Whyarewenotabletoseebeyondthreedimensions?
RogérioMartins1
CentrodeMatemáticaeAplicações(CMA),FCT,UNL
DepartamentodeMatemática,FCT,UNL
FaculdadedeCiênciaseTecnologiadaUniversidadeNovadeLisboa
Abstract:Thisisperhapsaphilosophicalquestionratherthanamathematicalone,wedo
notexpecttogiveafullanswer,eventhoughwehopetoclarifysomeideas.Inaddition,
wewouldliketoprovideanewperspectiveonthesubject.Wewillfindcuriousanalogies
withthewayweperceivecolorandmakesomeimaginaryexperimentsshowingthat,
evenlivingimprisonedinthreedimensions…itcouldbedifferent.
———
Didyoueverseeahypersphere?Iguessnot.So…whynot?Thisisthequestionweall,
soonerorlaterhaveposedtoourselves.Whydowehavethisblockadewhenpassing
fromthreetohigherdimensions?Couldthishavebeendifferent?Istherehopethatinthe
futurewecanovercomethiscondition?
Letusstartbygivinganattempttoclarifywhatweusuallymeanwhenwesaythatweare
“seeing”or“visualizing”ageometricalobject,forexampleasphere.
Ononehand,thereisthepuremathematicalobject,thatweallknowasthesphere,this
spherehasamathematicalcharacterizationandlivesinanabstractspace,inthePlatonic
senseifyouwill.Ontheotherhand,thereisourphysicalexperienceofarealsphere,
somethingthatwecanperceivewithoursensesandspacialintuition.Whatdowemean
by“seeing”amathematicalsphere?Itis,inmyopinion,thispossibilitytoimaginethe
mathematicalsphereinourthreedimensionalphysicalworld,somethingthatcouldhave
beenreal,likeasoccerball.Evenasaproductofourimagination,wecanimaginesome
physicalinteractionwithit,holdingit,rotatingitorchangingitsposition.Somehow,we
canuseourthreedimensionalphysicalintuitiontounderstandthesphere’sproperties
andthewayitinteractswithothergeometricalobjects.Weuseourthreedimensional
physicalspacetounderstandtheabstractEuclideanthreedimensionalvectorspace.
1
This work was partially supported by the Fundação para a Ciência e a Tecnologia
(Portuguese Foundation for Science and Technology) through the project
UID/MAT/00297/2013 (Centro de Matemática e Aplicações)
It’sfunnytothinkthisway,certainly
theideaofavectorspace,andthe
sphere,wascreateduponthephysical
sensibleexperienceofspace.Anyway,
theabstractideaofvectorspaceand
itsgeometricobjectsgainedalifeof
theirownanditspropertieswere
generalizedtohigherdimensional
spaces.Nowwehaveafourth
dimensionalobject,thatwecallan
hypersphere,withsimilarproperties
tothethreedimensionalcounterpart,
andwedonothavethe
Figure1:Wecanuseourthreedimensionalphysicalintuitionto
correspondentphysicalobjectto
understandthemathematicalsphere’sproperties.
“see”it.
Recentevidencefromneurobiologyshowsthatourbrainseemstohavebeenoriginally
createdtomanageourhomeostaticmechanismsandphysicalinteractionwithour
surroundings[5][6].Memory,conscience,planninganddecidingseemstobe
superimposedonthetopofacerebralstructureoriginallybuildtodealwithsensationand
movement.Almosteverypartofourbrain,evenbeingresponsibleforsomeother
cognitivefunction,hassomesensoryandmotorsignals.Apparentlywhenweimaginea
physicalsensation,forexampleholdingasphere,wearepartiallyactivatingtheverysame
tactileresponsesthatwouldbeactivatedifthespherewerereallyinourhands[7][8].
Theseideasarealsosomehowconnectedwithaphilosophicaltheorycalledembodied
cognition.
