512876395675648\pi^{12}}{573694705963134765625}$

$D_n$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $D_4^{*}\cong D_4$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$D_n$

$\Theta(L)$:

2.467401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058

comment: $\Theta(D_{4})=\frac{\pi^{2}}{4}$; $D_4^{*}\cong D_4$

$D_n$

$R/\rho$:

1.581138830084189665999446772216359266859777569662608413428752426396297219319619110672124054189650148

comment: $R/\rho=\frac{\sqrt{10}}{2}$; $R^2=\frac{5}{4}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#2,0,-5,2

$D_n$

$\Theta(L)$:

4.597730489972271939654968592204192888413062967792622812123160170349463570562389372938636500068838570

comment: $\Theta(D_{5})=\frac{5\sqrt{5}\,\pi^{2}}{24}$

$D_n$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$D_n$

$\Theta(L)$:

8.720515316334324424352713612622267400625862409155186538978151341695548570537229823065634665419561742

comment: $\Theta(D_{6})=\frac{9\pi^{3}}{32}$

$D_n$

$R/\rho$:

1.870828693386970692791874366158274650878009903889363473151872733660017578153469513988404947597189786

comment: $R/\rho=\frac{\sqrt{14}}{2}$; $R^2=\frac{7}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

16.74879150720249937623341649164595376584047882981056036581488544812874667846202831366940731906962842

comment: $\Theta(D_{7})=\frac{49\sqrt{7}\,\pi^{3}}{240}$

$D_n$

$R/\rho$:

comment: $R/\rho=2$; $R^2=2$ with $\mu=2$

$D_n$

$\Theta(L)$:

32.46969701133414574548011089623503708324252855756180723048928646332361848560906539670728907825099752

comment: $\Theta(D_{8})=\frac{\pi^{4}}{3}$

$D_n$

$R/\rho$:

2.121320343559642573202533086314547117854507813065422109765019606986098717693160558275581301491462359

comment: $R/\rho=\frac{3\sqrt{2}}{2}$; $R^2=\frac{9}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

63.40288157481051495121875225898752330451018746016756465453577633507924431073841609160610822690618890

comment: $\Theta(D_{9})=\frac{729\pi^{4}}{1120}$

$D_n$

$R/\rho$:

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{2}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#1,0,-5,2

$D_n$

$\Theta(L)$:

124.5197285096360080007899210788719101889244411735331179009979018905578581272843524112074790338155594

comment: $\Theta(D_{10})=\frac{625\pi^{5}}{1536}$

$D_n$

$R/\rho$:

2.345207879911714777282815056772233140294114176705868576802850945508512316376619860741057798030771568

comment: $R/\rho=\frac{\sqrt{22}}{2}$; $R^2=\frac{11}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

245.6997214891637038652516366755274286839542530324780275688313198892091407223667749134798224660769724

comment: $\Theta(D_{11})=\frac{14641\sqrt{11}\,\pi^{5}}{60480}$

$D_n$

$R/\rho$:

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $R/\rho=\sqrt{6}$; $R^2=3$ with $\mu=2$

$D_n$

$\Theta(L)$:

486.7032792474978712465485933490375738015288735910298493201698199138225433764397955839808383831573414

comment: $\Theta(D_{12})=\frac{81\pi^{6}}{160}$

$D_n$

$R/\rho$:

2.549509756796392415014112054511390994781885473049798203792485402212966816031112097794174425546966004

comment: $R/\rho=\frac{\sqrt{26}}{2}$; $R^2=\frac{13}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

967.2822325400784041861998603347598252633578565554118337372822829054336921706381920000638962698480395

comment: $\Theta(D_{13})=\frac{371293\sqrt{13}\,\pi^{6}}{1330560}$

$D_n$

$R/\rho$:

2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611

comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{2}$ with $\mu=2$

$D_n$

$\Theta(L)$:

1927.812922931379177251404512279908037561056929720208229225113573103208607163770313479427151985224367

comment: $\Theta(D_{14})=\frac{117649\pi^{7}}{184320}$

$D_n$

$R/\rho$:

2.738612787525830567284848914004010669763723474989916271134472248662466385613613669004292180819353129

comment: $R/\rho=\frac{\sqrt{30}}{2}$; $R^2=\frac{15}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

3851.540610753371660119237090711667214836194569918658352174187124324005311182715301583298012352092184

comment: $\Theta(D_{15})=\frac{84375\sqrt{15}\,\pi^{7}}{256256}$

$D_n$

$R/\rho$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $R/\rho=2\sqrt{2}$; $R^2=4$ with $\mu=2$

$D_n$

$\Theta(L)$:

7711.314095600212526428302631746450933188818872190848156127566356371837075774070894555805929259816738

comment: $\Theta(D_{16})=\frac{256\pi^{8}}{315}$

$D_n$

$R/\rho$:

2.915475947422650235437076438772791538260699167442985977225003372433905030998356313832620163226517699

comment: $R/\rho=\frac{\sqrt{34}}{2}$; $R^2=\frac{17}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

15468.09482832278955627140264900308815321696338382458716345567235102493863491392699045847774418313621

comment: $\Theta(D_{17})=\frac{410338673\sqrt{17}\,\pi^{8}}{1037836800}$

$D_n$

$R/\rho$:

comment: $R/\rho=3$; $R^2=\frac{9}{2}$ with $\mu=2$

$D_n$

$\Theta(L)$:

31079.10113302915160442958502494917366595996103006234869066190937050775585189181861015068273980671484

comment: $\Theta(D_{18})=\frac{4782969\pi^{9}}{4587520}$

$D_n$

$R/\rho$:

3.082207001484488225125096190727122112617812011722287272437286036229199825131832106483364899599449973

comment: $R/\rho=\frac{\sqrt{38}}{2}$; $R^2=\frac{19}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

62538.35872402732263916206252461473703794965889527943245429211254706998861378310229967593275660390010

comment: $\Theta(D_{19})=\frac{16983563041\sqrt{19}\,\pi^{9}}{35286451200}$

$D_n$

$R/\rho$:

3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295

comment: $R/\rho=\sqrt{10}$; $R^2=5$ with $\mu=2$

$D_n$

$\Theta(L)$:

