Tammes problem: the minimum angle of the best known spherical codes

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Numbers

$n$

normalisation

$\theta_n$ or $d_n$

$\theta_n$:

109.4712206344906913692459993399624359630068431009079482881711063560397464380603955633618712883605876

comment: The regular tetrahedron has $\cos(\pi\theta_4/180)=-1/3$ and $d_4=\sqrt{8/3}$. Optimality is proven [1].

$d_n$:

1.632993161855452065464856049803927594643964987104446752288461711500640251638210017693239622069760157

comment: The regular tetrahedron has $\cos(\pi\theta_4/180)=-1/3$ and $d_4=\sqrt{8/3}$. Optimality is proven [1].

$\theta_n$:

comment: $\theta_n=90$ and $d_n=\sqrt2$. Optimality is proven [2] [1].

$d_n$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $\theta_n=90$ and $d_n=\sqrt2$. Optimality is proven [2] [1].

$\theta_n$:

comment: $\theta_n=90$ and $d_n=\sqrt2$. Optimality is proven [2] [1].

$d_n$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $\theta_n=90$ and $d_n=\sqrt2$. Optimality is proven [2] [1].

$\theta_n$:

77.86954215542338680591828838037453919424590740642133591865073074314973538311781413484159439425902080

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $3x^{3}-9x^{2}-3x+1$ in $(0.210138312731-10^{-10},0.210138312731+10^{-10})$ [9]. Optimality is proven [2].

$d_n$:

1.256870468480660034904024544082533192335643253156436631660011061050399906071854844332135412434113098

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $3x^{3}-9x^{2}-3x+1$ in $(0.210138312731-10^{-10},0.210138312731+10^{-10})$ [9]. Optimality is proven [2].

$\theta_n$:

74.85849218561553500792611345605364079751084282049393387781847245319275920507463481339684227896154942

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{2}+2x-1$ in $(0.261203874964-10^{-10},0.261203874964+10^{-10})$ [9]. Optimality is proven [2].

$d_n$:

1.215562524131324605953884044732586661891605201667569759533735165691857679343105190624294468375968250

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{2}+2x-1$ in $(0.261203874964-10^{-10},0.261203874964+10^{-10})$ [9]. Optimality is proven [2].

$\theta_n$:

70.52877936550930863075400066003756403699315689909205171182889364396025356193960443663812871163941235

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $3x-1$ in $(0.333333333334-10^{-10},0.333333333334+10^{-10})$ [9]. Optimality is proven [2].

$d_n$:

1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $3x-1$ in $(0.333333333334-10^{-10},0.333333333334+10^{-10})$ [9]. Optimality is proven [2].

$\theta_n$:

66.14682198789054287220577533235752791223071708063365606081265231850570982361238460449340260549072983

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{3}-4x^{2}-2x+1$ in $(0.404394325217-10^{-10},0.404394325217+10^{-10})$ [9]. Optimality is proven [3].

$d_n$:

1.091426291403821042434682867757394743310099228970515891159497186619426980252598733761621464267204957

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{3}-4x^{2}-2x+1$ in $(0.404394325217-10^{-10},0.404394325217+10^{-10})$ [9]. Optimality is proven [3].

$\theta_n$:

63.43494882292201064842780627954670532879578570035477897201398368471193417851525938829142618939783528

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $5x^{2}-1$ in $(0.447213595500-10^{-10},0.447213595500+10^{-10})$ [9]. Optimality is proven [3].

$d_n$:

1.051462224238267212051338169695753214570995864486683563057871046482422292806428036743265257663105142

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $5x^{2}-1$ in $(0.447213595500-10^{-10},0.447213595500+10^{-10})$ [9]. Optimality is proven [3].

$\theta_n$:

63.43494882292201064842780627954670532879578570035477897201398368471193417851525938829142618939783528

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $5x^{2}-1$ in $(0.447213595500-10^{-10},0.447213595500+10^{-10})$ [9]. Optimality is proven [1].

$d_n$:

1.051462224238267212051338169695753214570995864486683563057871046482422292806428036743265257663105142

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $5x^{2}-1$ in $(0.447213595500-10^{-10},0.447213595500+10^{-10})$ [9]. Optimality is proven [1].