Therefore,ourphysicalsensibleintuitionmustbeincomparablystrongerthanour
mathematicalintuitioningeneral,thisisprobablywhatgivesusthissensationof“seeing”
asphereinoppositiontocomprehendasphere,whenwe“see”weareusingthisstrong
spacialintuition.Thisisprobablywhywefeelthispowerlessnesswhenmovingfromthree
tofourdimensions.
Thisisusuallytheendofthestory,wecannotseeanhyperspheresimplybecausewelive
inathreedimensionalphysicalspace.Wewilltrytoconvinceyouthatitcouldbe
different.
Thisisofcourseaquestionofmathematicalrelevance;thislimitsourdailymathematical
activity.So,whynotgiveitatry?Atleastwehopetoclarifysomeideas.Withthisinmind
wewilltrytofindoutsomeanalogieswiththewayweperceivecolorandmakesome
imaginaryexperimentsthatwehope,willconvinceyouthat,evenimprisonedinathree
dimensionalphysicalspace,wecouldhavebeendifferent.
Weproposethatthislimitationissimplyacharacteristicofourspecies,itisgivenbyour
biology,thatinturnwasshapedinordertosucceedinourenvironment,froman
evolutionarypointofview.Adifferentsensorysystemorbodycharacteristicscouldhad
equippeduswithotherperceptioncapabilitiesofthemathematicalgeometricobjects.
Thehigherdimensions
WeallremembertheadventuresofthesquarethatinhabitedtheworldofEdwinAbbott
Abbott’snovel,Flatland:ARomanceofmanydimensions[3][4].Thissquare,confinedto
itstwodimensionalworld,cannotimaginewhatthethirddimensionis,untilitestablishes
someconnectionwithaspherethatshowedhimthethirddimension.Thisspherenotonly
offersthesquareanexcursionthroughthethreedimensionalspacebutalsocallsthe
attentionforsomerelationsbetweenthetwoandthreedimensions,thatgiveus,
inhabitantsofathreedimensionalspace,someintuitiononthefourthdimension.This
leavesushumans,wonderingif,likethesquare,wereallyliveinasubspaceofafourth
dimensionalworld,andhowweperceiveafourthdimensionalobjectthatcrossesour
world.
Thereareofcourseseveraltrickstoapprehendthosehigherdimensions.Someadvantage
canbetakenfromthetemporaldimension,theuseofprojectionscanalsobehelpful,
analogieswithsmallerdimensions,andsoon.ItisworthtorefertheworkofCharles
HowardHintoninthisdirection.However,thisdoesnotgiveasatisfactorysolution,we
stillcannotseethehyperspherethesamewayweseethesphere.Ofcoursethatafairly
goodcomprehensionofthesehigherdimensionalobjectscanbeattained,understanding
theirproperties,andmanipulatingthem.But,thisdoesnotmeanthatwehaveaclear
visualizationofthosestructures,similartovisualizationwehaveofthecorrespondent
threedimensionalones:comprehendisdifferentfrom“seeing”it.Veryoftenwehaveno
muchmorethanintuitionbasedonthelowerdimensionalversionthatwecan“see.”
Allthesedimensionalquestionshavebecomerecurrentinseveralareasapartfrom
mathematics.Thefictionindustry,cinema,theatre,andliterature,recurrentlycastthe
issue.Pseudosciencealsousestheideasofhiddendimensionstobaseabunchofideas.
Ontheotherhand,thequestionofthenumberofdimensionsisraisedinquantum
physics’slasttheorytoexplainthesub-atomicworld,andshowthatafterallweliveina
ten,orso,dimensionalworld.However,evenifatendimensionalstringtheoryisvery
adequatetogiveamathematicalmodeltotheworldwelivein,theonlythingstheseextra
dimensionshaveincommonwiththethreespacialdimensionsweperceiveitsthename
andsomemathematicalsimilarities,indeedislikecomparingchalktocheese.
Inthispaperwedonotwanttospeculateabouttheexistenceofextradimensionsthatwe
cannot“see”.Theproblemisabitdifferent,weareconcernedwiththeproblemthat
everystudentfaceswhenlearninghigherdimensionalmathematics,theyfeelthatthereis
alimitationonwhattheycanvisualize.Surprisingly,almostnothinghavebeenwritten
aboutit.