126010.2118653030274052651086567326643417306691804037401943475220603852018356824016384215735021313726

comment: $\Theta(D_{20})=\frac{390625\pi^{10}}{290304}$

$D_n$

$R/\rho$:

3.240370349203930115482983718043998328852602153529173274855677198904808688922022185700180453302805118

comment: $R/\rho=\frac{\sqrt{42}}{2}$; $R^2=\frac{21}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

254208.7219781498555560951854170884174398226618680378193793482699273454753763321073707299213205108349

comment: $\Theta(D_{21})=\frac{1400846643\sqrt{21}\,\pi^{10}}{2364876800}$

$D_n$

$R/\rho$:

3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651

comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{2}$ with $\mu=2$

$D_n$

$\Theta(L)$:

513395.9878527280446916643806885169701221843324695595458265368406507334217321491427061587722598047905

comment: $\Theta(D_{22})=\frac{25937424601\pi^{11}}{14863564800}$

$D_n$

$R/\rho$:

3.391164991562634069532278163312984552597874161961644116375109791040364131993556294931320470468165022

comment: $R/\rho=\frac{\sqrt{46}}{2}$; $R^2=\frac{23}{4}$ with $\mu=2$

$D_n$

$\Theta(L)$:

1037886.845868297340940238884306113158695240503868522001794415835242396945444433517339734965164644836

comment: $\Theta(D_{23})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{56317176115200}$

$D_n$

$R/\rho$:

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $R/\rho=2\sqrt{3}$; $R^2=6$ with $\mu=2$

$D_n$

$\Theta(L)$:

2100131.636354929189889642229167777821181486755503260085892983530295843576549330947766615138450928655

comment: $\Theta(D_{24})=\frac{4374\pi^{12}}{1925}$

$E_n$

$R/\rho$:

1.632993161855452065464856049803927594643964987104446752288461711500640251638210017693239622069760157

comment: $R/\rho=\frac{2\sqrt{6}}{3}$; $R^2=\frac{4}{3}$ with $\mu=2$

$E_n$

$\Theta(L)$:

7.072190494000638853944463334968094461983569488941271478260699317992925336266585496130877091751642885

comment: $\Theta(E_{6})=\frac{32\sqrt{3}\,\pi^{3}}{243}$

$E_n$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$E_n$

$\Theta(L)$:

13.80971483069589033355048331569216302531970220742371493121084768611654259858484150003413232112575267

comment: $\Theta(E_{7})=\frac{9\sqrt{3}\,\pi^{3}}{35}$

$E_n$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $E_8^{*}=E_8$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$E_n$

$\Theta(L)$:

4.058712126416768218185013862029379635405316069695225903811160807915452310701133174588411134781374690

comment: $\Theta(E_{8})=\frac{\pi^{4}}{24}$; not a locally thinnest lattice covering [11]; $E_8^{*}=E_8$

$A_n^{*}$

$R/\rho$:

1.290994448735805628393088466594133203610973901763863608862524588704494363978993011173095792286224506

comment: $R/\rho=\frac{\sqrt{15}}{3}$; $R^2=\frac{5}{16}$ with $\mu=\frac{3}{4}$; $A_3^{*}$ is the body-centred cubic lattice
equals: Algebraic_numbers_of_degree_2#3,0,-5,2

$A_n^{*}$

$\Theta(L)$:

1.463503068966817998574244717334353083222189413972333837497858547123362891795664209396696336534534842

comment: $\Theta(A_{3}^{*})=\frac{5\sqrt{5}\,\pi}{24}$; the thinnest lattice covering in dimension 3 [5]; $A_3^{*}$ is the body-centred cubic lattice

$A_n^{*}$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=\frac{2}{5}$ with $\mu=\frac{4}{5}$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$A_n^{*}$

$\Theta(L)$:

1.765528508149352424827507939406410069150616179632367159855293505414194011095957519208436416026434011

comment: $\Theta(A_{4}^{*})=\frac{2\sqrt{5}\,\pi^{2}}{25}$; the thinnest lattice covering in dimension 4 [6]

$A_n^{*}$

$R/\rho$:

1.527525231651946668862682397909336162994818858922657300869080707968956141849256962201453853164481678

comment: $R/\rho=\frac{\sqrt{21}}{3}$; $R^2=\frac{35}{72}$ with $\mu=\frac{5}{6}$

$A_n^{*}$

$\Theta(L)$:

2.124285908991589721102606025324353057687042692239452369433234210002194459218544627794852930849704785

comment: $\Theta(A_{5}^{*})=\frac{245\sqrt{105}\,\pi^{2}}{11664}$; the thinnest lattice covering in dimension 5 [7]

$A_n^{*}$

$R/\rho$:

1.632993161855452065464856049803927594643964987104446752288461711500640251638210017693239622069760157

comment: $R/\rho=\frac{2\sqrt{6}}{3}$; $R^2=\frac{4}{7}$ with $\mu=\frac{6}{7}$

$A_n^{*}$

$\Theta(L)$:

2.551133828668911667945352901684633881153782697942228508150538867567655827742178406794412015042437601

comment: $\Theta(A_{6}^{*})=\frac{32\sqrt{7}\,\pi^{3}}{1029}$; a thinner lattice covering in dimension 6 is known [10]

$A_n^{*}$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{21}{32}$ with $\mu=\frac{7}{8}$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$A_n^{*}$

$\Theta(L)$:

3.059622898859479486587355087153620651585032899444023032172741646200423777967454039813256237342164178

comment: $\Theta(A_{7}^{*})=\frac{441\sqrt{21}\,\pi^{3}}{20480}$; a thinner lattice covering in dimension 7 is known [10]

$A_n^{*}$

$R/\rho$:

1.825741858350553711523232609336007113175815649993277514089648165774977590409075779336194787212902086

comment: $R/\rho=\frac{\sqrt{30}}{3}$; $R^2=\frac{20}{27}$ with $\mu=\frac{8}{9}$

$A_n^{*}$

$\Theta(L)$:

3.665847800000468057091580540030035742433405991358793231665146625494489715954440707063319060243864984

comment: $\Theta(A_{8}^{*})=\frac{20000\pi^{4}}{531441}$; a thinner lattice covering in dimension 8 is known [10]

