ilhan koman papers , mathematical sculpture

166 Pages·2021·56.699 MB·English

by  ilhan koman

Spiral Developable Sculptures of Ilhan Koman
TevfikAkgu¨nandIrfanKaya AhmetKoman ErgunAkleman
FacultyofArtandDesign MolecularBiologyand VisualizationSciencesProgram,
CommunicationDesign GeneticsDepartment DepartmentofArchitecture
Department Bog˜azic¸iUniversity TexasA&MUniversity
YildizTechnicalUniversity Istanbul,Turkey CollegeStation,Texas,USA
Istanbul,Turkey [email protected] [email protected]
[email protected]
[email protected]
Abstract
Inthispaper,wepresentspiraldevelopablesculpturalformsinventedbyIlhanKomanapproximately25yearsago.
WeidentifythemathematicalprocedurebehindKoman’sspiraldevelopableformsandshowthatusingthisproce-
dure,avarietyofspiraldevelopableformscanbeconstructed.
1 IntroductionandMotivation
This paper presents the spiral developable forms invented by sculptor Ilhan Koman during the 1970’s. Fig-
ure 1 shows Ilhan Koman in front of one of his spiral developable sculptures. For detailed photographs see
Figures2and4C.
Figure1: ACompositeofIlhanKomaninfrontofoneofhisSpiraldevelopablesculptures. (Thephotograph
ofIlhanKomanbyTayfunTunc¸elli,1980’s).
Ilhan Koman has not recorded his method to construct these spiral sculptures. By examining his sculp-
tures, we were able to reconstruct his method. In this paper, we demonstrate Ilhan Koman’s method that is
reconstructedbyus,hereinafterwesimplereferasourmethod,but,itis,infact,Koman’smethod.
To demonstrate that our method is viable, we show that using our method it is possible to construct
circular developable sculptures from one piece of rectangular paper. Note that using the same method we
can construct a spiral shape by changing the curvature. As shown in Figure 3B, a circular developable
sculpturecanbeconstructedusingourmethodfromonepieceofrectangularpapershowninFigure3B.
1
Figure2: AdetailedphotographofoneofthespiraldevelopablesculpturesofIlhanKoman.
Ourmethodcanalsoprovideaspiralbyusingaquadrilateralpieceofpaper,showninFigure4. Figure4C
shows one of Ilhan Koman’s spiral sculptures. Our method allows us to create a comparable form as seen
in this comparison. However, Koman’s metal sculpture looks much more elegant than ours as seen in this
comparison.
2 Overview
Ilhan Koman is one of the innovative sculptors of the 20th century who frequently used mathematical con-
cepts in creating his sculptures and discovered a wide variety of sculptural forms that can be of interest for
theart+mathcommunity. Komanwasbornin1921,Edirne,Turkey,studiedattheArtAcademyinIstanbul,
opened his first workshop and exhibition in Paris, 1948, moved to Sweden in 1958, where he taught at the
Konstfack School of Applied Art in Stockholm until his death in 1986. His works cover a wide spectrum
of styles and materials, including 12 public monuments in Sweden and 4 in Turkey. He is represented in
severalmuseumsincludingModernaMuseet,Stockholm,MuseumofModernArt,NewYork,Muse´ed’Art
Moderne de la Ville de Paris. During the last twenty years of his life, he worked on inventing diverging
geometricalformswhichhedevelopedasprototypestoberealizedinlarge-scaleprojects. Amongthegeo-
metricalshapeshedeveloped,tetraflexwasaflexiblepolyhedronwhichheregisteredattheSwedishPatent
Office in 1971. At the time he had created and proposed these structures as modules for architecture, con-
structions in space and aviation fuel tanks and published his method in Leonardo [8] For a detailed Ilhan
Komanbiographysee[8,9,1].
IlhanKomanalsoinventedavarietyofdevelopableforms, buthedidnotpublishedhismethods. Some
ofhisdevelopablesurfaceswerethesubjectofourearlierpaper[1]. Developablesurfacesaredefinedasthe
surfaces on which the Gaussian curvature is 0 everywhere [18]. The developable surfaces are useful since
they can be made out of sheets metal or paper by rolling a flat sheet of material without stretching it [14].
Most large-scale objects such as airplanes or ships are constructed using un-stretched sheet metals, since
sheetmetalsareeasytomodelandtheyhavegoodstabilityandvibrationproperties.
