MINIMAL SURFACES FOR ARCHITECTURAL CONSTRUCTIONS UDC 72.01(083.74)(045)=111

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1 FACTA UNIVERSITATIS Series: Archiecure and Civil Engineering Vol. 6, N o 1, 008, pp DOI: 10.98/FUACE V MINIMAL SURFACES FOR ARCHITECTURAL CONSTRUCTIONS UDC 7.01(083.74)(045)=111 Ljubica S. Velimirović 1, Grozdana Radivojević, Mića S. Sanković, Dragan Kosić 1 1 Universiy of Niš, Faculy of Science and Mahemaics, Niš, Serbia Universiy of Niš, The Faculy of Civil Engineering and Archiecure, Serbia Absrac. Minimal surfaces are he surfaces of he smalles area spanned by a given boundary. The equivalen is he definiion ha i is he surface of vanishing mean curvaure. Minimal surface heory is rapidly developed a recen ime. Many new examples are consruced and old alered. Minimal area propery makes his surface suiable for applicaion in archiecure. The main reasons for applicaion are: weigh and amoun of maerial are reduced on minimum. Famous archiecs like Oo Frei creaed his new rend in archiecure. In recen years i becomes possible o enlarge he family of minimal surfaces by consrucing new surfaces. Key words: Minimal surface, spaial roof surface, area, soap film, archiecure 1. INTRODUCTION If we dip a meal wire-closed space curve ino a soap soluion, when we pull i o a soap film forms. A naure solves a mahemaical quesion of finding a surface of he leas surface area for a given boundary. Among all possible surfaces soap film finds one wih he leas surface area. Deep mahemaical problems lie in he heory behind. The heory of minimal surfaces is a branch of mahemaics ha has been inensively developed, paricularly recenly. On he base of his heory we can invesigae membranes in living cells, capillary phenomena, polymer chemisry, crysallography. Minimal surfaces are also applied in archiecure. In spie of he fac ha i seems ha soap film easily solves mahemaical problem of finding minimal surface for he boundary curve, aemps o solve some basic problems as well as o give descripion of minimal surfaces was hard work in mahemaics for over 00 years. The main fields of mahemaics conribuing o minimal surface heory are differenial geomery, complex analysis, heory of parial differenial equaions and calculus of variaions. Received November 6, 008

2 90 LJ.S. VELIMIROVIĆ, G. RADIVOJEVIĆ, M.S. STANKOVIĆ, D. KOSTIĆ In he recen ime, as in many oher areas, grea progress was made by using compuers. This new echnology enabled researchers o enlarge he family of minimal surfaces as well as o confirm old ideas, o see old absrac known minimal surfaces, o aler hem and o check heir properies. Theoreical invesigaion of hese surfaces is useful for applicaion of his knowledge in furher invesigaion of forms in archiecure. One of he firs uses of compuers was for he analysis of srucures, using heories ha have been developed coninuously from he 16h cenury. Minimal surfaces are exremely sable as physical objecs, and his can be an advanage in many kinds of srucures. From archiecs' poin of view compuerized illusraions of some of minimal surfaces are inrigued by he possibiliy of adaping hem o srucures, boh inerior and exerior. 1. INFINITESIMAL DEFORMATION OF A SURFACE WITH A FIXED CONTOUR Minimal surfaces are defined as surfaces of he smalles area spanned by a given space curve. The Plaeau's problem is he problem in calculus of variaions o find he minimal surface for a boundary wih specified consrains (having no singulariies on he surface). In 1873 a physicis named Joseph Plaeau observed ha soap film bounded by wire appeared o form minimal surfaces. The problem named afer him, Belgian physicis experimenally solved for some special cases. Jess Douglas 1931 solved his problem. In general, here may be one, muliple, or no minimal surfaces spanning a given closed curve in a space. Soap film mus go o he sae a which he surface area is minimized in order o minimize surface ension and reach equilibrium. Le us consider infiniesimal deformaion of a surface S : r = r, D R, including his iniial surface in a family of surfaces r r r S : = + z(, 0, = (, S S, (1) 0 = where deformaion field is surface normal. Differeniaing wih respec o u and v, we obain r r r r = + z( + z, u u u u r r r r = + z( + z. v v From here, neglecing erms of higher order hen he firs, we have E = E e + o( ), F = F f + o( ), G = G g + o( ). Inroducing he mean curvaure Eg Ff + Ge H =, we ge ( EG F ) v v

3 Le Minimal Surfaces for Archiecural Consrucions 91 E G F = ( EG F )(1 4H ) + o( ), and E G F = ( EG F )(1 H ) o( ). S (1) be a regular surface in + 3 R, hen he area enclosed by a fixed conour, A( ) = E G F dudv, (1) and A(0) = EG F dudv, () area on he surface S enclosed by a same fixed conour. The firs variaion ' A( ) A(0) EG F (1 H ) EG F + o( ) A (0) = lim = lim 0 0 dudv (3) = H EG F dudv = HdA. In he case when he mean curvaure vanishes H=0, we have minimal surface i.e. he surface of minimum area passing hrough a closed curve. A Monge pach r r = v, h( ), (4) is a minimal surface if (1 + uu u v uv vv h v ) h h h h + (1 + h u ) h = 0, (5) which is Lagrange's equaion of he minimal surface. I follows as an immediae consequence of he fac ha for a Monge equaion (4) mean curvaure is (1 + h v ) h h h h + (1 + h u ) h uu u v uv vv H = and H = 0. 3 (1 + h u + h v ). EXAMPLES OF MINIMAL SURFACES We will here noe some of he ypes of minimal surfaces suiable for applicaion a civil engineering and archiecure. Picures are made using program package Mahemaica. 1. A plane is a rivial minimal surface:

4 9 LJ.S. VELIMIROVIĆ, G. RADIVOJEVIĆ, M.S. STANKOVIĆ, D. KOSTIĆ. Enneper surface: 3. Higher order f Enneper surfaces: 4. The helicoid: 5. The caenoid: Meusnier found caenoid and helicoid. Helicoid and caenoid are he only wo ruled minimal surfaces. Caenoid can isomericaly be ben o helicoid hrough isomerical minimal surfaces

5 Minimal Surfaces for Archiecural Consrucions Jorge-Meeks surfaces wih n ends: n=4 7. Richmond surface: Spheres wih one planar and wo caenoid ends: 9. Chen-Gacksaer surface: 10. Cosa surface:

6 94 LJ.S. VELIMIROVIĆ, G. RADIVOJEVIĆ, M.S. STANKOVIĆ, D. KOSTIĆ 11. Cosa-Hoffman-Meeks surfaces of genus k: k=3 1. The singly-periodic Scherk surface: 13. The doubly-periodic Scherk surface: 14. The singly-periodic Riemann's saircase:

7 Minimal Surfaces for Archiecural Consrucions APPLICATION OF MINIMAL SURFACES IN ARCHITECTURE Increasing number of designers and archiecs are aware of he fac ha knowledge of form is a very imporan aspec of design of srucures. The main aim of his work is o poin ou o a class of surfaces ha are suiable for applicaion in archiecure. The main reason for applicaion of minimal surfaces in archiecure lies in he definiion. Having he leas area propery minimal surface is used for ligh roof consrucions, form-finding models for ens, nes and air halls. Among he surfaces having he same boundary minimal surface is he surface of he leas area. I's weigh is herefore less and he amoun of maerial is reduced on minimum. Form of huge soap films are spanned by he boundary and fixed a some poins. Balanced surface ension sabilizes he whole consrucion since he ension is in equilibrium a each poin on he roof, as on a soap film. Famous archiecs and among hem Oo Frei creaed minimal roofs. German Pavillon for Expo 1968 a Monreal is one of hem. The Munich Olimpic Sadium and Kongreshall in Berlin are he ohers. Oo Frei was also experimening wih hanging chain nes and soap films. Hyperbolic paraboloid is a ruled surface. Someimes i is menioned o be a minimal surface, bu i is no. The only ruled surfaces among minimal surfaces are caenoid and helicoid, and plane. However hyperbolic paraboloid a some condiions can be used as good and simple approximaion of minimal surface. 15. Experimenal building by archiec Michael Bur The archiec Michael Bur, called he 'Hexahyp', a he Israel Insiue of Technology, Haifa, Israel, and he picure he made on pars of is surface. The fiberglass covering consiss of saddle-back shaped surfaces of he ype ha would be assumed by soap films sreched beween oulines of he supporing srucure. These surfaces, called minimal surfaces, provide he maximum of srengh for he minimum amoun of maerial.

8 96 LJ.S. VELIMIROVIĆ, G. RADIVOJEVIĆ, M.S. STANKOVIĆ, D. KOSTIĆ REFERENCES 1. Monge G. Applicaion de l'analyse a'la geomerie. Paris, 1807 e Caalan E. Memoire sur les surfaces gauches a plan direceur. Journ. Ecole poly. XVII, cah. 9, Gray, A. Modern Differenial Geomery of Curves and Surfaces wih Mahemaica, nd ed. Boca Raon, FL: CRC Press, Velimirovic Lj. S., Radivojevic G., Kosic D. Analysis of Hyperbolic Paraboloids a Small Deformaions Faca Universiais, Series Arhiecure and Civil Engineering,vol 1. No.5,1998, Velimirovic Lj. S., Radivojevic G., Kosic D. Geomeric Analysis of Hyperbolic Paraboloid as Building Technique Elemen, Bull. For Applied Mah., Budapes. MINIMALNE POVRŠI U ARHITEKTONSKIM KONSTRUKCIJAMA Ljubica S. Velimirović, Grozdana Radivojević, Mića S. Sanković, Dragan Kosić Minimalne površi su površi najmanje površine za dau granicu. Ekvivalenna je definicija da su o površi sa nulom srednjom krivinom. Teorija minimalnih površi se rapidno razvija u novije vreme.konsruišu se mnogi novi primeri. Osobina da su o površi sa minimalnom površinom čini ih pogodnim za primenu u arhiekuri. Glavni razlog za o je da su ežina i količina maerijala svedeni na minimum. Poznai arhieke kao Oo Frei su kreirali novi rend u arhiekuri. U poslednje vreme je posalo moguće uvećai familiju površi koje se primenjuju konsrukcijom novih.