$A_n^{*}$

$R/\rho$:

1.914854215512676219950203822739643106073421485994264122566625823521966907152468209009041788532262707

comment: $R/\rho=\frac{\sqrt{33}}{3}$; $R^2=\frac{33}{40}$ with $\mu=\frac{9}{10}$

$A_n^{*}$

$\Theta(L)$:

4.388947936151627216724177374225484490372238023249542626136580827006776588271738702298506165338957413

comment: $\Theta(A_{9}^{*})=\frac{43923\sqrt{33}\,\pi^{4}}{5600000}$; a thinner lattice covering in dimension 9 is known [12]

$A_n^{*}$

$R/\rho$:

comment: $R/\rho=2$; $R^2=\frac{10}{11}$ with $\mu=\frac{10}{11}$

$A_n^{*}$

$\Theta(L)$:

5.251713602604193934516201482794718369208850423491812986732239821522037611820492864207216652920346488

comment: $\Theta(A_{10}^{*})=\frac{2500\sqrt{11}\,\pi^{5}}{483153}$; a thinner lattice covering in dimension 10 is known [12]

$A_n^{*}$

$R/\rho$:

2.081665999466132735282297706979931487024319992663885436150990643536537066861171500211112037407113529

comment: $R/\rho=\frac{\sqrt{39}}{3}$; $R^2=\frac{143}{144}$ with $\mu=\frac{11}{12}$

$A_n^{*}$

$\Theta(L)$:

6.281306221903178204445738766734193692744792119654676997808894382475618320972943181572755816112586911

comment: $\Theta(A_{11}^{*})=\frac{5436100813\sqrt{429}\,\pi^{5}}{5485491486720}$; a thinner lattice covering in dimension 11 is known [12]

$A_n^{*}$

$R/\rho$:

2.160246899469286743655322478695998885901734769019448849903784799269872459281348123800120302201870079

comment: $R/\rho=\frac{\sqrt{42}}{3}$; $R^2=\frac{14}{13}$ with $\mu=\frac{12}{13}$

$A_n^{*}$

$\Theta(L)$:

7.510113769576549303636204735308724672374067327234787264502812055299385599129591809206003226170731784

comment: $\Theta(A_{12}^{*})=\frac{470596\sqrt{13}\,\pi^{6}}{217206405}$; a thinner lattice covering in dimension 12 is known [12]

$A_n^{*}$

$R/\rho$:

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $R/\rho=\sqrt{5}$; $R^2=\frac{65}{56}$ with $\mu=\frac{13}{14}$
equals: Algebraic_numbers_of_degree_2#1,0,-5,2

$A_n^{*}$

$\Theta(L)$:

8.976768394939004134904601717190052647825931149143188591614851194586406664698145765595222254395956857

comment: $\Theta(A_{13}^{*})=\frac{1160290625\sqrt{65}\,\pi^{6}}{1001849942016}$; a thinner lattice covering in dimension 13 is known [12]

$A_n^{*}$

$R/\rho$:

2.309401076758503058036595122007829822590407005080507504074409305935910689211733382774861582342998101

comment: $R/\rho=\frac{4\sqrt{3}}{3}$; $R^2=\frac{56}{45}$ with $\mu=\frac{14}{15}$

$A_n^{*}$

$\Theta(L)$:

10.72735849771784183757477485114728938415858358994865094218626033274480879714128256474420237148041224

comment: $\Theta(A_{14}^{*})=\frac{15420489728\sqrt{15}\,\pi^{7}}{16815125390625}$; a thinner lattice covering in dimension 14 is known [12]

$A_n^{*}$

$R/\rho$:

2.380476142847616665999799937122421759588723719967577944402810294255273468813340628752996071177606806

comment: $R/\rho=\frac{\sqrt{51}}{3}$; $R^2=\frac{85}{64}$ with $\mu=\frac{15}{16}$

$A_n^{*}$

$\Theta(L)$:

12.81687351505401476768881764529366219416442403965739985163645692129908816829392641620599011438092602

comment: $\Theta(A_{15}^{*})=\frac{1282308353125\sqrt{85}\,\pi^{7}}{2785921946615808}$; a thinner lattice covering in dimension 15 is known [12]

$A_n^{*}$

$R/\rho$:

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $R/\rho=\sqrt{6}$; $R^2=\frac{24}{17}$ with $\mu=\frac{16}{17}$

$A_n^{*}$

$\Theta(L)$:

15.31092684557848389880230889593003306620873327451895708293779990606852187747162902088353319479976479

comment: $\Theta(A_{16}^{*})=\frac{95551488\sqrt{17}\,\pi^{8}}{244151510435}$; the thinnest lattice covering in dimension 16 listed by Dutour Sikirić, Schürmann and Vallentin [12]

$A_n^{*}$

$R/\rho$:

2.516611478423583232412228268982039019407398234874460046099542301903573984639270398527484024715841324

comment: $R/\rho=\frac{\sqrt{57}}{3}$; $R^2=\frac{323}{216}$ with $\mu=\frac{17}{18}$

$A_n^{*}$

$\Theta(L)$:

18.28781095600886607219344412662745149952538841637398685811818043343032577853446429618361313411570534

comment: $\Theta(A_{17}^{*})=\frac{6969012721055784593\sqrt{969}\,\pi^{8}}{112556454284898931507200}$; a thinner lattice covering in dimension 17 is known [12]

$A_n^{*}$

$R/\rho$:

2.581988897471611256786176933188266407221947803527727217725049177408988727957986022346191584572449012

comment: $R/\rho=\frac{2\sqrt{15}}{3}$; $R^2=\frac{30}{19}$ with $\mu=\frac{18}{19}$

$A_n^{*}$

$\Theta(L)$:

21.84094905010656848391139715130307628863297661807842350305105978725322896206257903289677991734096805

comment: $\Theta(A_{18}^{*})=\frac{379687500\sqrt{19}\,\pi^{9}}{2258813884453}$; the thinnest lattice covering in dimension 18 listed by Dutour Sikirić, Schürmann and Vallentin [12]