$\theta_n$:

57.13670307757834201767700274322533597760711636528224555516565534861346950278936664449157529803315396

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $x^{8}-24x^{7}-12x^{6}+8x^{5}+38x^{4}+24x^{3}-12x^{2}-8x+1$ in $(0.542636486830-10^{-10},0.542636486830+10^{-10})$ [9]. Optimality is proven [4].

$d_n$:

0.9564136272244990720875205665808557947269876360823935179447247145199889402438175360251024534129268150

$\theta_n$:

55.67056999565952572422946524468126528134983339933129590091570352836575510701258072290020366120459492

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $4x^{4}-2x^{3}+3x^{2}-1$ in $(0.563950300361-10^{-10},0.563950300361+10^{-10})$ [9]. Optimality is proven [5].

$d_n$:

0.9338626233440278688077841691123048999950999946977422766327774075751129850007292358699087911906913833

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $4x^{4}-2x^{3}+3x^{2}-1$ in $(0.563950300361-10^{-10},0.563950300361+10^{-10})$ [9]. Optimality is proven [5].

$\theta_n$:

53.65785012993267380552604148370299303169577018942255647465603609403395050000327059497239044377493759

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $13x^{5}-x^{4}+6x^{3}+2x^{2}-3x-1$ in $(0.592605902926-10^{-10},0.592605902926+10^{-10})$ [9]. Cohn marks this configuration as optimal, citing the 2026 preprint [7]; that proof claim is distinguished here from the established published cases.

$d_n$:

0.9026561882299663511276264760307219938239215614122841771344488235999651726273727841908295218026690448

$\theta_n$:

52.24439575272065377204341696894192408372344517847959576403435169874430504823078187364332134385647256

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $23x^{6}+6x^{5}+5x^{4}+4x^{3}-3x^{2}-2x-1$ in $(0.612294616483-10^{-10},0.612294616483+10^{-10})$ [9].

$d_n$:

0.8805741121760318749621097470943480162399443231658259154473708725447978848627326948539161468949108172

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $23x^{6}+6x^{5}+5x^{4}+4x^{3}-3x^{2}-2x-1$ in $(0.612294616483-10^{-10},0.612294616483+10^{-10})$ [9].

$\theta_n$:

51.09032855162777854657552387628875512696140109538242717714956916962016320304792956793687517468126625

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.8624448793169084073408845262983338825653241623297785110129646186099117287685098483793368096561656861

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

49.55665476843424246091024211381574183273081464780093027384950985046921349197025762588699604800221768

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.8382173557936908656657105762753363318752035308224810456154185138534639771572551133316250127721611622

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

47.69191410876624840141630606242048882455884334136696319819462117952532994830283236056716513807861562

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.8085581147202085326048189225383308759118591759714600683835929868359128245208954494958900105844163022

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

47.43103622671058794874110898739208056495251708472259489802058097854051803072851876177063163448720652

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $21x^{3}-9x^{2}-5x+1$ in $(0.676477138130-10^{-10},0.676477138130+10^{-10})$ [9].

$d_n$:

0.8043915239115197786798331800377488477071546465562427822662686942369865463034830685161557357134335496

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $21x^{3}-9x^{2}-5x+1$ in $(0.676477138130-10^{-10},0.676477138130+10^{-10})$ [9].

$\theta_n$:

45.61322311

$d_n$:

0.7752439215

$\theta_n$:

44.74016116716264539249224387535237040533603841148933744771629301307369978526494619388751795720465498

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.7611750671433078531679774887287235136368490028296560006502358555380840221519648322824553029290166432

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

43.70996421

$d_n$:

0.7445173136

$\theta_n$:

43.69076710767579618171644921518231448443934426057622589508560623880706548956391472616352325219452075

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{3}+x^{2}-3x-1$ in $(0.723078468334-10^{-10},0.723078468334+10^{-10})$ [9]. Optimality is proven [6].

$d_n$:

0.7442063311562075967551192250788305032776327199015088815823724075583781520771015804522782613296565679

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $7x^{3}+x^{2}-3x-1$ in $(0.723078468334-10^{-10},0.723078468334+10^{-10})$ [9]. Optimality is proven [6].