Sheet metal is not only excellent for stability, fluid dynamics and vibration, but also one can construct
aesthetic buildings and sculptures using sheet metal or paper. Developable surfaces are frequently used by
(A) (B)
Figure 3: Constructing a circular developable sculpture from one piece of rectangular paper shown in (A)
usingourmethod.
contemporaryarchitects,allowingthemtodesignnewforms. However,thedesignandconstructionoflarge-
scaleshapeswithdevelopablesurfacesrequiresextensivearchitecturalandcivilengineeringexpertise. Only
afewarchitecturalfirmssuchasGehryAssociatescantakeadvantageofthecurrentgraphicsandmodeling
technologytoconstructsuchrevolutionarynewforms[11].
Developable surfaces are particularly interesting for sculptural design. It is possible to find new forms
by physically constructing developable surfaces. Recently, very interesting developable sculptures, called
D-forms,wereinventedbytheLondondesignerTonyWillsandintroducedbySharp,PottmanandWallner
[15,12]. D-formsarecreatedbyjoiningtheedgesofapairofsheetmetalorpaperwiththesameperimeter
[15, 12]. Pottman and Wallner introduced two open questions involving D-forms [12, 6]. Sharp introduced
anti-D-forms that are created by joining holes [16]. Ron Evans invented another related developable form
calledPlexagons[5]. PaulBourkehasrecentlyconstructedcomputergeneratedD-formsandplexons[3,5]
usingEvolverdevelopedbyKenBrakke[2].
Ilhan Koman’s spiral developable sculptures are the result of a procedure that transforms a single flat
developablesurfacetospiralshapesin3D.Spiralsareoneofthemostcommonshapesinnature,mathemat-
ics, and art [23]. Behind the beauty of many natural objects such as snail shells, seashells and rams’ horns
lies their spiral shapes [22]. The spiral forms exist in almost all cultures as artistic and mystical symbols.
This widespread usage of the spiral form may imply that humans innately find it aesthetically pleasing and
interesting.
Spiralsarealsopopularinmathematicalart. SpiralsfrequentlyappearsFractalart[21]. Spiralformscan
beseenCharlesPerry’smathematicallyinspiredsculptures[13]. DanielErdely’sSpidronsalsoshowsspiral
structures in 3D [19, 20]. Spirals are among the most studied curves since ancient Greek times. Although
spiralsareusuallyrepresentedbyparametricequations,thereareawidevarietyofmethodsthatcanbeused
to construct and represent spirals. There exist a wide variety of spirals such as the Archimedean spiral, the
Fermat’sspiral,theLogarithmicspiralandtheFibonaccispiral[23].
3 OurMethodtoConstructIlhanKoman’sDevelopableSpiralForms
Inthispaper,wepresentourreconstructionofIlhanKoman’smethodtobuilddevelopablespiralsculptures.
Wehavedevelopedfirstamethodtobuildcircularsculptures. Spiralsculpturescanbeobtainedbychanging
the curvature of circular sculptures. The way we change the curvature can directly affect the shape of the
spirals.
(A) (B) (C)
Figure 4: Constructing a spiral developable sculpture (B) from one piece of a quadrilateral paper shown
in (A) using our method. We traced the boundary with pen to show the the structure better. (C) shows a
photographofaspiralsculptureofIlhanKoman.
Figure 5 shows the the basic procedure to create a circular sculpture. Figure 5A differentiates two
different parts of the rectangular paper strip that is used to construct circular sculpture. Dark-blue pieces
always stay planar and light-yellow pieces eventually become curved and create the 3D structure of the
sculpture.
(A) (B)
Figure 5: Square paper that is used to create circular sculpture. We color coded paper to differentiate the
two types of pieces. Blue pieces stays planar and yellow pieces becomes curved. The values of d, h and l
canbechosenindependently.
Ourmethodstartsbygluingorstaplingthedark-bluepieceintheboundarytoaplanarsurface. Onceit
isflattened,thenextneighboringdark-bluepieceisflattenedbyrotatingwithanangleaandtranslatingsuch
thattheseconddark-bluepieceentersunderthenow-curvedlight-yellowpiecethatisbetweentwodark-blue
pieces as shown in Figure 6A. Note that this procedure just makes the yellow piece shorter and forces it to
be curved. If we continue this procedure using the same angle a, we start to see a circular arc as shown in
Figure6C.Notethatifa=2π/nwherenisaninteger,thecirculararcisclosedasannsidedregularpolygon
andnbecomesthenumberofbluepieces.