Thewayweperceivecolor
Whenwewerekidssomeonetoldusthateverycolorcouldbeobtainedfromthemixing
ofthreecolors,theprimariesones.Let’sassumewearedealingwithanadditivecolor
system,forexamplebeamsoflight,theprimariescommonlyusedarered,greenandblue,
alltheotheronescanbeproducedasacombinationofdifferentintensitiesofthesethree
colors.
Indeed,experimentalevidenceshowsthat,whilethreecolorsisenough,nosetoftwo
colorscanbemixedtoproducethecompletepaletteweperceive.Sinceeverycolorwe
candistinguishismadeofacertaincombinationofintensitiesofthesethreecolors,we
canexpecttoarrangethefullpaletteofcolorsinathreedimensionalvectorspace.The
setofprimarycolorsbeingabasisofthisspace,multiplybyascalarcorrespondingtothe
applicationofacertainintensitytoacertaincolorandsummationoftwocolors
correspondstothemixtureofthesecolors.Aslongaswekeepourselvesfarfromextreme
valuesofluminosities,extremedimorbrightlights,itturnsoutthatourperceptionof
colorisconsistentwiththisthreedimensionalvectorspacemodel.Actuallythepositive
octantofathreedimensionalvectorspace,sincethereisnorealinterpretationfor
multiplicationofacolorbyanegativenumber.ThisisessentiallytheRGBcolormodel.
However,thisisjustthewayweperceivecolor,
therealityisabitdifferent.Inrealitythereisno
purebeamoflight,withjustone
electromagneticfrequency.Eachbeamoflight
iscompoundedofamixtureofwavelengths
andcanbedescribedbyawavelengthdensity
distributionfunctiondefinedoverthevisible
spectrum.Thesetofrealbeamsoflightcanbe
putincorrespondencewiththepositivereal
continuousfunctionsdefinedoveraninterval
oftherealline.Mathematically,thesetofreal Figure2.Eachbeamoflightiscompoundedofa
mixtureofwavelengthsandcanbedescribedbya
beamsoflightisaninfinitedimensionalvector wavelengthdensitydistributionfunctiondefinedon
thevisiblespectrum.
space.
Whenabeamoflightreachesourretina,itstimulatesacertaingroupofcells,thesocalledconecells.Anhealthypersonhasthreetypesofconecells,eachtypeismore
sensitivetocertainfrequencies.Thesepeaksofsensitivitydonotcorrespondtothe
frequenciesoftheprimarycolorsweuse,howeverthefactthatweuseonlythreeprimary
colorsisaconsequenceoftheexistenceofonlythreetypesofconecells.Eachbeamof
lightstimulateseachtypetoacertaindegree,sooureyeseventuallyretainthreevalues
fromarealbeamoflightanditisfromthosethreevaluesthatourbrainassociatesanidea
ofacertaincolor.
Thefactthatwe,humans,havethreetypesofconecellsandliveinathreedimensional
vectorspaceseemstobeacoincidence,actuallyintheanimalworldthereissome
diversityconcerningthenumberoftypesofconecells:itseemstobeanevolutivefeature
ofthespecies.Thekangaroosandhoneybeesalsohavethreetypesofconecells,weare
theso-calledtrichromats.Mostofnon-primatemammalsaredichromats,onlyhavetwo
typesofconecells,thisisalsothecaseofsomecolorblindhumans.Adichromateonly
needstwoprimarycolorstobuilditstwodimensionalchromaticspace.Monochromacyis
veryrareamonghumansbutthisisthecaseofmarinemammalsandsomesealions,they
onlyhaveonesensor,theirchromaticspaceisunidimensional.Ontheotherextreme
therearethetetrachromats,withfourtypesofconecells,thisisthecaseofsomereptiles,
birds,insects…andsomewomen[1].Thefactthatamonghumansonlywomenhavethis
characteristicisrelatedtothefactthatthegeneticinformationconcerningthisconecells
iscontainedintheXchromosome.Therearesomespeciesofmonkeyswherethefemales
havetrichromaticvisionwhilethemalesaredichromat.Therearealsopentachromats,
somebutterfliesandpigeons,andotherspecieswithmorethanfivesensors,however
theybecomemoreandmorerare.
