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N-clusters

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General information about nm-clusters

An n2-cluster is n > 1 lattice points in R2 such that no 3 are co-linear and no 4 are co-circular and all mutual distances between points are integers > 0.

In other words, a 2-dimension n2-cluster is a collection of n lattice (grid points with integer (x,y) coordinates) on a flat plane such that no 3 lie on a straight line and no 4 lie on a circle and all of the distances between each pair of points are whole numbers > 0.

One need not be restricted to the R2. We define nm-clusters in Rm as follows:

nm-cluster related terminology:


n2-clusters

An n2-cluster is: Here is an example of a 62-cluster:
(0,0) (546,272) (132,720) (960,720) (546,-1120) (1155,-540)

The following are the smallest n2-clusters:

(0,0) (1,0)

(0,0) (3,0) (0,4)

(0,0) (3,4) (3,-4) (6,0)

(0,0) (16,30) (-16,30) (0,-33) (56,0)

(0,0) (546,272) (132,720) (960,720) (546,-1120) (1155,-540)

A picture of the smallest 62-cluster is available.

UPDATE: 72-clusters DO exist!

Chuck Simmons and Landon Curt Noll discovered a number of 72-clusters on 2006-May-18 15:26:55 PDT:

first 7-cluster in R^2 discovered 2006-May-18 15:26:55 PDT

(0,0) (327990000,0) (238776720,118951040) (222246024,-103907232)
(243360000,21896875) (198368352,50379264) (176610000,-94192000)

NOTE: This 72-cluster is not be the smallest. The above mentioned 72-clusters simply prove that these strutures exist. The smallest 72-cluster is:

smallest 7-cluster in R^2

(0,0) (374400,-2230800) (1081600,-1488240) (-453024,-1630200)
(426725,-1630200) (569088,-1291680) (-439040,-1308720)

As recent as 2010-August-26, using an improved search algorithm, 25 nonequivalent (new) 72-clusters were discovered:

(68634995347500,46759948729375) (85941805950000,71210832840000) (68634995347500,4632341698125)
(130072970722500,48916330870000) (133822723450416,98216877284288) (137269990695000,0) (0,0)

(68634995347500,46759948729375) (51328184745000,71210832840000) (68634995347500,4632341698125)
(7197019972500,48916330870000) (3447267244584,98216877284288) (0,0) (137269990695000,0)

(8742402935396,-10550893265055) (12491072799192,-7329880708800) (1251972023796,-12174625385247)
(8742402935396,590045900847) (17484805870792,0) (0,0) (5153771043792,-23774836660350)

(8742402935396,-10550893265055) (4993733071600,-7329880708800) (16232833846996,-12174625385247)
(8742402935396,590045900847) (0,0) (17484805870792,0) (12331034827000,-23774836660350)

(57074946589137,-29602439091140) (68119373772450,-49995038226600) (57074946589137,6361294356684)
(108165052231587,-40677455540084) (96768547697250,-80181802070200) (114149893178274,0) (0,0)

(57074946589137,-29602439091140) (46030519405824,-49995038226600) (57074946589137,6361294356684)
(5984840946687,-40677455540084) (17381345481024,-80181802070200) (0,0) (114149893178274,0)

(127760562354100,174076664586825) (189715358500200,126876727286400) (18296188244100,177918702348075)
(127760562354100,14239567321200) (255521124708200,0) (0,0) (12519023151384,356682343821888)

(127760562354100,174076664586825) (65805766208000,126876727286400) (237224936464100,177918702348075)
(127760562354100,14239567321200) (0,0) (255521124708200,0) (243002101556816,356682343821888)

(1481541457500,-537780709375) (1665923415000,0) (832961707500,-1005271678125)
(832961707500,432023150000) (1550362069464,1036842072448) (0,0) (1523288063400,-2113353748800)

(184381957500,-537780709375) (0,0) (832961707500,-1005271678125)
(832961707500,432023150000) (115561345536,1036842072448) (1665923415000,0) (142635351600,-2113353748800)