Thevalueofacanbeuniquelydeterminedbythevaluesofhandd as
a=arctan(2d/h)
(SeeFigure6B).Inpractice,2dischosenmuchsmallerthanhasintheexampleshowninFigure3B.If2dis
muchsmallerthanh,thenthevalueofabecomesverysmall. Forverysmallvaluesofa,theregularpolygon
closelyapproximatesacirclewitharadiusrwhere
nd d d h
r≈ = = =
2π a 2d/h 2
sincetana≈a=2d/h. Notethatr≈h/2isalsovisibleinFigure3B.Foracircle,curvatureisconstantand
canbecomputed1/r. Therefore,wecanchangethecurvaturebychangingeitherd orhorboth.
(A) (B) (C)
Figure6: Basicprocedure. Thetriangledrawnin(B)showstherelationshipbetweenanglea,d andh. The
circularstructurein(C)isdrawnbyusingexaggeratedvalues: d =4andh=21. Therefore,theangleais
notsmall,whichisapproximately200.
One possible approach is to change only the d value and keep a constant. The value of a can be kept
constant by choosing h/d constant. We have created our spiral sculptures this way by changing only the d
valueandkeepingaconstantasseeninFigure4B.Inthisexample,sincethed valueischangedlinearly,our
spiral is not a logarithmic spiral, which is more common in nature and considered more aesthetic [22]. We
thinkthatKoman’ssculptureismorelikealogarithmicspiralandthereforeitlooksmoreaesthetic.
Although,itispossibletocreatelogarithmicspiralsbychangingonlythed value,forlogarithmicspirals
d values quickly becomes smaller and it is very hard to cut such a thin slits by hand with scissor. We have
not developed a software for this purpose, but, it is possible to automatically create such a drawing and cut
the paper or metal using a laser cutter. On the other hand, since laser cutters were not available in 1980,
this was not an option for Ilhan Koman. Observing his sculptures it is clear that he did not change d value
drastically,insteadhemadehexponentiallysmaller. Wearesurethatheusedasimpleproceduretochange
thevaluesofhandd. However,wedonotknowhowhechangedthevaluesandwhatkindofprocedurehe
usedtochangethem.
4 ConclusionandFutureWork
Inthispaper,wepresentedIlhanKoman’sspiraldevelopablesculptures. Wehaveidentifiedthemathematical
ideasbehindthesespiralforms. Wehavealsointroducedamethodthatcanallowtheconstructionofvariety
ofaspiralformsbychangingonlytwoparameters.
Usingthesametypeofcuts,IlhanKomanalsocreatedmorecomplicateddevelopablesurfaces. Ournext
goalistounderstandhowhecreatedthosemorecomplicatedsculpturesanddisseminateourfindings.
References
[1] TevfikAkgunandAhmetKomanandErgunAkleman. DevelopableSculpturalFormsofIlhanKoman.
InBridges’2006,MathematicalConnectionsinArt,MusicandScience,pages343–350,2006.
[2] KennethA.Brakke,SurfaceEvolver,http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html
[3] PaulBourke,D-Forms,http://astronomy.swin.edu.au/pbourke/surfaces/dform/
[4] C.R.Calladine,”TheoryofShellStructures”,CambridgeUniversityPress,Cambridge,1983.
[5] RonEvans,PlexonscreatedbyPaulBourke,http://astronomy.swin.edu.au/pbourke/geometry/plexagon/
[6] Erik D. Demaine and Joseph O’Rourke, “Open Problems from CCCG 2002,” in Proceedings of the
15th Canadian Conference on Computational Geometry (CCCG 2003), pp. 178-181, Halifax, Nova
Scotia,Canada,August11-13,2003.
[7] Ilhan Koman Foundation For Arts & Cultures, ”Ilhan Koman - Retrospective”, Yapi-Kredi Cultural
Activities,ArtsandPublishing,Istanbul,Turkey,2005.
[8] IlhanKomanandFranoiseRibeyrolles,”OnMyApproachtoMakingNonfigurativeStaticandKinetic
Sculpture”,Leonardo,Vol.12,No1,pp.1-4,PergamonPressLtd,NewYork,USA,1979.
[9] KomanFoundationweb-site;http://www.koman.org
[10] JunMitaniandHiromasaSuzuki,“MakingPapercraftToysFromMeshesUsingStrip-BasedApproxi-
mateUnfolding.ACMTrans.Graph.23(3).pp.259-263,2004.