$A_n^{*}$

$R/\rho$:

2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611

comment: $R/\rho=\sqrt{7}$; $R^2=\frac{133}{80}$ with $\mu=\frac{19}{20}$

$A_n^{*}$

$\Theta(L)$:

26.08182004558247299120570192465693329051955846418696703022882245563458643273853955842322756512379959

comment: $\Theta(A_{19}^{*})=\frac{97906861202319841\sqrt{133}\,\pi^{9}}{1290475929600000000000}$; a thinner lattice covering in dimension 19 is known [12]

$A_n^{*}$

$R/\rho$:

2.708012801545320120153294522755346782834734695575104275883858531510972267613557564999719200982645626

comment: $R/\rho=\frac{\sqrt{66}}{3}$; $R^2=\frac{110}{63}$ with $\mu=\frac{20}{21}$

$A_n^{*}$

$\Theta(L)$:

31.14344838176163833619977115559577386653874347128074657846682049365825855052181144007269401192031499

comment: $\Theta(A_{20}^{*})=\frac{40527225939062500\sqrt{21}\,\pi^{10}}{558455475496975411383}$; a thinner lattice covering in dimension 20 is known [12]

$A_n^{*}$

$R/\rho$:

2.768874620972691617528087581635830670077404749651885259810729456345861677807717191289132836077452350

comment: $R/\rho=\frac{\sqrt{69}}{3}$; $R^2=\frac{161}{88}$ with $\mu=\frac{21}{22}$

$A_n^{*}$

$\Theta(L)$:

37.18456782695306450528575888269467295961046865148534255944523223839591639905882526228820219248192479

comment: $\Theta(A_{21}^{*})=\frac{238815593270954968849\sqrt{161}\,\pi^{10}}{7631529844136427572428800}$; a thinner lattice covering in dimension 21 is known [12]

$A_n^{*}$

$R/\rho$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $R/\rho=2\sqrt{2}$; $R^2=\frac{44}{23}$ with $\mu=\frac{22}{23}$

$A_n^{*}$

$\Theta(L)$:

44.39458951818174359637317864436976908494816749456870258215781640358423760362071239838701606838887097

comment: $\Theta(A_{22}^{*})=\frac{424958764662784\sqrt{23}\,\pi^{11}}{13506078318429915225}$; a thinner lattice covering in dimension 22 is known [12]

$A_n^{*}$

$R/\rho$:

2.886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626

comment: $R/\rho=\frac{5\sqrt{3}}{3}$; $R^2=\frac{575}{288}$ with $\mu=\frac{23}{24}$

$A_n^{*}$

$\Theta(L)$:

52.99952960374942460718670124536522878256088982965332566626670899771059990599080305978934662983591161

comment: $\Theta(A_{23}^{*})=\frac{19753699881386280059814453125\sqrt{69}\,\pi^{11}}{910855701610617424251753685057536}$; a thinner lattice covering in dimension 23 is known [12]

$A_n^{*}$

$R/\rho$:

2.943920288775948951588014242319751321391546493643805942070067657519046646078480946790145179804923839

comment: $R/\rho=\frac{\sqrt{78}}{3}$; $R^2=\frac{52}{25}$ with $\mu=\frac{24}{25}$

$A_n^{*}$

$\Theta(L)$:

63.26908185551182732920800286985663862751481384433217155794137106718604632314067105845120015526323247

comment: $\Theta(A_{24}^{*})=\frac{381715826646728704\pi^{12}}{5576312541961669921875}$; the Leech lattice $\Lambda_{24}$ is a thinner lattice covering [12]

$D_n^{*}$

$R/\rho$:

3/2

comment: $R/\rho=\frac{3}{2}$; $R^2=\frac{9}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

2.498243614025743900392480534343650756126280162457825127310879060855698845674861327134823892816258534

comment: $\Theta(D_{5}^{*})=\frac{81\pi^{2}}{320}$

$D_n^{*}$

$R/\rho$:

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{4}$ with $\mu=1$
equals: Algebraic_numbers_of_degree_2#1,0,-3,2

$D_n^{*}$

$\Theta(L)$:

4.360257658167162212176356806311133700312931204577593269489075670847774285268614911532817332709780871

comment: $\Theta(D_{6}^{*})=\frac{9\pi^{3}}{64}$

$D_n^{*}$

$R/\rho$:

1.802775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359

comment: $R/\rho=\frac{\sqrt{13}}{2}$; $R^2=\frac{13}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

4.568694211108658270461274542464035977395587936863368093527406036754464973587660675764016284625116151

comment: $\Theta(D_{7}^{*})=\frac{2197\sqrt{13}\,\pi^{3}}{53760}$

$D_n^{*}$

$R/\rho$:

comment: $R/\rho=2$; $R^2=1$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

8.117424252833536436370027724058759270810632139390451807622321615830904621402266349176822269562749379

comment: $\Theta(D_{8}^{*})=\frac{\pi^{4}}{12}$

$D_n^{*}$

$R/\rho$:

2.061552812808830274910704927987038512573599612686810217199316786547477173168810796793931825405342148

comment: $R/\rho=\frac{\sqrt{17}}{2}$; $R^2=\frac{17}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

8.666183496864602951956953728581011447868407247910625205443812669853112064185933898873460151579225058

comment: $\Theta(D_{9}^{*})=\frac{83521\sqrt{17}\,\pi^{4}}{3870720}$

$D_n^{*}$

$R/\rho$:

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{4}$ with $\mu=1$
equals: Algebraic_numbers_of_degree_2#1,0,-5,2

$D_n^{*}$

$\Theta(L)$:

15.56496606370450100009874013485898877361555514669163973762473773631973226591054405140093487922694493

comment: $\Theta(D_{10}^{*})=\frac{625\pi^{5}}{12288}$

$D_n^{*}$

$R/\rho$:

2.291287847477920003294023596864004244492228288383985951303621061953434212773885443302180779746722516

comment: $R/\rho=\frac{\sqrt{21}}{2}$; $R^2=\frac{21}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

16.81438765664326948399572725958410291735661041459349226719451898728610243097161146497441592488378607

comment: $\Theta(D_{11}^{*})=\frac{21609\sqrt{21}\,\pi^{5}}{1802240}$

$D_n^{*}$

$R/\rho$:

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $R/\rho=\sqrt{6}$; $R^2=\frac{3}{2}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

30.41895495296861695290928708431484836259555459943936558251061374461390896102748722399880239894733384

comment: $\Theta(D_{12}^{*})=\frac{81\pi^{6}}{2560}$

$D_n^{*}$

$R/\rho$:

5/2

comment: $R/\rho=\frac{5}{2}$; $R^2=\frac{25}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

33.12848111029900414134935618485423494982948311926463455542741825336278393102503355236808612898849115

comment: $\Theta(D_{13}^{*})=\frac{244140625\pi^{6}}{7084965888}$

$D_n^{*}$

$R/\rho$:

2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611

comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{4}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

60.24415384160559928910639100874712617378302905375650716328479915947526897386782229623209849953826148

comment: $\Theta(D_{14}^{*})=\frac{117649\pi^{7}}{5898240}$

$D_n^{*}$

$R/\rho$:

2.692582403567252015625355245770164778147560080822394418840194335008322981413829346438316890839917742

comment: $R/\rho=\frac{\sqrt{29}}{2}$; $R^2=\frac{29}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

66.00017379164128675001190474046585794739681999537693375619032686007967782185855351027627905501274891

comment: $\Theta(D_{15}^{*})=\frac{17249876309\sqrt{29}\,\pi^{7}}{4250979532800}$

$D_n^{*}$

$R/\rho$:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $R/\rho=2\sqrt{2}$; $R^2=2$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

120.4892827437533207254422286210382958310752948779820024394932243183099543089698577274344676446846365

comment: $\Theta(D_{16}^{*})=\frac{4\pi^{8}}{315}$

$D_n^{*}$

$R/\rho$:

2.872281323269014329925305734109464659110132228991396183849938735282950360728702313513562682798394061

comment: $R/\rho=\frac{\sqrt{33}}{2}$; $R^2=\frac{33}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

132.5987860734753264674647135793810769032044662970110755899386017708173024920346027870024405243423392

comment: $\Theta(D_{17}^{*})=\frac{1578460851\sqrt{33}\,\pi^{8}}{648858828800}$

$D_n^{*}$

$R/\rho$:

comment: $R/\rho=3$; $R^2=\frac{9}{4}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

242.8054776017902469096061330074154192653121955473620991457961669570918425929048328918022089047399597

comment: $\Theta(D_{18}^{*})=\frac{4782969\pi^{9}}{587202560}$

$D_n^{*}$

$R/\rho$:

3.041381265149109844499842122601033531042485047393205593209576523243166362659455119901533213978924332

comment: $R/\rho=\frac{\sqrt{37}}{2}$; $R^2=\frac{37}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

268.1598305964027984638471678342584620723772999172741870372569777982262612334257923046849407128604662

comment: $\Theta(D_{19}^{*})=\frac{129961739795077\sqrt{37}\,\pi^{9}}{87876248902041600}$

$D_n^{*}$

$R/\rho$:

3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295

comment: $R/\rho=\sqrt{10}$; $R^2=\frac{5}{2}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

492.2273900988399508018168306903619700848854264859521101341700080483796946706343814000842714927006742

comment: $\Theta(D_{20}^{*})=\frac{390625\pi^{10}}{74317824}$

$D_n^{*}$

$R/\rho$:

3.201562118716424343244108837310906632260210066310509442764636313334091379098438037144677151124934982

comment: $R/\rho=\frac{\sqrt{41}}{2}$; $R^2=\frac{41}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

545.1906894082681901536000204863069502751248347526646049925881541063922970915301692187996890996322921

comment: $\Theta(D_{21}^{*})=\frac{13422659310152401\sqrt{41}\,\pi^{10}}{14763209815542988800}$

$D_n^{*}$

$R/\rho$:

3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651

comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{4}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

1002.726538774859462288406993532259707269891274354608487942454766895963714320603794347966352069931232

comment: $\Theta(D_{22}^{*})=\frac{25937424601\pi^{11}}{7610145177600}$

$D_n^{*}$

$R/\rho$:

3.354101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412

comment: $R/\rho=\frac{3\sqrt{5}}{2}$; $R^2=\frac{45}{16}$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

1113.253048406610343788994768380183759750388160034924922992343454564108130521569127272261428047522681

comment: $\Theta(D_{23}^{*})=\frac{756680642578125\sqrt{5}\,\pi^{11}}{447149070956363776}$

$D_n^{*}$

$R/\rho$:

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $R/\rho=2\sqrt{3}$; $R^2=3$ with $\mu=1$

$D_n^{*}$

$\Theta(L)$:

2050.909801127860537001603739421658028497545659671152427629866728804534742723956003678335096143485015

comment: $\Theta(D_{24}^{*})=\frac{2187\pi^{12}}{985600}$

$E_n^{*}$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=\frac{2}{3}$ with $\mu=\frac{4}{3}$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$E_n^{*}$

$\Theta(L)$:

2.652071435250239570229173750613035423243838558352976804347762244247347001099969561049078909406866082

comment: $\Theta(E_{6}^{*})=\frac{4\sqrt{3}\,\pi^{3}}{81}$

$E_n^{*}$

$R/\rho$:

1.527525231651946668862682397909336162994818858922657300869080707968956141849256962201453853164481678

comment: $R/\rho=\frac{\sqrt{21}}{3}$; $R^2=\frac{7}{8}$ with $\mu=\frac{3}{2}$

$E_n^{*}$

$\Theta(L)$:

4.187197876800624844058354122911488441460119707452640091453721362032186669615507078417351829767407105

comment: $\Theta(E_{7}^{*})=\frac{49\sqrt{7}\,\pi^{3}}{960}$

$\Lambda_n$

$R/\rho$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $R/\rho=\sqrt{2}$; $R^2=2$ with $\mu=4$; $\Lambda_{24}$ is the Leech lattice
equals: Algebraic_numbers_of_degree_2#1,0,-2,2

$\Lambda_n$

$\Theta(L)$:

7.903536371318468804212103428857682494130060554241242530753116640589806117892036736972176171770445469

comment: $\Theta(\Lambda_{24})=\frac{4\pi^{12}}{467775}$; a locally thinnest lattice covering [11]; $\Lambda_{24}$ is the Leech lattice

Definition

For a lattice $L\subset\mathbb{R}^n$ with covering radius $R$, the largest distance from a point of $\mathbb{R}^n$ to $L$, packing radius $\rho$, half the length of a shortest nonzero vector, and determinant $\det L$: the normalised covering radius $R/\rho$ and the covering density $\Theta(L)=V_nR^n/\sqrt{\det L}$ [1] of the covering of $\mathbb{R}^n$ by the balls of radius $R$ centred at the points of $L$, $V_n$ the volume of the unit ball, for $\mathbb{Z}^n$, the root lattices, their duals and the Leech lattice.