(5272996965444,-1914030845545) (5929236112488,0) (2964618056244,-4039359357033)
(2964618056244,2019747908633) (6173748624888,3622814603200) (0,0) (6745851981888,-7326859050450)

(5272996965444,-1914030845545) (5929236112488,0) (2964618056244,-4039359357033)
(2964618056244,2019747908633) (6173748624888,3622814603200) (0,0) (6745851981888,-7326859050450)

(656239147044,-1914030845545) (0,0) (2964618056244,-4039359357033)
(2964618056244,2019747908633) (-244512512400,3622814603200) (5929236112488,0) (-816615869400,-7326859050450)

(1481541457500,-537780709375) (1665923415000,0) (832961707500,-1005271678125)
(832961707500,432023150000) (1550362069464,1036842072448) (0,0) (1523288063400,-2113353748800)

(184381957500,-537780709375) (0,0) (832961707500,-1005271678125)
(832961707500,432023150000) (115561345536,1036842072448) (1665923415000,0) (142635351600,-2113353748800)

(68634995347500,46759948729375) (85941805950000,71210832840000) (68634995347500,4632341698125)
(130072970722500,48916330870000) (133822723450416,98216877284288) (137269990695000,0) (0,0)

(68634995347500,46759948729375) (51328184745000,71210832840000) (68634995347500,4632341698125)
(7197019972500,48916330870000) (3447267244584,98216877284288) (0,0) (137269990695000,0)

(8742402935396,-10550893265055) (12491072799192,-7329880708800) (1251972023796,-12174625385247)
(8742402935396,590045900847) (17484805870792,0) (0,0) (5153771043792,-23774836660350)

(8742402935396,-10550893265055) (4993733071600,-7329880708800) (16232833846996,-12174625385247)
(8742402935396,590045900847) (0,0) (17484805870792,0) (12331034827000,-23774836660350)

(57074946589137,-29602439091140) (68119373772450,-49995038226600) (57074946589137,6361294356684)
(108165052231587,-40677455540084) (96768547697250,-80181802070200) (114149893178274,0) (0,0)

(57074946589137,-29602439091140) (46030519405824,-49995038226600) (57074946589137,6361294356684)
(5984840946687,-40677455540084) (17381345481024,-80181802070200) (0,0) (114149893178274,0)

(127760562354100,174076664586825) (189715358500200,126876727286400) (18296188244100,177918702348075)
(127760562354100,14239567321200) (255521124708200,0) (0,0) (12519023151384,356682343821888)

(127760562354100,174076664586825) (65805766208000,126876727286400) (237224936464100,177918702348075)
(127760562354100,14239567321200) (0,0) (255521124708200,0) (243002101556816,356682343821888)

(214219880550,6301947600) (224858052864,-26545742352) (175084785954,0)
(121350268260,-110224651680) (134680604580,-179574139440) (0,0) (345929626949,-494479514400)

(617451851250,-875785365000) (780757128000,-819794984400) (525254357862,-700339143816)
(804949411500,-154727118000) (1122338371500,0) (0,0) (3015706938447,99720035404)

(1009529065706589428,0) (1203398391990949336,45860386955466720) (0,0)
(676939921408158408,985583027337548640) (53761902907451508,-563219935220920560)
(1246841707833420832,1354229751267244425) (2606346791263681648,-1834662143896632960)

(220244176831120,0) (246555468595800,12985596405600) (165138740488800,35797817293056)
(187901542064600,73409269700325) (242381054588400,120746605948800) (95836927172560,134624270161920) (0,0)

(1541709237817840,0) (1725888280170600,90899174839200) (1155971183421600,250584721051392)
(1315310794452200,513864887902275) (1982197591480080,1057172048789376) (670858490207920,942369891133440) (0,0)

(0,0) (-1203015538725,-2544411343500) (10749021956400,0) (19954499237400,4909587883200)
(11374103163150,-17184050629200) (23379026610072,-4158083198304) (24966478323150,2041480914200)