[11] FrankGehry,http://www.gehrytechnologies.com/
[12] HelmutPottmannandJohannesWallner,“ComputationalLineGeometry”,Springer-Verlag,4,p.418,
2001.
[13] CharlesPerry. Sculptures. www.charlesperry.com,2006.
[14] http://www.rhino3.de/design/modeling/developable/index.shtml
[15] JohnSharp,D-formsandDevelopableSurfaces,Bridges2005,pp.121-128,Banff,Canada,2005.
[16] JohnSharp,D-forms: Surprisingnew3Dformsfromflatcurvedshapes,Tarquin2005.
[17] Meng Sun and Eugene Fiume, A technique for constructing developable surfaces, Proceedings of the
conferenceonGraphicsinterface’96,Toronto,Ontario,Canada,pp.176-185,1996.
[18] Eric W. Weisstein. ”Developable Surface.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/DevelopableSurface.html.
[19] Daniel Erdelyi. Some surprising properties of the spidrons. In Bridges’2005, Mathematical Connec-
tionsinArt,MusicandScience,pages179–186,2005.
[20] Daniel Erdelyi. Spidron domain: The expending spidron universe. In Bridges’2006, Mathematical
ConnectionsinArt,MusicandScience,pages549–550,2006.
[21] B.Mandelbrot. TheFractalGeometryofNature. W.H.FreemanandCo.,NewYork,1980..
[22] D’ArcyWentworthThompson. OnGrowthandForm(OriginallyPublishedin1917). Dover,1992.
[23] EricW.Weisstein. Wolframmathworld: Spirals. mathworld.wolfram.com/Spiral.html,2006.
mathematics
Article
Turning Hild’s Sculptures into Single-Sided Surfaces
CarloH.Séquin
EECSComputerScience,UniversityofCalifornia,Berkeley94720,CA,USA;[email protected]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:15December2018;Accepted:17January2019;Published:25January2019
Abstract: EvaHildusesanintuitive,incrementalapproachtocreatefascinatingceramicsculptures
representing2-manifoldswithinterestingtopologies. Typically,thesefree-formshapesaretwo-sided
andthusorientable. HereIamexploringwaysinwhichsimilar-lookingshapesmaybecreatedthat
aresingle-sided. Somedifferencesinourtwoapproachesarehighlightedandthenusedtocreate
somecomplex2-manifoldsthatareclearlydifferentfromHild’srepertoire.
Keywords: EvaHild;2-manifolds;computer-aideddesign;3Dprinting
1. EvaHild’sSculptures
EvaHildisaSwedishartist[1],whocreateslargeceramicsculptures(Figure1)inanintuitive,
incrementalprocess[2,3]. Ihavebeenfascinatedbyherworkforseveralyears[4],beforeIwasfinally
abletomeetherinherstudioinSparsör,SwedeninJuly2018(Figure1b).
Figure1.EvaHild:(a)Aportrait;(b)workinginherstudio;and(c)withthesculptureHollow[1].
Hercreationsaretypicallythinsurfaces,whichmaytakeonthestructureofbulbousoutgrowths
in plant-like assemblies (Figure 2a), or configurations of intricately nested funnels (Figure 2b).
These surfaces may be bordered by simple circular openings or by long, sensuously undulating
curvedrims,whichareconnectedbyasystemofcrisscrossingtunnels(Figure2c). Thesesculptures
invitementalexplorations,raisingquestionssuchas:howmanytunnelsarethere?Howmanyseparate
rimsarethere? Isthisa1-sidedor2-sidedsurface? Thesesculpturesalsohaveinspiredmetocreate
similar shapes [4]. I do not possess the skills to create large ceramic pieces myself, so I have tried
tocreatecomputer-aideddesign(CAD)modelsofsuchsurfaces, andthenhaverealizedthemore
promisingonesas3Dprintedmodels. MystudentsandIfounditratherdifficulttocaptureHild’s
sculpturesinCADtoolssuchasBlender[5]orMaya[6]. Wehaddifficultiesindefiningtheproper
topologies—oftenjuststartingfromafewphotographsfoundontheWorldWideWeb. Wealsofound
itdifficulttofine-tunethesetopologicallycorrectmodels,sothattheywouldreproducetheorganic
Mathematics2019,7,125;doi:10.3390/math7020125 www.mdpi.com/journal/mathematics
Mathematics2019,7,125 2of17
flowandnaturaleleganceofHild’screations. Wethusendedupdevelopingsomeadditionalmodeling
toolstomakethedesignofsuchshapeslesscumbersomeandfrustrating.
