escher´s universe 1. introduction · closer to mathematicians than to my own colleagues." ......

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ESCHER´SUNIVERSE

1. INTRODUCTION

When the night is upon the Earth, a beautiful and unknown Universe is open to our eyes,

leavingtotheimaginationanunlimitedfreedomtoexploreanddiscover,tothinkanddream,

towriteandpaint...

InthePlanetarium,acleansky, freeof lightsandpollution,transportsustoothertimesand

places.

This fascination that overwhelms us when we look at the night sky is something innate to

humansandthroughouthistoryweareawareofmanydifferentinterpretationsthatattempt

todescribe,withmoreorlesssuccess,whatwefeelandsee.

Philosophy,Art,ScienceorPoetry,allofthemhavewovendifferenttheoriesabouttheorigin

and evolution of a Universe, where besides energy, space and time, we ourselves and

everythingsurroundingusbelongto.

2. WITHESCHER´SEYES

"Myworkisagame,averyseriousgame".

WhoisthisMauritsCornelisEscherforwhomhisworkisagame?

Isheamathematician,anarchitect,anartist,atraveler...?

"Allmyworksweredonewiththeintentionofexpressingaparticularidea.Theseideasare

basedonmyaweandadmirationforthelawscontainedintheworldaroundus".

"Forthesephenomenatomoveus,weneedwithoutdoubttheadmiringtalentofchildren.

AndIpossessitinabundance.Admirationisthesaltoftheearth".

"WhowondersstillabouttheMoon,suspendedhighinthesky?







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Escher, very found of Astronomy, feels the attraction of the night sky and wonders about

celestialphenomena.

...showsOrionconstellation...

...andplaysadmirabilywithSunandMoon...

...orrecreatesotherworlds'landscapeswithimpossiblearchitectures.

Hedisplaysauniverseofobjectsandlandscapes,whoseformsseemtomoveontheirown.

3. POLYHEDRAINTHEUNIVERSE

Asthepolyhedra,whichPlatoassociatedtotheoriginofHumanityandCosmos.

ThePythagorean cosmology associates them with the the four fundamental elements of

Nature.

ThisPythagoreanmodelinspiredKepleraCosmologywhichrelatedthefiveregularpolyhedra

tothesixknownplanets.Hecalledit"theCosmographicMystery".

EscherconstructedamodelofUniverseinwhichthebodieswereinscribedoneineachother.

Hewastremendously interested intheregularityofpolyhedraandhowitalsoshowedup in

thecrystalstructures,sensingthefractalconcept.

Escher is marveled by crystals in Nature, their perfect symmetries, their transparency and

colordiversity.

“Longbeforemanexistedonearth,crystalsgrewintheEarth'scrust...There issomething

thrillinginthelawsgoverningthecrystallineformations.Theyarenotproductofthehuman

work.Theyareindependentofus‐theysimply'are'."

Escher,borninHollandin1898,travelstoItalyandisexcitedaboutitslandscapes.Villagescut

onhighcliffs,shorehamletswithsteeproadsandstairedstreets.







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OnhisfirsttriptoSpain,theAlhambraimpresseshimdeeply,especiallythegeometricshapes

oftilesandplasterwork.Hewroteinhisdiary:

"ThismorningIwasattheAlhambra.Ithoroughlyenjoyedthissublimeandnobleworkof

art.IntheafternoonIwentbackthereagainandbegantocopytheornamentsofthetiles.

TheArabsdevelopedagreatskillintheartoffillingsurfaceswithfiguresthatarerepeated

withoutleavingonefreeslot...".

Thesevisitswillgivea turntohiswork.Fromthemhewilldevotehimself towhathecalled

"theregularpartitionoftheplane"breathinglifeintopuregeometry.

"The regular partition of the surface is the richest source of inspiration I have ever

encountered,andisstillfarfromexhausted...."

HowdoesEscherachievethiseffect?Itisnotdifficult,withregularpolygons.

Heappealstomathematicsbyperforminglinearshifts,simmetriesandrotations.

Butifthepolygonisdeformedandtheprocedureisrepeatedtheresultsyieldgreaterbeauty.

"ItisunfortunatethatIslamprohibitsimages.Therefore,themosaicswerelimitedtotheuse

ofabstractgeometric shapes.NoArabartistdared touse recognizable figures (birds, fish,

reptiles,men)asdecorativeelements."