(1163564931724079313,-269477403542383960) (834417531683024619,0) (1455548309962875744,134645123549696760)
(270089852186798469,-710440427520830440) (945507656505731319,829472932009543360) (0,0)
(540705461390553459,-3959673836535537120)

(2839035903168,420142221576) (2716494421875,0) (3488742000000,411865375000)
(678132000000,-760880250000) (1889735250000,-1984222012500) (3053812164000,-2179591564500) (0,0)

(888913935,1185218580) (732147500,1624350000) (750638434,291746112)
(0,0) (1899388750,0) (1508338125,-938521500) (3000817152,354263136)

(90559954461600,-56348416109440) (66832836431400,-65671968304800) (99758004738288,0)
(62274616513200,7351864449475) (0,0) (76736926721760,102315902295680) (219297144627600,85232898316800)

(645746712,1432660320) (1133937120,1511916160) (920228400,108638075)
(0,0) (1474118256,0) (1338199200,-832657280) (3240541200,1259481600)

(46093983313228774560,7133592655618738920) (51205858532566473645,14766053479192164552) (30836021244266471870,0)
(37713496340872872000,-14317464299263259160) (48652889618418976800,-16680990726315077760)
(23093827308239198310,-11696944424489686080) (0,0)

(204525733500,0) (181222195380,31071384160) (116714131380,-89881235840)
(64902832764,110138140448) (107446130880,167222282535) (0,0) (195966456000,492121242000)

(1537489668,2659073040) (720014022,3200062320) (0,0)
(3448272282,0) (-1907771943,1360265200) (8837240832,1733306400) (970646382,-5946302160)

(1861671240,2343685344) (1416176720,1989335040) (2776610200,1084764525)
(3581650800,1784265600) (2440245600,528982272) (3254535440,0) (0,0)

(13031698680,16405797408) (9913237040,13925345280) (19436271400,7593351675)
(17081719200,3702875904) (29290818960,15621770112) (22781748080,0) (0,0)

(18880367688,-8041182240) (19189941309,-15711063060) (20689633692,0)
(28850942484,-6681773280) (26355138250,5713113840) (37802741526,-10726662720) (0,0)

(5722762500,7204470000) (6540315210,9187326720) (6160119966,3068777712)
(8604820224,3361726368) (10778796875,2336565000) (8461703250,0) (0,0)

(5722762500,7204470000) (6540315210,9187326720)
(5123868750,2732730000) (8604820224,3361726368) (10778796875,2336565000) (8461703250,0) (0,0)

(154949507036870777,-230980631607757680) (69052004404618851,-181633390270652760) (91144391325948576,-327981711210799800)
(176997658235793101,0) (-102311826723971249,-274457955444334200) (0,0) (906675315979444011,209982703452642120)

(4959471309432841342089,0) (5145703589278133161956,-1817658122369063017680) (2728622318116179176640,819077651069860662480)
(3631649471438236223040,2723737103578677167280) (5383609759475803626444,3592702165069798173360) (0,0)
(50220507702021572864,1908048894599832784800)

(335600575349000,223959939060000) (460196641548696,345147481161522) (526540779804000,0)
(451186570951000,-159376248780000) (0,0) (877918993526536,263533660291902) (35997201279000,1367656824909375)

(220540073481,238652060192) (164939775000,277533828000) (270450480300,0)
(0,0) (-140026136250,336062727000) (299378956800,679511448000) (760141882500,493806456000)

(3749181249177,4057085023264) (2803976175000,4718075076000) (4597658165100,0)
(0,0) (-1487477641650,8113514409000) (5089442265600,11551694616000) (12922412002500,8394709752000)

(21945462903,10830898260) (21142072098,0) (37836432900,12972491280)
(0,0) (8666936388,38519717280) (54627962688,-22761651120) (60508055894,68284953360)

(122766861254427,47715055386336) (139881033166875,0) (0,0)
(100323715366875,205792236650000) (274032673366875,100613730150000) (-22482696967185,190441668427920)
(-38698090055625,-377699815350000)