Parameters

family

— family of the lattice

$n$

— dimension ($n\geq 1$ for $\mathbb{Z}^n$ and $A_n$; $n\geq 4$ for $D_n$; $n\in\{6,7,8\}$ for $E_n$; $n\geq 3$ for $A_n^{*}$; $n\geq 5$ for $D_n^{*}$; $n\in\{6,7\}$ for $E_n^{*}$; $n=24$ for $\Lambda_n$)

Formulas

(1)

$R/\rho=2R/\sqrt{\mu}$ and $\Theta(L)=V_n\,R^n/\sqrt{\det L}=(R/\rho)^n\,\Delta(L)$, where $\Delta(L)=V_n\rho^n/\sqrt{\det L}$ is the packing density of $L$, in the table of the classical lattices, and $V_n=\pi^{n/2}/\Gamma(n/2+1)$ is the volume of the unit ball.

(2)

$R^2=n/4$ for $\mathbb{Z}^n$ at $\mu=1$; $R^2=a(n+1-a)/(n+1)$ with $a=\lfloor (n+1)/2\rfloor$ for $A_n$, $R^2=n/4$ for $D_n$ ($n\geq 4$) and $R^2=4/3$, $3/2$, $1$ for $E_6$, $E_7$, $E_8$, all at $\mu=2$; $R^2=n(n+2)/(12(n+1))$ for $A_n^{*}$ at $\mu=n/(n+1)$ [8] [9]; $R^2=n/8$ for even $n$ and $R^2=(2n-1)/16$ for odd $n$ for $D_n^{*}$ at $\mu=1$ ($n\geq 4$); $R^2=2/3$ for $E_6^{*}$ at $\mu=4/3$ and $R^2=7/8$ for $E_7^{*}$ at $\mu=3/2$; $R^2=2$ for $\Lambda_{24}$ at $\mu=4$ [2].

(3)

$(R/\rho)^2=n$ for $\mathbb{Z}^n$; $2a(n+1-a)/(n+1)$ for $A_n$; $n/2$ for $D_n$; $8/3$, $3$, $2$ for $E_6$, $E_7$, $E_8$; $(n+2)/3$ for $A_n^{*}$; $n/2$ for even $n$ and $(2n-1)/4$ for odd $n$ for $D_n^{*}$; $2$ and $7/3$ for $E_6^{*}$ and $E_7^{*}$; $2$ for $\Lambda_{24}$.

(4)

$\Theta(\mathbb{Z}^n)=V_n\,(n/4)^{n/2}$, $\Theta(A_n)=V_n\,\bigl(a(n+1-a)/(n+1)\bigr)^{n/2}/\sqrt{n+1}$ with $a=\lfloor (n+1)/2\rfloor$, $\Theta(D_n)=V_n\,(n/4)^{n/2}/2$, $\Theta(A_n^{*})=V_n\sqrt{\bigl(n(n+2)/(12(n+1))\bigr)^n\,(n+1)}$ [8] [9], $\Theta(E_8)=V_8=\pi^4/24$ [19] and $\Theta(\Lambda_{24})=2^{12}\,V_{24}=4\pi^{12}/467775$ [11]; in particular $\Theta(A_2)=2\pi/(3\sqrt{3})$ [18] and $\Theta(A_3^{*})=5\sqrt{5}\,\pi/24$.

Comments

(5)

The covering radius $R=\max_{x\in\mathbb{R}^n}\min_{v\in L}|x-v|$ is the circumradius of the Voronoi cell of $L$, the points of $\mathbb{R}^n$ at distance $R$ from $L$ are its deep holes, and the packing radius $\rho=\sqrt{\mu}/2$, with $\mu=\min\{v\cdot v : v\in L,\ v\neq 0\}$ the minimal norm, is the inradius of the cell; $\det L$ 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. $\Theta(L)$ is the number of balls of radius $R$ covering a point of $\mathbb{R}^n$ on average, at least $1$, and is what Conway and Sloane [1], chapter 2, call the covering density or thickness of the lattice covering; $R/\rho\geq 1$, and $(R/\rho)^2=4R^2/\mu$ is a rational number for every lattice listed. Both quantities are unchanged when $L$ is scaled, so no scaling of a named lattice has to be chosen. The packing density $\Delta(L)$, centre density and Hermite number of the same lattices are in the table of the classical lattices, whose families and notation this table follows; $\Theta(L)=(R/\rho)^n\,\Delta(L)$.

(6)

$\mathbb{Z}^n$ is the cubic lattice; $A_n$, $D_n$ and $E_n$ are the root lattices [1], chapter 4, with $A_2$ the hexagonal lattice, $A_3=D_3$ the face-centred cubic lattice and $A_3^{*}$ the body-centred cubic lattice; $\Lambda_{24}$ is the Leech lattice [3]. A lattice with two names is listed once: $D_3$ under $A_3$, and $A_1^{*}$, $A_2^{*}$ and $D_4^{*}$ not at all, being similar to $A_1$, $A_2$ and $D_4$; $E_8^{*}=E_8$. The laminated lattices $\Lambda_9$ to $\Lambda_{23}$ and the Coxeter–Todd lattice $K_{12}$ of the table of the classical lattices are not listed. Every family stops at $n=24$.