Mathematically,thesetofcolorswecanperceivecanbeseenastheimage,overathree
dimensionalspace,ofaprojectionappliedonthisinfinitedimensionalspaceofthereal
colors.ThisisarealoccurrenceoftheallegoryofPlato’scave.Asaconsequence,inreality
therearemanymore“colors”thanwecandistinguish,therearedifferentwavelength
distributionsthatwehumansperceiveasthesamecolor,thisistheso-called
metamerism.Insomesenseweareallcolorblind,indeedtherearedifferentrealbeams
oflightthatwecannotdistinguish.
Eventhoughthereissomehope,in2009agroupofscientistssucceedininjectingavirus
intheeyesofadichromatemonkeyandtransformedsomeofthoseconecells[2].Of
coursethisisgoodenough,howevertheresultsovercametheexpectationsofthegroup,
sincethebrainofthisadultmonkeysomehowmanagedtodealwiththeadditional
informationsentbythemodifiedconecells,something,theybelieve,couldonlybe
possibleinearlystagesofthedevelopmentofthebrain.Thiswasconfirmedsincethe
monkeywastrainedtoidentifycolorspotsonascreen,thismonkeyreallystartedto
distinguishcolorsthathedidnotdifferentiatebeforetheoperation.
Thisexperienceraisesaninterestingthought:whatdidthismonkeyfeelwhenhestarted
toseethenewcolors?IntheFig.3.,ontheright,weseeanimagethewayitisseenbya
red-greencolorblind:thecaseexhibitedbythismonkey.Ontheleft,weseetheimageas
seenbyatrichromat.Suddenlythismonkeystartedtodistinguishcolorshedidnot
experiencebefore.Itwouldbeimpossibletoexplaintothismonkey,beforetheoperation,
howthecolorofthet’shirtwasdifferentfromthegreencoloroftheboard.Thisis
probablyoneofthemostsimilarsensationswecanthinkofwhenseeingafourth
dimension.
Figure3.Asimulationofanimageseenwithnormalcolorvisionontheleftandred-greencolorblindnesson
theright.PhotobyRodrigodeMatos.
Thisexperimentgivessomehopetocolorblindpeople,thisgroupofscientistsbelieve
thatthisopennewpossibilitiesinthetreatmentofcolorblindnessinhumans.Whoknows
ifthesametechnologycouldalsobeusedtocreateafourthconecellinatrichromat.
Perhapsinthefuturewecanaskforanextrasensorwiththesameeasinessweasktoget
atattoo.
Abunchofideasraisedbythecolorexample
Theexampleaboveisinterestinginitsown,butitisalsousefultoclarifytheproblemwe
aretryingtoaddress.Itraisessomequestionsandallowssomecuriousanalogies.So,let
ustrytoanswersome:
Arethereotheroptionsforafamiliarvectorspacewherewecan“see”?
Inthebeginningofthispaperwesawhowdoweuseourthreedimensionalphysicalspace
toseeorvisualizeageometricobject,forexampleasphere,thatlivesinathree
dimensionalabstractvectorspace.Wetakeadvantageofourstrongspatialintuitionto
understandthepropertiesofthisobject,whenweimaginethisobjectassomethingreal
thatwecanfeel.Whatifthereisanotherintuitivevectorspacethatwecanuseto
visualizegeometricobjects?
