Landon Noll looking up Fremont Peak Observatory 0.8m telescope Leonid 2001 meteor squall count at Fremont Peak

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:

  • m and n are integers > 1
  • n lattice points in Rm
  • for all integer 0 < k < m, no k+2 points lie in k-dimensional plane (a k-dimensional affine subspace of Rm)
  • for all integer 0 < l < m, no l+3 points lie on the surface of an l-dimensional sphere
  • all mutual distances between the n lattice points are non-zero integers

nm-cluster related terminology:

  • prime nm-cluster: an nm-cluster where the greatest common divisor (gcd) of the mutual distances = 1
  • primitive nm-cluster: A prime nm-cluster that has been rotated, reflected and translated into canonical form
    (Note to web page editor: define the canonical form here)
  • equivalent nm-cluster: two nm-clusters are equivalent if and only if they can be made identical under the combined operations of rotation, reflection and translation
  • nonequivalent nm-cluster: two nm-clusters that are not equivalent
    (all primitive nm-clusters are both prime and nonequivalent)
  • n-cluster: shorthand for n2-cluster
  • n(m)-cluster: alternate notation for nm-cluster used in places where sub-scripting is not available or desirable


n2-clusters

An n2-cluster is:
  • n > 1 lattice points (points with integer (x,y) coordinates) in R2
  • no 3 points lie on a straight line
  • no 4 points lie on a circle
  • all mutual distances between points are integers > 0
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:
  • n lattice points (points with integer (x,y,z) coordinates) in R3
  • no 3 points lie on a straight line
  • no 4 points lie on a flat plane
  • no 4 points lie on a circle
    NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.
  • no 5 points lie on the surface of a sphere
  • all mutual distances between points are integers > 0.
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:

  • n lattice points (points with integer (x,y,z,w) coordinates) in R4
  • no 3 points lie on a straight line
  • no 4 points lie on a plane
  • no 5 points lie on a hyper-plane (an R4 plane)
  • no 4 points lie on a circle
    NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.
  • no 5 points lie on the surface of a sphere
    NOTE: If no 5 points lie on a hyper-plane (an R4 plane), then no 5 points lie on the surface of a sphere and therefore the above test may be skipped.
  • no 6 points lie on the surface of a hyper-sphere (an R4 sphere)
  • all mutual distances between points are integers > 0.

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:

  • n lattice points (points with integer (x,y,z,w,v) coordinates) in R5
  • no 3 points lie on a straight line
  • no 4 points lie on a plane
  • no 5 points lie on a hyper-plane (an R4 plane)
  • no 6 points lie on a 5-flat (an R5 plane)
  • no 4 points lie on a circle
    NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.
  • no 5 points lie on the surface of a sphere
    NOTE: If no 5 points lie on a hyper-plane (an R4 plane), then no 5 points lie on the surface of a sphere and therefore the above test may be skipped.
  • no 6 points lie on the surface of a hyper-sphere (an R4 sphere)
    NOTE: If no 6 points lie on a 5-flat (an R5 plane), then no 6 points lie on the surface of a hyper-sphere and therefore the above test may be skipped.
  • no 7 points lie on the surface of a 5-sphere (an R5 sphere)
  • all mutual distances between points are integers > 0.

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:

  • Erdös/Noll infinite-or-bust nm-cluster conjecture:
    For any m > 1, n > 2, there exists either 0 or an infinite number of primitive nm-clusters.
  • Noll infinite nm-cluster conjecture:
    For any m > 1, n > 2, there exists an infinite number of primitive nm-clusters.
  • Noll/Rathbun computation observation:
    For any m > 1:
    • One will be able to find many (m+3)m-clusters;
    • With bit more effort, one will also find some: (m+4)m-clusters;
    • However, finding an (m+5)m-cluster will be a significant computational challenge.

Open questions about nm-clusters:

  • No 82-cluster has ever been found. Do 82-clusters exist?
  • No 83-cluster has ever been found. Do 83-clusters exist?
  • No 94-cluster has ever been found. Do 94-clusters exist?
  • No 105-cluster has ever been found. Do 105-clusters exist?

For more information see:



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