(7)

The covering radii of $\mathbb{Z}^n$, $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ and $A_n^{*}$ are those of Conway and Sloane [1], chapter 4, and that of the Leech lattice is the theorem of Conway, Parker and Sloane [2] [16], $R=\sqrt{2}$ at minimal norm $4$, so that $R/\rho=\sqrt{2}$ as for $E_8$, whose deep holes at minimal norm $2$ are the halves of its vectors of norm $4$, at distance $1$ from the lattice [17]. For $D_n^{*}=\mathbb{Z}^n\cup(\mathbb{Z}^n+(\tfrac12)^n)$ at minimal norm $1$ the distance of $x\in[0,\tfrac12]^n$ to the lattice satisfies $d(x,D_n^{*})^2=\min\bigl(\sum_i x_i^2,\ \sum_i(\tfrac12-x_i)^2\bigr)$, and the maximum of this convex function over the polytope $\sum_i x_i\leq n/4$ is attained at the deep hole $(\tfrac12,\ldots,\tfrac12,0,\ldots,0)$ with $n/2$ halves for even $n$ and $(\tfrac12,\ldots,\tfrac12,\tfrac14,0,\ldots,0)$ with $(n-1)/2$ halves for odd $n$, which gives $R^2=n/8$ and $R^2=(2n-1)/16$. The covering radii of $E_6^{*}$ and $E_7^{*}$, $R^2=2/3$ at minimal norm $4/3$ and $R^2=7/8$ at minimal norm $3/2$, were computed exactly from their Voronoi cells.

(8)

$A_n^{*}$ is the thinnest lattice covering of $\mathbb{R}^n$ for $n\leq 5$: the hexagonal lattice $A_2=A_2^{*}$ is the thinnest covering of the plane by equal discs of any kind [4], and the lattice covering problem was solved by Bambah for $n=3$ [5], by Delone and Ryshkov for $n=4$ [6] and by Ryshkov and Baranovskii for $n=5$ [7]. In every dimension $6\leq n\leq 24$ other than $16$ and $18$ a lattice covering thinner than $A_n^{*}$ is known, and Table 2 of Dutour Sikirić, Schürmann and Vallentin [12] lists the thinnest known in each dimension $n\leq 24$: lattices found by Schürmann and Vallentin in dimensions $6$, $7$ and $8$ [10] and by Dutour Sikirić, Schürmann and Vallentin in dimensions $9$ to $15$ [12], the Coxeter lattices $A_{17}^{9}$, $A_{19}^{10}$, $A_{20}^{7}$ and $A_{21}^{11}$, sublattices of $A_n^{*}$ containing $A_n$, in dimensions $17$, $19$, $20$ and $21$, the duals $\Lambda_{22}^{*}$ and $\Lambda_{23}^{*}$ of the laminated lattices, and the Leech lattice in dimension $24$. Before them the Coxeter lattices $A_9^{5}$ [13], $A_{11}^{4}$ and $A_{14}^{5}$ [14], $A_{13}^{7}$ and $A_{15}^{8}$ [15] were the thinnest known in their dimensions. The Leech lattice is a locally thinnest lattice covering and $E_8$ is not [11].

(9)

The comment on each entry gives its value in closed form, $R/\rho$ as the square root of a rational number and $\Theta(L)$ as $\pi^{\lfloor n/2\rfloor}$ times the square root of a rational number; on $R/\rho$ also $R^2$ and the minimal norm $\mu$ in the scaling of Conway and Sloane [1], chapter 4, which is $\mu=2$ for the root lattices, $\mu=1$ for $\mathbb{Z}^n$ and $D_n^{*}$, $\mu=n/(n+1)$ for $A_n^{*}$, $\mu=4/3$ and $3/2$ for $E_6^{*}$ and $E_7^{*}$ and $\mu=4$ for $\Lambda_{24}$; and on $\Theta(L)$, where it is known, whether the lattice is the thinnest lattice covering of its dimension or a thinner one is known.

Programs

(P1)

Sage

G = CartanMatrix(['D', 4]); n = G.nrows(); mu = 2
vecs = pari(G).qfminim(4, None, 0)[2].sage().columns()             # lattice vectors of norm <= 4, one of each pair +-v
V = Polyhedron(ieqs=[[v*G*v/2] + list(s*G*v) for v in vecs for s in (1, -1)])   # the Voronoi cell
R2 = max(x.vector()*G*x.vector() for x in V.vertices())           # 1, the squared covering radius
sqrt(4*R2/mu)                                                     # sqrt(2) = R/rho
pi^(n/2)/gamma(n/2 + 1) * sqrt(R2^n/G.det())                      # pi^2/4 = Theta(D_4)

References

[1]

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

[2]

J. H. Conway, R. A. Parker and N. J. A. Sloane, The covering radius of the Leech lattice, Proceedings of the Royal Society of London A 380 (1982), 261–290.

[3]

J. Leech, Notes on sphere packings, Canadian Journal of Mathematics 19 (1967), 251–267.

[4]

R. Kershner, The number of circles covering a set, American Journal of Mathematics 61 (1939), 665–671.

[5]

R. P. Bambah, On lattice coverings by spheres, Proceedings of the National Institute of Sciences of India 20 (1954), 25–52.

[6]

B. N. Delone and S. S. Ryshkov, Solution of the problem of the least dense lattice covering of a four-dimensional space by equal spheres, Soviet Mathematics Doklady 4 (1963), 1333–1334; translation from Doklady Akademii Nauk SSSR 152 (1963), 523–524.

[7]

S. S. Ryshkov and E. P. Baranovskii, Solution of the problem of least dense lattice covering of five-dimensional space by equal spheres, Soviet Mathematics Doklady 16 (1975), 586–590; translation from Doklady Akademii Nauk SSSR 222 (1975), 39–42.

[8]

A. F. Gameckii, On the theory of covering Euclidean $n$-space by equal spheres, Soviet Mathematics Doklady 3 (1962), 1410–1414; translation from Doklady Akademii Nauk SSSR 146 (1962), 991–994.