Imaginesomeonewhohassuchacomplete
masteryovercolorthathecanimagine
geometricobjectsinhischromaticspace.For
example,theFig.4.representsthesetofcolors
thatformacubeinourthreedimensional
chromaticspace.Wearealsorepresentingthis
setofcolorsinapicturethatrepresenteach
coloratapointinathreedimensionalphysical
Figure4.Thesetofcolorsthatforma
space.However,theideaistoconceivesomeone tridimensionalcubeintheRGBcolorsystem.
whohasasodeepcontrolovercolorthathedoesnotfeeltheneedtoarrangethecolors
aspointsinathreedimensionalphysicalspace.Eventhough,thispersonhasaclearidea
ofthepropertiesofthevectorspacesatisfiedbythecolors(points),whichcolorsareina
neighborhoodofacertaincolor,thedistancebetweencolorsinitschromaticvector
space,theresultofthe‘sum’oftwocolors,multiplicationbyascalarandsoon.This
personcouldimaginedirectlythesetofcolorsthatformthiscubeandhowthesecolors
setoutinhischromaticspace.
Inprinciplethispersoncanvisualizeanygeometricalobjectinthisspace.However,since
thechromaticspaceofanormalpersonisthreedimensional,thesameblockadehappens
inpassingfromthethirdtothefourthdimension.Butmaybesometetrachromatwoman
wantstogiveitatry,ormaybeinthefutureoneofuscandothesamewithartificialextra
conecells.
Coulditbedifferentevenlivinginathreedimensionalspace?
Thecolorexampleshowsusthatourbrainseemstobeformatteddependingontheway
weinteractwithreality,eachbrainisadaptedtoacertainnumberoftypesofconecells.
Inthesameway,aswesawintheintroduction,ourbrainwasoriginallybuiltessentiallyto
dealwithourphysicalinteractionwithphysicalsurroundings,andsoitwasformatted
accordingly.
Weintuitivelysplitandgroupourphysicalstateinaspatialposition,plusandirectionof
thebodyinthisphysicalspace,plusapositionintime,andmanyothermeasures,like
thermalsensationornoiselevel(Fig.5.).Forus,thisisthenaturalsplit,eachofthese
characteristicshavetheirownphysicalandmathematicalproperties.Thethreespatial
coordinatesaresimilarinnatureandavectorspace.Then,directionofthebodyisno
longeravectorspace,mathematicallyitisagroup,theusualSO(3),thegroupofall
rotationsabouttheorigin.Timehasdifferentphysicalproperties,andsooneandsoforth.
Figure5.Wesplitandgroupourphysicalstateinatridimensionalspacialposition,plusanorientationinthisphysical
space,plusapositionintime,andsoon.
However,itcouldbedifferent.Imaginethatwewereananimalthatonlymovesina
relativelyflatsurface,sayakindoflizardinthedesert,allourlifehappensinthisplane,
predators,food,mating,etcetera.Itisnotdifficulttoimaginethatmaybeinthiscaseour
braincannotfindarelationbetweenthetwoplanarcoordinates,neededtoposition
ourselvesinthisplane,andaltitude.
Probablytheorientationortimecouldbemuchmoreimportantforusthanthealtitude,
thatmightnotbeevenrecognized(Fig.6.).Inthiscase,wewouldhaveaproblemtrying
toseeasphere,probablythisbeingwouldnotseebeyondtwodimensions.
Figure6.However,itcouldbedifferent.
Inamoreextremesituationwecanimaginethatsomebrainscanonlybeawareofone
spatialdimension.Imagineforexamplelice,maybetheonlyimportantspatialdatatotake
intoaccountisthedistanceneededtoreachthenexthead,anunidimensionalquantity.If
thisbrainattainsareasonableunderstandingofmathematics,thenitcannoteven“see”
thecircle.
Whataboutseeingmorethanthreedimensions?
Considerthefollowingimaginaryexperiment:whatifourbodyweremadeoftwoparts?
Twopartsbutjustonebrain(Fig.7).Weareaccustomedtotheideathattoeachbody
correspondsonebrain,thatinsomecasescanevenhaveaconscienceofitself.However,
thisisnothingbutjustonemorecharacteristicofthespecies,likehavingtwoarms.Iam
notconcernedwithwhetherthismakesensefromthebiologicalandevolutionarypointof
view.Surelytherearesomeexplanationsmissing.Wouldthisbrainbeattachedtojust
onepartorwoulditbesplit?Howisitlinkedtotheparts?Nonetheless,donotworry
abouttechnicaldetailsandforthepresentmomentjustimagineabeingmadeofone
brainandabodyseparatedintotwodisconnectedparts.