(90026022240,262575898200) (0,0) (473881430880,419911601252)
(-110715848880,631802398500) (427936195995,0) (680445131520,779676713200) (1105723338720,460718057800)
Chuck Simmons made images of the 25 72-clusters found on 2010-August-26 in a blog posting:
Pictures and coordinates of twenty-five recently found 7-clusters
For more information see:
N-Cluster Search: Progress Update (as of 2010-Oct)

It should be noted that Tobias Kreisel and Sascha Kurz reported the discovery of two 7-element integral heptagons. While these 7-element integral heptagons are 7 points in R2, no 3 co-linear and no 4 are co-circular and all mutual distances between points are integers > 0, their coordinates are not lattice points in R2. Because some of their points are not lattice points in R2 (i.e., some points do not have integer (x,y) coordinates) they are not 72-clusters:

There are Integral Heptagons, No Three Points on a Line, No Four on a Circle

Even so, this above work and results of Kruz and Kreisel is impressive and is worth reading.


n3-clusters

An n3-cluster is: Here are some 73-clusters recently discovered by Randall Rathbun:
(0,0,0) (54,28,12) (30,-20,60) (12,-56,-72) (-3,-76,-12) (-30,-84,12) (-72,0,96)

(0,0,0) (21,20,0) (-87,-4,-72) (108,24,-72) (126,-120,0) (-3,-172,96) (24,192,96)

The following are the smallest n3-clusters:

(0,0,0) (1,0,0)

(0,0,0) (2,2,1) (2,2,-1)

(0,0,0) (2,2,1) (2,2,-1) (0,4,0)

(0,0,0) (6,6,3) (6,-6,3) (8,0,0) (8,0,6)

(0,0,0) (12,12,6) (-8,0,15) (-4,12,18) (20,0,15) (8,24,12)

(0,0,0) (54,28,12) (30,-20,60) (12,-56,-72) (-3,-76,-12) (-30,-84,12) (-72,0,96)


n4-clusters

A n4-cluster is:

Randall and I (well Randall did all of the coding and I kibitzed and theorized on the side :-)) found these 74-clusters on 8 June 2001:

(0,0,0,0) (8,4,1,0) (0,8,-4,-8) (8,-4,5,-8) (-4,-4,11,4) (4,0,-6,-12) (12,8,-10,4)

(0,0,0,0) (8,8,4,0) (8,-8,4,0) (0,12,6,-4) (0,0,-9,12) (12,0,9,8) (8,4,10,12)

(0,0,0,0) (4,4,4,1) (4,-4,4,1) (-8,-4,4,10) (0,0,0,12) (0,12,-4,6) (0,12,12,6)

(0,0,0,0) (5,5,5,5) (8,-8,8,8) (-3,9,3,1) (5,1,11,-7) (0,-6,12,4) (0,-6,12,-12)

(0,0,0,0) (4,4,4,1) (-4,4,4,1) (-4,-8,4,10) (0,0,0,12) (12,0,-4,6) (12,0,12,6)

(0,0,0,0) (10,8,2,1) (10,-8,2,1) (0,0,0,12) (12,0,4,-6) (-6,-12,10,9) (12,0,-12,6)

and this 84-clusters on 10 June 2001:

(0,0,0,0) (8,8,8,8) (8,-10,4,-4) (0,-6,-12,4) (12,-3,-6,-6) (0,-9,-6,-18) (-12,-15,18,-6) (8,-25,-22,14)

The following are the smallest n4-clusters:

(0,0,0,0) (1,0,0,0)

(0,0,0,0) (1,1,1,1) (1,1,1,-1)

(0,0,0,0) (1,1,1,1) (1,1,-1,1) (0,0,0,2)

(0,0,0,0) (4,3,0,0) (-4,3,0,0) (0,2,4,-4) (4,1,4,4)

(0,0,0,0) (4,2,2,1) (-4,2,2,1) (0,0,0,6) (0,6,2,3) (0,6,-6,3)

(0,0,0,0) (8,4,1,0) (0,8,-4,-8) (8,-4,5,-8) (-4,-4,11,4) (4,0,-6,-12) (12,8,-10,4)