$\theta_n$:

41.63446125970606406767024214741580439206152387118051048191563247810318655999535762806209426164383366

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.7107761552303330599427222043473561053795135178734944886026131510343220056811048133280791217219247887

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

41.03766160675250218608721374440023245934261213702983507114556637137177434726900940710701626633877561

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.7010304171431628833141504526901419067964211066931217182959544662804611207665636461719674450599193749

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

40.67760068512674658008426875219181826237202623047945250561214975101720910728948644177296019178585591

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.6951414089570876888852481195205176391528617110516831083289331865074355279714766615139851438523810348

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

39.35514357

$d_n$:

0.6734533951

$\theta_n$:

38.71365119

$d_n$:

0.6629005718

$\theta_n$:

38.59711595386765606658303351176410722068094792113867161902240174454545275938897731504661824649471709

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.6609812779572696382718134325081396164782068706854392768941802836820873489371734658926068123699771659

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

37.70982914370989458237011117017694655887335581557542773980966980224966084907075333286193910627414486

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.6463457073444907495878761799859959746226221969240235534291369756195496148457204439825550738103842041

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

37.47521397487154131177437208730674048457529757692086861566553292771657980069388220402500232544887019

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.6424692757303458159434634750035208230179756704754858423655962813857269946389544403723583796118628602

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

36.25455298

$d_n$:

0.6222578024

$\theta_n$:

35.80778440

$d_n$:

0.6148425210

$\theta_n$:

35.31980759

$d_n$:

0.6067326075

$\theta_n$:

35.18973226

$d_n$:

0.6045689617

$\theta_n$:

34.42240801

$d_n$:

0.5917896920

$\theta_n$:

34.25066067222863122268183736668475604863630632810011398334707877088116648114477850843863955089954015

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.5889257008796497051926213773453604457703941908248557329352616866483774123559375006793547196752065076

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

33.48904658

$d_n$:

0.5762094716

$\theta_n$:

33.15835626

$d_n$:

0.5706801688

$\theta_n$:

32.72909442

$d_n$:

0.5634956202

$\theta_n$:

32.50638634974149291094556186886744874622303767263225252061844035564707386377695798244050963969050693

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.5597650370666593981132760330640500424180665121937637350403067449216294517833435746995413308835462916

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

32.09062441

$d_n$:

0.5527949612

$\theta_n$:

31.98342303

$d_n$:

0.5509965911

$\theta_n$:

31.32308143

$d_n$:

0.5399083736

$\theta_n$:

30.95916349

$d_n$:

0.5337899084

$\theta_n$:

30.78181596

$d_n$:

0.5308062520

$\theta_n$:

30.76278555063470255272279475166525256460496249397517001625032210918509447938903220881838309834643441

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $34x^{5}+15x^{4}-12x^{3}-14x^{2}-6x-1$ in $(0.859292295076-10^{-10},0.859292295076+10^{-10})$ [9].

$d_n$:

0.5304860128680869911358365318309447191565187367298696885292842813336946860025663196956216413275569364

comment: With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $34x^{5}+15x^{4}-12x^{3}-14x^{2}-6x-1$ in $(0.859292295076-10^{-10},0.859292295076+10^{-10})$ [9].

$\theta_n$:

29.92358511

$d_n$:

0.5163497283

$\theta_n$:

29.75295639697365612330731744297445625024915826185666920679022507054042258853757167946710208712158170

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.5134720846805647881992541205653954233345131537543003582529503063769833734110944962677356138605248128

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

29.36840688

$d_n$:

0.5069825146

$\theta_n$:

29.19475790534553237106281307477898182637601182239826492917397555907891150912476625291156229181265596

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$d_n$:

0.5040501784938189268309565236802392569339516330211263126733164376069960843951079997625282798520778085

comment: The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py [9].

$\theta_n$:

28.81389721

$d_n$:

0.4976147024

$\theta_n$:

28.71692053

$d_n$:

0.4959751882

$\theta_n$:

28.26279142

$d_n$:

0.4882928496

$\theta_n$:

28.14804665

$d_n$:

0.4863505353

$\theta_n$:

27.82667595

$d_n$:

0.4809080211

$\theta_n$:

27.55641596

$d_n$:

0.4763281533

$\theta_n$:

27.39497567

$d_n$:

0.4735910955

$\theta_n$:

27.19283000

$d_n$:

0.4701625940

$\theta_n$:

26.87327787

$d_n$:

0.4647398341

$\theta_n$:

26.68399700

$d_n$:

0.4615260544

$\theta_n$:

26.48692251

$d_n$:

0.4581786097

$\theta_n$:

26.23504331

$d_n$:

0.4538982980

$\theta_n$:

26.06982995

$d_n$:

0.4510895508

$\theta_n$:

25.94744369

$d_n$:

0.4490082906

$\theta_n$:

25.68398135

$d_n$:

0.4445262022

$\theta_n$:

25.46382446

$d_n$:

0.4407790342

$\theta_n$:

25.33363644

$d_n$:

0.4385624098

$\theta_n$:

25.17091998

$d_n$:

0.4357911500

$\theta_n$:

24.98793806

$d_n$:

0.4326736951

$\theta_n$:

24.92648608

$d_n$:

0.4316264926

$\theta_n$:

24.55377925

$d_n$:

0.4252725511

$\theta_n$:

24.42093978

$d_n$:

0.4230068002

$\theta_n$:

24.30172251

$d_n$:

0.4209729097

$\theta_n$:

24.12819444

$d_n$:

0.4180116432

$\theta_n$:

24.00128368

$d_n$:

0.4158452965

$\theta_n$:

23.93102542

$d_n$:

0.4146457795

$\theta_n$:

23.62399170

$d_n$:

0.4094019799

$\theta_n$:

23.55306720

$d_n$:

0.4081902479

$\theta_n$:

23.34763768

$d_n$:

0.4046796417

$\theta_n$:

23.19460741

$d_n$:

0.4020636457

$\theta_n$:

23.08299764

$d_n$:

0.4001552655

$\theta_n$:

23.05173064

$d_n$:

0.3996205728

$\theta_n$:

22.77916207

$d_n$:

0.3949581589

$\theta_n$:

22.67436939

$d_n$:

0.3931650346

$\theta_n$:

22.54665743

$d_n$:

0.3909792903

$\theta_n$:

22.46788105

$d_n$:

0.3896308186

$\theta_n$:

22.31660235

$d_n$:

0.3870407572

$\theta_n$:

22.15402326

$d_n$:

0.3842564682

$\theta_n$:

22.05179633

$d_n$:

0.3825053589

$\theta_n$:

22.02758147

$d_n$:

0.3820905227

$\theta_n$:

21.81038013

$d_n$:

0.3783687833

$\theta_n$:

21.72371348

$d_n$:

0.3768833725

$\theta_n$:

21.59455006

$d_n$:

0.3746691942

$\theta_n$:

21.52060993

$d_n$:

0.3734014642

$\theta_n$:

21.40065028

$d_n$:

0.3713443829

$\theta_n$:

21.37106074

$d_n$:

0.3708369155

$\theta_n$:

21.13596738

$d_n$:

0.3668041349

100

$\theta_n$:

21.03120200

100

$d_n$:

0.3650064961

Definition

The Tammes problem maximises the minimum separation of $n$ points on the unit sphere in $\mathbb R^3$ [5]. Best known separations are recorded in two invertible normalisations: angle $\theta_n$ in degrees and chord length $d_n$.

Parameters

$n$

— number of points ($n\geq2$)

normalisation

— angular or chord separation (either angle or chord)

Formulas

(1)

$d_n=2\sin(\pi\theta_n/360)$ and $\theta_n=(360/\pi)\arcsin(d_n/2)$, with $0<\theta_n\leq180$ and $0<d_n\leq2$.

(2)

For the selected configuration $x_1,\ldots,x_n\in S^2$, let $c_n=\max_{i<j}\langle x_i,x_j\rangle$. Then $\theta_n=(180/\pi)\arccos(c_n)$ and $d_n=\sqrt{2-2c_n}$.

Comments

(3)

The snapshot was retrieved on 25 October 2026. For $7\leq n\leq100$, the configurations and algebraic descriptions are from Henry Cohn's table [9] [10]. For $4\leq n\leq6$, the source is the table of N. J. A. Sloane, R. H. Hardin, W. D. Smith and collaborators [11]. The attached source data preserve the coordinates, source addresses and file hashes.

(4)

Angles are measured in degrees. The angular separation in radians is $\pi\theta_n/180$. Chord lengths are measured on the sphere of radius one; the two normalisations describe the same separation.