Figure 2. Eva Hild’s ceramic creations [1]: (a) a bulbous surface with oculus-like openings;
(b)variousnestedfunnels;and(c)hyperboloidtunnelsandundulatingrims.
To place the results of this paper into proper perspective, I start with a brief review of the
developmentsthatledmetothiscurrentinvestigationandsomeofthemanyothercontributionsin
thisdomain,madebyartistsaswellasmathematicians.
2. BackgroundandPreviousWork
For much of my life I have been fascinated by abstract, geometrical sculpture. As a student,
some of my heroes were Alexander Calder, Naum Gabo, and Max Bill. In the 1980’s, when I
startedtoteachcoursesincomputergraphicsandcomputer-aideddesign[7],Iemployedthehelp
of such techniques to analyze and synthesize topologically interesting sculptural shapes. The key
stimulationformystronginvolvementinthisfieldcamefrommyinteractionwithBrentCollins[8].
Thisconnectionwaspromptedbyananalyticalwrite-upbyGeorgeFrancis[9]inaspecialissueon
visualmathematicsinLeonardoin1992,whichclearlyshowedtheconnectionbetweentheintuitive
sculptural work of Collins and the geometry of minimal surfaces, the topology of interconnected
tunnelsandhandles,andbasicknottheory. Istartedtowriteaspecialized,parametrizedcomputer
program,calledSculptureGeneratorI(Figure3a)[10]thatallowedmenotonlytocapturesomeinspiring
shapescreatedby Collins, butalsoenabled meto makeseveral derivativedesignsthat seemedto
belongtothe“samefamily”.Collinswasfascinatedbythisprospectandwaseagertosculptadditional,
morecomplexshapesbyrelyingheavilyonmydetailed,computer-generatedprintouts. Ourfirsttwo
collaborativepieceswereHyperbolicHexagonII(Figure3b)[11]andHeptoroid,(Figure3c)[12],bothof
whichwereageneralizationofa6-or7-storyScherk-Tower[13]bentintoaclosed,possiblytwisted
loop. Heptoroidwasparticularlyexcitingtous,becauseitintroducedanon-orientable,single-sided
surfacewithaheavilyknottedborder.
SculptureGeneratorIwasaveryspecificprogramthatcouldonlygeneratesculpturesbasedon
Scherk’sSecondMinimalSurface[13]. TocaptureanyotheroneofCollins’inspirationalsculptures,
suchasPaxMundi(Figure4a)[14],Ineededanewprogramwithdifferentgeometricalprimitives:
inparticular,versatileprogressivesweepsthatwouldallowmetomoveanarbitrarycrosssection
alongany3Dsweeppath. ThisgraduallyledtothedevelopmentoftheBerkeleySLIDEprogram
(Figure4b)[15]. Thismodelingtoolhasavarietyofparameterizedgeometricmodulesthatcanbe
combinedinmanydifferentways. Byusingitssweepconstruct,IwasabletocapturePaxMundiand
designalargecollectionofViaeGlobisculptures[16],inwhicharibbontravelsalongthesurfaceof
asphere.WiththisprogramwescaledupPaxMundi,andwiththehelpofSteveReinmuth[17]realized
it as a large Bronze sculpture [18]. The SLIDE program then allowed us to create other Viae Globi
sculptures,someofwhichhadknottedsweeppaths,suchasMusicoftheSpheres(Figure4c)[19].
Mathematics2019,7,125 3of17
Figure3.SculptureGeneratorI:(a)Theuserinterface;(b)HyperbolicHexagonII;and(c)Heptoroid.
Figure4.RoadsonaSphere:(a)PaxMundi;(b)SLIDEuserinterface;and(c)MusicoftheSpheres.
Collinscreatedhisinspirationalsculpturesinanintuitiveway,drivenbytheaestheticsofthe
emerging form, not by thinking about mathematics or topological issues. Other artists start out
quite consciously with mathematics on their mind. Helaman Ferguson [20] has created stone and
bronzesculpturesbasedonmathematicalequations,givenknotstructures,ortopologicallyintriguing
surfaces[21]. GeorgeHart[22]isoftendrivenbysomehigh-ordersymmetrygroupandthenfinds
ways to interlink many identical pieces in an intricate manner, while maintaining the envisioned
symmetry[23]. Hereliesoncomputer-aidedtoolstodefinetheseshapes[24],andeventofabricate
themwithacomputer-drivenpapercutter[25]. AsimilarwayofusingCADtoolsalsoenablesthe
workofmanyotherartists,suchasBathshebaGrossman[26],VladimirBulatov[27],RinusRoelofs[28],
and Hans Schepker [29]. Charles O. Perry [30,31], even though he often starts by defining crucial
elementsofhissculpturebyshapingphysicalmaterialssuchassteelcables,isalsoquiteconsciousof
themathematicalunderpinningsofsomeofhissculptures[32,33].