Sinceeachpartofthisbodylivesinourthree
dimensionalphysicalworld,itneedsthreespatial
coordinatestospecifyitsposition.Eventuallythis
braincouldsplitthepositionofeachpart,saythree
dimensionalpositionforpartAplusthree
dimensionalpositionforpartB,however,thefactis
thatfromamathematicalpointofview,asix
dimensionalEuclideanvectorspacewouldbethe
rightspacetodefinethepositionofthisbeing.Since
thisbrainwouldhaveevolvetogetherwiththisbody,
itispossiblethatitwoulddealdirectlywiththesix
coordinatesasawhole.Inthiscasethisbeingwould
hadanintuitivesixdimensionalvectorspaceto“see”
thegeometricalobjects.Probably,ifthisbeingwere
theauthorofthispaperthetitlewouldbe:whyarewe Figure7.Whatifweweremadeoftwoparts?
notabletoseebeyondsixdimensions?
Doesthenumberofdimensionsweperceivecoincidewiththenumberofreal
dimensions?
Thelastexampleshowedusthatweperceiveathreedimensionalchromaticspace,even
ifthe“real”chromaticspaceisbetterdescribedasaninfinitedimensionalvectorspace.
Thefirstthoughtthatcomestomindisthatsomesimilarphenomenacouldhappenwith
ourphysicaldimensions.Maybewesomehowonlyperceivethreedimensionsofareality
thatishighdimensional,perhapseveninfinitedimensional.
Well,insomesense,wereallyperceiveathreedimensionalphysicalspacewhilethereare
higherdimensionalmathematicalmodelsthatbetterdescribeourreality,thisisfor
examplethecaseofthegeneraltheoryofrelativity.Thefactthatthe“real”setofcolorsis
modeledbyaninfinitedimensionalspaceis,likethetheoryofrelativity,justanabstract
modelthatgoesfurtherinthedescriptionofourreality.Thechromaticspacegenerated
bythethreeprimariescolorsorthethreephysicaldimensionsweperceiveareafirst
approachbaseddirectlyonoursenses.
However,thehigherdimensionalmathematicalmodelsthatdescribetherealityseemto
showdifferentqualitativepropertiesamongthosedimensions.Forexample,thequantum
theories,involvemodelswithmoredimensions,atanyrateallthishappensinscales
whereoursensesarehelpless.Itwouldbedifficulttocomparethephysicaldimensions
weperceivewiththeremainingones.Ontheotherhand,intheinfinitedimensional
modelofcolorallthedimensionshaveasimilarrule.
Theideaofsuddenlyseeafourthspacialdimensioncertainlygiveagreatstorylinefora
filmorabook,howeverwedonotknowhowitcouldmakesense.
Conclusion
Finally,itseemsthat“seeing”uptothreedimensionsisanevolutionarycharacteristiclike
havingtwoears,onenose,orhavingthreeconecellsandhencetrichromaticvision.The
brainseemstobeformattedwithacertainviewofreality,thatcouldbemoreorless
inevitableduetoourphysicalcondition.
Butitcouldbedifferent.Ialwayswonderedhowdifferentwewouldbeifwecouldsee
morethenthreedimensions.Wouldthemathematicswehavedevelopedbethesame?
Probably,notcompletely.Forexample,ifwewerecapableofseeinginfourdimensions,
theroleoccupiedbyrealanalysiswouldprobablyhavebeentakenbycomplexanalysis.It
isinsomesensemuchsimplerapartfromthehandicapthatthecomplexfunctionsof
complexvariablehavefourdimensionalgraphs,andsoatanyratenoteasilyhandledby
someonethathasaprobleminvisualizingbeyondthreedimensions.
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