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?

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.

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.

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.

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”

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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