Forotherauthors,thekeymotivationistogenerateavisualizationofsomemathematicalconcept,
ratherthancreatingapieceofart.JarkevanWijkusescomputergraphicstogeneratetheSeifertsurfaces
supportedbymathematicalknots[34]. Hisdemonstrationtoolisreadilyavailableontheweb[35],
andbychoosingtherightsmoothingoption,onecanproducevisualizationmodelsthatcanalsostand
ontheirownaspleasingsculpturalshapes. Othermathematicalconcepts,wherevisualizationmodels
canreadilyturnintoart,concernnon-orientablesurfaces—suchasBoy’ssurface(Figure5a)[36,37]or
Kleinbottles(Figure5b)[38,39]—andthedepictionofregularmaps[40–42],regularmeshes[43,44],
wovensurfaces[45,46],knottheory[47–49],orhigh-genusobjects(Figure5d)[50,51].
Mathematics2019,7,125 4of17
Figure5.Mathematicalvisualizationsasart:(a)aBoy’ssurface[36];(b)aKleinbottle[38];(c)aregular
map[40];and(d)ahigh-genusobject[50].
For most of the above mathematical concepts and mentioned artists, Berkeley SLIDE [15] or
readilyavailableprogramssuchasRhino3D[52]orTopMod[53]areadequatetoolstocapturethe
corresponding models, because these shapes typically can be depicted with a few clearly defined
geometricalprimitives. ThisistrueevenfortheworksofartistslikeBrentCollins[8],KeizoUshio[54],
or Bob Longhurst [55], who intuitively sense some underlying mathematical principles, but do
not explicitly use a mathematical formulation or a computer program to define their shapes.
Also,the“D-Forms”introducedbyTonyWills[56]orthedevelopablesurfacesbyIlhanKoman[57,58]
haveaclearlydefinedgeometricalstructurethatcanbecapturedinaproceduralspecification.
EvaHild’swork[1],ontheotherhand,presentsanewmodelingchallenge. Herceramicsurfaces
aretrulyfree-form[2,3],andiftheunderlyingtopologywouldallowforsomeregularityorsymmetry,
Hildwoulddeliberatelyremoveit[4]. Thus,thepredefinedmodulesinSLIDEarenolongersufficient,
andmodelingherworkrequiressomeadditionalcapabilities,whichwillbedescribedinSection4.
Moreover,asdiscussedinSections7and8,Ifacedanothernovelchallenge: howtomakeasignificant
topologicalchangetosomeintuitivelyconceivedgeometries,whilekeepingthevisuallook-and-feel
thesame.
3. Classificationof2-Manifolds
From a mathematician’s point of view, Eva Hild’s sculptures ( Figures 1 and 2) are smooth
2-manifoldswithoneormoreborders[59]. Asabriefreminder: everyinteriorpointina2-manifold
has a neighborhood that is topologically equivalent to a small disk; and every border point has
aneighborhoodequivalenttoahalf-disk(Figure6a,b). Branchingofsurfacesheets,asinthespineof
abook,isnotallowed. Thus,ifthe2-manifoldsurfaceisfinelytessellated,theneveryfacetedgeis
usedbyeitheroneorbytwoadjacentfacets.
Figure6.Various2-manifolds:(a)a2-sidedcylinder;(b)a1-sidedMöbiusband;(c)agenus-2handle
body;(d)adisk:Eulercharacteristicχ=1,genusg=0;and(e)adiskwith5holesand5cuts.
All possible 2-manifolds can be classified by just three numbers: b, the number of borders; s,
itssidedness;anditsconnectivitydescribedbyitsgenus,g. Foragiven2-manifold,thenumberof
bordersiseasilydetermined: Westartwithsomeborderpoint;wefollowalongthisrimlineuntilwe
arrivebackatthestartingpoint;wemarkthisrimas‘counted’;andwerepeatthisuntilthereareno
furtherunmarkedborders. Todeterminethesidedness,s,westartwithaninteriorpointandpaintits