(0,0,0,0) (8,8,8,8) (8,-10,4,-4) (0,-6,-12,4) (12,-3,-6,-6) (0,-9,-6,-18) (-12,-15,18,-6) (8,-25,-22,14)


n5-clusters

A n5-cluster is:

Randall and I (as before) co-discovered these 85-clusters:

(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (13,6,2,8,4) (8,6,6,2,2)

(0,0,0,0,0) (6,6,6,6,0) (4,4,8,8,3) (10,2,10,6,4) (8,4,12,4,4) (4,8,8,12,1) (4,4,16,8,3) (8,4,12,8,1)

(0,0,0,0,0) (5,4,2,2,0) (0,8,4,8,0) (0,10,2,6,2) (5,14,0,8,2) (1,6,12,12,6) (8,10,2,14,6) (1,8,2,6,4)

(0,0,0,0,0) (5,4,2,2,0) (0,8,4,8,0) (0,10,2,6,2) (5,10,8,8,6) (1,12,6,6,12) (8,10,2,14,6) (1,6,4,8,2)

(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (9,14,2,8,4) (8,6,6,2,2)

(0,0,0,0,0) (8,4,4,4,3) (8,0,0,8,4) (6,10,2,6,7) (16,0,0,0,0) (10,2,10,6,7) (8,4,16,4,3) (8,4,4,8,6)

(0,0,0,0,0) (6,6,6,6,0) (0,12,4,8,1) (10,10,2,6,4) (8,12,4,8,1) (8,12,0,12,3) (4,16,4,8,3) (4,8,4,8,3)

(0,0,0,0,0) (6,6,6,6,0) (4,4,8,8,3) (10,2,10,6,4) (4,8,8,12,1) (8,0,12,12,3) (4,4,16,8,3) (8,4,12,8,1)

(0,0,0,0,0) (12,8,4,4,4) (12,5,10,2,4) (12,10,4,0,8) (12,6,12,0,0) (16,7,2,6,4) (4,13,10,10,12) (8,7,2,6,4)

(0,0,0,0,0) (8,0,0,0,0) (4,12,4,4,8) (8,10,10,4,3) (4,10,2,0,13) (12,8,4,0,10) (4,4,12,12,16) (4,8,4,8,6)

as well as these 95-clusters on 26 July 2001:

(0,0,0,0,0) (2,0,-2,4,-5) (4,-4,-4,4,0) (4,4,-8,0,-2) (-4,-4,0,8,-2) (-8,8,-4,4,6)
(-4,4,-4,12,8) (-8,-8,-8,0,-8) (0,2,-4,6,-5)

(0,0,0,0,0) (0,8,0,0,0) (4,4,0,4,-4) (0,6,-3,6,0) (0,6,5,-2,-4) (8,4,-10,0,4) (8,4,-6,-8,-4)
(0,-4,-10,-8,-4) (-8,4,-6,-8,-4)

The following are the smallest n5-clusters:

(0,0,0,0,0) (1,0,0,0,0)

(0,0,0,0,0) (1,1,1,1,0) (0,0,0,2,0)

(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2)

(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2) (0,0,0,0,4)

(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2) (0,0,0,0,4) (0,4,1,2,2)

(0,0,0,0,0) (5,4,2,2,0) (5,2,0,4,2) (0,4,4,4,4) (0,0,0,8,0) (5,6,0,4,2) (6,4,4,4,4)

(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (13,6,2,8,4) (8,6,6,2,2)

(0,0,0,0,0) (2,0,-2,4,-5) (4,-4,-4,4,0) (4,4,-8,0,-2) (-4,-4,0,8,-2) (-8,8,-4,4,6) (-4,4,-4,12,8) (-8,-8,-8,0,-8)
(0,2,-4,6,-5)


open nm-cluster questions, conjectures & observations

nm-cluster conjectures:

Open questions about nm-clusters:

For more information see:







© 1994-2022 Landon Curt Noll
chongo (was here) /\oo/\
$Revision: 8.1 $ $Date: 2022/07/07 23:59:44 $