Somaybe I havetotellyouwhatthetryinglatestfuriousWell, intuitivelyitis a sphere.
Butitissmadecombinatorialsothateveryfaceis a triangle.
Soyouwanttoattachedbunchoftrianglesinsuch a waythattheentireshapeisessentiallyidenticalto a sphere.
Youcancontinuouslymoveittoform a makeitround.
Somaybetheeasiestexample a ce a simplex.
Sosimplexhasfourvorticesonetoo 34 Andifyoujustlookattheboundaryofitthetetrahedron, thenitisessentially a spherecancontinuouslydeforminto a sphere.
Everyfaceofitis a triangle.
Infact, youseefourofthem 123 andfour.
So, asyoucanimagine, therearemany, manydifferentwaystotriangulatethisfear.
Anotherexample, maybe, isthesecondeasiestexampleistheboundaryofoctahedronisessentially a sphereand I cangoon.
MaybethethirdformulaexampleistheboundaryofIkewas a hydrantwhere I gluedtogether 20 trianglestomakeusfeel e Soyou'relessformulasayssomethingabouttryinglatestfears.
Weknowverywellhowtypicaltriangulatedonedimensionalsphereslooklikethis, thisonetriangulatedonespherewhere I usedfivearetoceaseandfivebudgets.
Somynumberswillbe F zeroequalsfiveand F oneequalsfiveandtheoilistrelation.
Inthiscase, we'lllooklikethis.
Andagain, thenumberzerothatweseehere, ISSindependentoffthetriangleizationofftheonedimensionalspherethat I havechosen.
Forexample, ifyouhaveusedsevenvortices, youareforcedtousesevenedgestoo, becauseyouwanttheentirethingtobe a onedimensionalspherewhichis a circle.
Sowiththealwaysrelationinonedimensionissimplysayingthatabzerominus f 10 Solet's goevenlowerevenandthinkaboutzerodimensionalspears.
Sothisis a littlebittrickyifyouhaveneverthoughtaboutzerodimensionalspheres.
Butwhatisreally a spheresphereissupposedtobethatsomethingthatlivesinourspace, whichconsistsof a llpointsthatareoffthesamedistancefromtheoriginorwhateverpointthatyouhavechosen.
Sothezerodimensionalsphereshouldlivein a onedimensionalspace, whichistheline.
Andif I collectallpointsthatareoff, saydistanceonefromBurrageinthen I wouldgetexactly 2.0 dimensionalspearistwopoint, andthat's well, inmysense, that's a trianglewiththisfear.
Andthereisreallyonenumber F zeroThenumberofearthisisandthatnumberofartistsshouldbealwaystoo.
So, inthiscase, theoldersformulawilllooklikeAPzerobecausetwoit's goingupagain.
That's right.
So I hadtoinDimensionzerozeroinDimensionOneandtwoinDimensiontosowecanimaginewhatwouldbethenextnumberandsayinDimensionthree.
Whatisitgonnashow?
I guessyoushouldguess, Isthisgonnaalsolike, Isitgonnagobacktozero?
Fourdimensionalsimplexwilllooklike a fivedryitinthispieceofpaper, andsimilarly, everytripleuppointsthatyouseeinthispicturewillform a trianglebecausethefivepointsinfourspaceorinternalpositionandeveryfourcollectionoffthefivepointswillform a TetraHydro.
Faceoffthatsimplexsowecancomputethesenumbers F.
So f zeroThat's fiveand F onewillbefive.
Chooseto.
That's oneofthebinomialcoefficients, whichis 10 and f, too.
We'llbe 5 to 3 nextentryinPascal's triangleand F threewillbefivechoosefor, whichisfive.
Solet's testourpredictionifyoustartfromfivesubstruck 10 at 10 substratefiveandyougotzero.
Resultinthissequencethat, youseeiswhatisknownasthe h vectoroffthethisparticulartryingtodisappear.