(5)

Optimality is established for $4\leq n\leq14$ and $n=24$: by Fejes Tóth for $n=4,6,12$ [1], Schütte and van der Waerden for $n=5,7,8,9$ [2], Danzer for $n=10,11$ [3], Musin and Tarasov for $n=13,14$ [4] [5], and Robinson for $n=24$ [6]. Cohn also marks $n=15$ as optimal, citing the 2026 preprint [7]; this newer claim is recorded separately from the established published cases. The other entries are best known configurations, without an assertion of global optimality.

(6)

An exact algebraic description of an attainable separation does not prove its optimality. Cohn supplies such descriptions as minimal polynomials of $c_n=\cos(\pi\theta_n/180)$ [10]. These descriptions support the high precision rows even for configurations whose global optimality is not established.

(7)

For $n=97$, the snapshot uses the configuration credited to Lai and collaborators [8], with $\theta_{97}\approx21.40065028$ degrees. This improves the value $21.4006197$ degrees in the older Sloane table [11].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
R = RealBallField(numberdb.bits(100, losing=64))
# The regular tetrahedron, n = 4.
c = -R(1)/3
theta = 180*c.arccos()/R.pi()
d = (2-2*c).sqrt()
theta, d

References

[1]

L. Fejes Tóth, Über eine Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jahresbericht der Deutschen Mathematiker-Vereinigung 53 (1943), 66–68.

[2]

K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Mathematische Annalen 123 (1951), 96–124. (doi)

[3]

L. Danzer, Finite point-sets on S² with minimum distance as large as possible, Discrete Mathematics 60 (1986), 3–66. (doi)

[4]

O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete & Computational Geometry 48 (2012), 128–141. (arXiv) (doi)

[5]

O. R. Musin and A. S. Tarasov, The Tammes problem for N = 14, Experimental Mathematics 24 (2015), 460–468. (arXiv) (doi)

[6]

R. M. Robinson, Arrangements of 24 points on a sphere, Mathematische Annalen 144 (1961), 17–48. (doi)

[7]

mt0-svg, The Tammes problem for fifteen points, preprint (2026). (doi)

[8]

X. Lai, D. Yue, J.-K. Hao, F. Glover and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Computers & Operations Research 151 (2023), 106121. (doi)

Links

[9]

Henry Cohn, Table of spherical codes (archived dataset)

[10]

Table of spherical codes: current coordinates and minimal polynomials

[11]

N. J. A. Sloane, with R. H. Hardin, W. D. Smith and others, Tables of Spherical Codes

Similar tables

Best known packings of equal circles in a circle — maximises separation in a disk rather than on a sphere

Best known packings of equal circles in an equilateral triangle — maximises separation in a triangular planar container

Global minimum energies of Lennard-Jones clusters — optimises a pair-potential energy for finite point configurations rather than their minimum separation

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds both normalisations for every $4\leq n\leq100$, a contiguous reference range of small spherical codes from the cited snapshot; larger configurations are not included)

Sources of data: [9], [10], [11]

How they were obtained:

The 27 cases with a source minimal polynomial, and the elementary cases $n=4,5,6$, are evaluated to 100 significant digits. Polynomial roots are isolated by 400 exact rational bisections after a rational derivative bound certifies uniqueness within $10^{-10}$ of the printed source cosine. The angle and chord conversions use 768-bit real ball arithmetic; the smallest measured relative accuracy is 431 bits. The exact angles for $n=5,6$ are stored as integers. For the remaining 67 configurations, the source cosine is enclosed by one unit in its twelfth decimal place and converted to ten significant digits. Every written decimal contains the whole transformed input ball. This is a source-accuracy model supported by agreement, not a bound on the global optimum.

All 97 configurations were independently checked from their published coordinates after radial normalisation, by computing every pairwise inner product and Euclidean distance. For $n\geq7$, the recomputed cosines differ from the source decimals by less than $10^{-12}$; the lower precision Sloane coordinates for $n=4,5,6$ agree with the exact cosines within $10^{-8}$. Known-answer controls use antipodal points, a regular tetrahedron and an octahedron. Both angle/chord conversions were checked on all 194 written values, together with closed forms for the tetrahedron, square antiprism, $n=9$, octahedron and icosahedron. The coordinate check supports the available source digits; the higher precision of algebraic rows comes from the root enclosures. These computations do not reproduce the source optimisations or their optimality proofs.