[9]

M. N. Bleicher, Lattice coverings of $n$-space by spheres, Canadian Journal of Mathematics 14 (1962), 632–650.

[10]

A. Schürmann and F. Vallentin, Computational approaches to lattice packing and covering problems, Discrete and Computational Geometry 35 (2006), 73–116;. (arXiv)

[11]

A. Schürmann and F. Vallentin, Local covering optimality of lattices: Leech lattice versus root lattice $E_8$, International Mathematics Research Notices 2005, no. 32, 1937–1955;. (arXiv)

[12]

M. Dutour Sikirić, A. Schürmann and F. Vallentin, A generalization of Voronoi's reduction theory and its application, Duke Mathematical Journal 142 (2008), 127–164, Table 2;. (arXiv)

[13]

E. P. Baranovskii, The perfect lattices $\Gamma(\mathfrak{A}^n)$, and the covering density of $\Gamma(\mathfrak{A}^9)$, European Journal of Combinatorics 15 (1994), 317–323.

[14]

M. M. Anzin, On the density of a lattice covering for $n=11$ and $n=14$, Proceedings of the Steklov Institute of Mathematics 239 (2002), 13–44; translation from Trudy Matematicheskogo Instituta imeni V. A. Steklova 239 (2002), 20–51.

[15]

M. M. Anzin, On lattice covering density for $n=13$ and $n=15$, Mathematical Notes 79 (2006), 721–725; translation from Matematicheskie Zametki 79 (2006), 781–784.

Links

[16]

Wikipedia: Leech lattice

[17]

Wikipedia: E8 lattice

[18]

OEIS A248897: decimal expansion of $2\pi/(3\sqrt{3})$, the covering density of the hexagonal lattice

[19]

OEIS A164108: decimal expansion of $\pi^4/24$, the covering density of $E_8$

[20]

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

Similar tables

Packing densities and Hermite numbers of the classical lattices — the packing side of the same lattices, $\Delta(L)=\Theta(L)/(R/\rho)^n$

Volume of the $d$-dimensional unit ball — $V_n$, the factor in $\Theta(L)=V_nR^n/\sqrt{\det L}$; $\Theta(E_8)=V_8$ and $\Theta(\Lambda_{24})=2^{12}V_{24}$

Rational multiples of pi — holds $\Theta(\mathbb{Z}^2)=\pi/2$ and $\Theta(A_3)=2\pi/3$

Algebraic numbers of degree 2 — holds $R/\rho$ where it is $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $2/\sqrt{3}$, $\sqrt{5/2}$ or $\sqrt{5/3}$, which those entries link

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (every family to $n=24$ is here: $\mathbb{Z}^n$ and $A_n$ for $1\leq n\leq 24$, $D_n$ for $4\leq n\leq 24$, $E_6$, $E_7$, $E_8$, $A_n^{*}$ for $3\leq n\leq 24$, $D_n^{*}$ for $5\leq n\leq 24$, $E_6^{*}$, $E_7^{*}$ and $\Lambda_{24}$, 117 lattices and 234 entries)

How they were obtained:

For each lattice the determinant of its integral Gram matrix is an exact integer, the minimal norm and the number of shortest vectors come from PARI's qfminim and must equal the known values, and $R^2$ is the exact rational of (2), so that $(R/\rho)^2=4R^2/\mu$ and $\Theta^2/\pi^{2\lfloor n/2\rfloor}$ are exact rationals; a value that is rational is written exactly, and the rest are balls in arb, a square root and for $\Theta$ the product with $\pi^{\lfloor n/2\rfloor}$, computed with 64 guard bits beyond the 100 digits written; the widest ball relative to its value, $\Theta(A_{24})$, has radius $4.5\cdot 10^{-119}$.

For every lattice of dimension $n\leq 6$ the generator recomputes $R^2$ before use as the largest vertex norm of the Voronoi cell, built as a rational polytope from all lattice vectors of norm at most $B$ with $B$ raised until $4R^2\leq B$, which proves that every Voronoi-relevant vector was used; a disagreement with the formula is an error rather than an entry. Outside the generator, before any entry was written, $R^2$ was recomputed for every lattice of dimension $7$ and $8$ except $A_7^{*}$ and $A_8^{*}$ as the largest vertex norm of the polytope cut by the lattice vectors of norm at most $2\mu$ (the cell of $E_8$ has $19440$ vertices), and the vertex attaining it was proven a deep hole by an exact enumeration, with qfminim on an augmented form, of the lattice vectors nearer to it than $R$, of which there is none; and $R/\rho$ was recomputed from the catalogue's own Gram matrices [20] of $A_3^{*}$, $D_5^{*}$, $D_7^{*}$, $E_6^{*}$ and $E_7^{*}$ (two versions), giving the same values. The closed form of $\Theta(A_n^{*})$ holds in balls for $2\leq n\leq 24$; $\Theta(A_n^{*})$ for $n=2,4,5,10,12$ and $16\leq n\leq 21$ and $\Theta(\Lambda_{24})$ agree with the six decimals of Table 1 of Schürmann and Vallentin [10], and the table of the thinnest lattice coverings known to dimension $24$ of Dutour Sikirić, Schürmann and Vallentin [12] agrees with $\Theta(A_n^{*})$ where it lists $A_n^{*}$ ($n=2,4,5,16,18$) and is smaller than it in every other dimension from $6$ to $24$; both tables print $1.463505$ for $\Theta(A_3^{*})=5\sqrt{5}\,\pi/24=1.4635031\ldots$, which the Voronoi cell of the catalogue's Gram matrix confirms; $\Theta(A_2)$ and $\Theta(E_8)$ agree with OEIS A248897 and A164108 to every digit those entries give; $\Theta=(R/\rho)^n\Delta$ holds in balls for all 117 lattices against the stored densities of the table of the classical lattices, $\Theta(\mathbb{Z}^n)=V_n(n/4)^{n/2}$, $\Theta(E_8)=V_8$ and $\Theta(\Lambda_{24})=2^{12}V_{24}$ against the stored digits of the table of unit-ball volumes, and the entries linked to other tables agree with the stored digits there. Controls that must fail did: a point of $A_3$ that is not a hole is refused by the enumeration, $E_7^{*}$ offered with $R^2=3/4$ and $D_7^{*}$ with the even-dimensional formula are refused by the Voronoi cell, and $\Theta(A_4^{*})$ moved by $2\cdot 10^{-6}$ no longer matches the tables.