Butifyouthinkaboutatleastverysmallpartofthesymmetrywhichpredictsthatthelastcenturyinyour H factorshouldbeone.
Thenagain, ifthisoil s formulaforthisthreedimensionalsphere, becausethinkabouthowyougotthisonestartingfromtheoriginaldata, youhaveyour F zero.
Youhaveyour F zerominusoneandthisnumberis F oneminus F zeroplusone.
Thisnumberis F twominusoneplus F zeroinSwanandthisnumberhereISS F threeminus F to F oneminuszeroplusoneandthesymmetryoftheeightDirectordictatesthatthisshouldbeone.
I canwritethisagainin a tripleinform, sayingthatthealternatingsomeofthenumberoffacesofvariousdimensionzeroforthistriangulated, threedimensionalsphereway, we'realmostthere.
Soifyouthinkoffallitsformulainanydimensionas a smallpartofthislargersymmetrythatyouseeinthe H vectorinthebottomrowoffthissubstructureinPaschalTriangle, thenthedependenceontheparityofthedimensiongoesaway.
Allthisformalreallyassessthatinanydimension.
Thelastentryofthistrianglethatyouseeshouldbeonewhichisjust a verytinypartofftheentiresymmetry.
It's a highlynonobvious, butin a certainsenseisnaturalsymmetry.
Sotheothernumbersarealsointeresting, andtheyarebettingnumbersoffsomehigherdimensionalspace, whichisnot a spherebutisconstructedfrom a sphere.
Andthefactthatthesequencespalindrome E, forexample, thisfourhereisequaltotheotherfourthat, youseegivesyouanotherformulaonthenumberoffacesoffvariousdimensions, whichisdifferentfromallthis.
Wilma.
It's issomethinglikeorlessformula.
Butisthisreallyknewthatthefactthatthe H vectoroftryingLeetosphereisalwayspullingdrawmakeitwasdiscoveredthensomewhatjustifiedabout 100 yearsagobymathematiciansKnocksthenandDuncanSomerville.
Theydidn't dothis.
Theyhad, ah, somewhatmorecomplicatedwayofexpressingthissymmetryoffwhatwenowcall H vector.
Butlatermathematicianshaverealizedthatthisisthebestwaytounderstand, and I thinkthisparticularwayoffviewingthisMetreonisduetoRichardstanding.
Theyarecalledthe F victorbecauseweusuallycall F C o F.
One f 23 f orthefaceandsayWhatandwheredowejumptoage?
Well, thereissomethingcalledthe G Victor, too, anditisreallyessentiallyrelatedtotheunsoldproblemthat I promiseyoutointroduce.
Sothisis a factweknowseveraldifferentwaysupjustifyingthefactthatthis H vectorispalindromeIQinwhateverdimensionforwhatevertryinglatestfearthatyoustartwith.
Ifyouproduce h vectorintermsofthenumberoffaces, thenyoualwaysseethismehtree.
Andthiscyst, theunsoldproblemthat I havetrytoexplainyouanditgoesunderthenamethe G conjecture G conjecture.
So G itwasthemissingletterand g ississ.
Thedifferenceoffthe H istwosuccessivehs.
Sothe G herewouldbethree g herewouldbetoandthe G conjecturesaysthatforeveryeye G, I isnonegative, whichisanotherwayoffsayingthatthe H factorincreasesatleastuptothemiddleWeeklyincreasesweeklyincreases.
Soitispossible, forexample, inthissequencehere 11111 itdoesn't strictlyincrease, butstill, the G vectorisnonegativeinthereasonwhy I likethisconjectureisthatyoucandotheexperimentfairlyconcretely, aswehavedonerightnow, anexplicitexample.
Sowheneveryoucomeupwith a newwaytotranslatethisfear, youhaveyourchancetodisprovethisconjecture.
Anotherreasonwhy I lovethisconjectureisthatnotonlyisveryelementaryandconcrete, itactuallypredicts a verydeepunlockwithseveraldifferentpartsofmathematics.