However,Eschersucceedesadmirabily.

"(...)Although Idonothaveatraining inmathematicalsciencesorotherexpertise, Ioften

feelclosertomathematiciansthantomyowncolleagues."

What was his profession? He was neither an architect, nor astronomer, geologist or

mathematician.







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4. COSMOLOGY

TheancientPythagoreanconceptionofaUniversemadeupof fiveregularpolyhedracanbe

consideredasagoodstartingpointforacosmologicalmodel.Althougholdandinaccurate,itis

basedonobservation,whichistheessenceoftheScientificMethod.

Theuniversewas formed about 14billions years ago. TheBig Bang is somewhat difficult to

explain formathematics and it is called a singularity. From her theUniversewas born, and

spaceandtimewithit.Andspacebegantoexpandwithoutacenter.

With Escher's work "Cubic Space Division" we can understand this concept. If we place

ourselves inanyof thecubes in the jointsof thisexpanding lattice, it seemsthateverything

shiftsawayfromus.

Itlookslikewearethecenterofexpansion.ThesituationisverysimilarintheUniverse:each

groupofgalaxiesismovingawayfromusandfromeachother.

Someof themajor issuesraisedabout theUniverse, likewether it is finiteor infinite, flator

curved, can be found them in some Escher prints inwhich the same structure is replicated

cyclicylyreducingtoinfinity,whilekeepingitsshapeunchanged.

With the same shapes, as described by the astrophysicist Penrose, Escher recreates the

geometryofflatspace,onefiniteandcurvedandoneinfiniteandcurvedspace.

"InmyworksIattempttotestifythatweliveinaneatandbeautifulworld...andnotina

shapelesschaos,asitsometimesseems."

4b. PERCEPTION

Escherplaysbeautifullywiththeperspective,asinhis"Waterfall"inwhich,ifitwaspossibleto

build,itwouldconstituteaperpetualsourceoffreeenergy...

...Or inhis famous"Belvedere"wheretwofloorsareparallelandperpendicularatthesame

time...

…andhiswork"UpandDown"which joins inasingle imagethesimultaneousviewsof two

observerswatchingfromthefloorandtheroof.







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5. SIMULTANEOUSWORLDS,IMPOSSIBLEWORLDS

HeisalsointerestedinthenewideasinPhysicsofhistime,likerelativityandgravitation,and

heusedtotalkaboutthemwithscientistsfriends.

In "OtherWorld"he showsus the relativityof thevertical.Which is the floor?Which is the

ceiling?Whicharethewalls?

Usingthissameidea,Escherpresentshislithography"Relativity"

"Threeflatgravitationalsurfaces influenceeachotherrespectively intheverticaldirection.

Threeflatsurfacesoneachofthemlivingpeople,intersectatrightangles.Twopeoplefrom

different worlds cannot walk, sit or stand on the same floor, as their ideas of what is

horizontalorverticaldonotagreewitheachother.Howevertheycanusethesameladder.

Intheupperladder,twopersonsmoveinthesamedirection.Butonegoesupandtheother

down.Itseemsimpossiblethattheycangettoestablishsomecontactwitheachother:they

liveindifferentworldsandknownothingabouttheother'sexistence."

Regardinghis"Portraitgallery"hetellsus:

"Twoveryeruditepersons,(...),havelongtriedtoconvincemethatIhaddrawnaRiemann

surface. Idoubttheyareright, (...).Anyhow,Riemann isbeyondmycapability, (...)What I

triedtorepresentisasurfacethatwasjustswelling,ring‐shaped,withoutbeginningorend.

"

6. FINAL

Escher was, thus, an artist marveling at nature around him, amazed by the beauty of

landscapes, by crystals and their perfect order, bymathematics andastronomy,by thenew

physicsdevelopingintheearlytwentiethcentury,anartistabletoexpresssimultaneouslythe

regularityandthediversity.

“Stillnessandmovement”

thebeginningandtheend

“Nightandday”







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Letuslistentohimoncemorebeforesayinggoodbye:

"(...)SobeautifulandpurearethethingsItrytoexpress."

"So let us try to climb the mountain, not to set foot on what is below us, but to lift us

towardswhatisaboveus,whichinmycasearethestars."

escher´s universe 1. introduction · closer to mathematicians than to my own colleagues." ......

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