A PARAMETRIC REPRESENTATION OF RULED SURFACES

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1 A PARAMETRIC REPRESENTATION OF RULED SURFACES ELENA PROUSALIDOU, SEAN HANNA Univesiy College London, Unied Kingdom Asa. This pape poposes a simple paamei sysem o geneae an almos omplee se of uled sufaes ha may e used o desie uilding geomey. The majo lasses of egula, named uled sufaes an e geneaed fom a limied se of uves. Eah of hese is shown o e eduile o a ansfomaion of a single sandad uve, a helix, and heefoe epesened y a limied se of six paamees. Six exa paamees an posiion eah sufae on a gloal oodinae sysem. The epesenaion is designed o e flexile enough o epesen oh plana and uvilinea foms, poduing a desipion of fom fom minimal daa. 1. Inoduion Geneaive digial ehniques aemp o exploi he powe of infomaion ehnology as a die ool o epesen and geneae ahieual foms. Demonsaive appoahes in his new field of design ehniques ae paamei and evoluionay design, ofen used in ominaion. Thei ommon feaue is he design of he end podu as an insane of a lass of possile soluions. Anohe similaiy hey shae is he desipion of he podu y a seies of paamees ha ake on diffeen values. These values an eihe e inpu y he designe, in he fis ase, o e evolved y a pogam, in he lae. Paamei and geneaive epesenaions of uildings, whehe ased on ohogonal (Mah and Seadman 1971; Rosenman and Geo 1999; Seadman and Waddoups 2; Seadman 21) o uvilinea (DeCOi 2) geomey, ae poweful y viue of hei ailiy o apue a high degee of vaiaion in a few numeial values. This pape poposes a paamei epesenaion of uled sufaes ha uses a minimal se of vaiales o epesen a wide vaiey of sufaes, inluding hose mos ommonly used in ahieue. A seial omposiion of he sufaes an e applied in he epesenaion of uilding geomey.

2 2 E. PROUSALIDOU, S. HANNA 1.1. ARCHITECTURAL STRUCTURES AS NUMERICAL SYSTEMS Configuaion of uil foms as epesened y Lionel Mah in his isomophi Boolean appoah (Mah 1972) and Philip Seadman in his aheypal uilding (Seadman 1998) ae indiaive ealy aemps o epesen ahieue using a numeial desipion of fom. Boh use inay enoding ha oesponds o a maix of uoids. Alhough hey do ahieve a ompa epesenaion of eangula foms, he undelying elaions of he uilding elemens ae esied o poximiy and veial/hoizonal paking REPRESENTATIONS IN EVOLUTIONARY DESIGN Repesenaion of fom in sings is widely applied in Evoluionay Design ehniques whee he geneaive poess opeaes wih vaious paamees enoded ino sing-like suues. Appoahes foused on ohogonal spaial onfiguaions, suh as he leaning appoah developed y John Geo and he hieahial gowh appoah developed y Mike Rosenman (Rosenman and Geo 1999) o he poedue poposed y Jakson (Jakson 22), use genes o epesen simple design aions as poduion ules o dawing insuions whih when exeued esul in shapes o podue pas of design soluions (Benley 1999). Repesenaions of 3dimensional fom ae ofen ased on modula assemlies of pimiive foms, suh as he Repile sysem poposed y Faze (Faze 1995), lipped sehed uoids developed y Pee Benley and Jonahan Wakefield (Benley 1999a), Fom Gow y Sephen Todd and William Laham (1999) o he poduion of ojes epesened hough he inseion of sphees a he isospaial gid as poposed y Paul Coaes, Teene Boughon and Helen Jakson (Coaes Boughon and Jakson 1999). While hese modules ae quie simple, a paamei desipion of fom may e eaed ha is flexile enough o epesen omplex uves and sufaes. This is aply demonsaed y Buy s exended appliaion of paamei ehniques on he analysis of he uled sufaes used y Anonio Gaudi. He applies paamei design sofwae a he Sagada Familia wih he inenion o emodel and esolve he use of hese sufaes ino a measuale inepeaion ha an e used o advane he uilding wok, a fom of aionalisaion useful in design (Buy e al. 21). The poposed model inopoaes Buy s wok, and pesens a simple sufae desipion ha may e used in evoluionay design and ohe paamei appoahes. I an e evaluaed y he numes of paamees needed and he ange of foms i is apale of epesening.

3 A PARAMETRIC REPRESENTATION OF RULED SURFACES 3 2. Ruled sufaes A uled sufae an e geneaed y he moion of a line in spae, simila o he way a uve an e geneaed y he moion of a poin. A 3D sufae is alled uled if hough eah of is poins passes a leas one line ha lies eniely on ha sufae. Simple o define and onsu a a loal level, uled sufaes ae ale o desie high levels of fom omplexiy, espeially y hei ineseions when assemled (Buy 23). Ruled sufaes have een exensively applied o ahieue, wih hei poenial exploied y new ehnologial means. The Paamoph (DeCOi 2) is an example. 2.1 ARCHITECTURAL STRUCTURES Tadiionally he plane, ylindial and onial sufaes have een of he oades use in ahieue, and hei familiaiy ofen osues hei definiion as uled. The hypeoloid and he hypeoli paaoloid ae he mos ypial examples of moe omplex uled sufaes, followed y he helioid and, eenly, he moeius sip. Consuion of new suual sysems using hypeoloids and hypeoli paaoloids was fis aomplished y Vladimi Shukhov, a he end of 19h enuy (English 25). The douly-uved sufaes wee fomed of a laie of saigh ion as. Gaudi exended he appliaion of uled sufaes y using hee sufaes ha ould e onsued a his ime: hypeoli paaoloid, helioid and hypeoloid, all geneaed hough saigh lines. The enie design of he nave in he Sagada Familia is an assemly of hese hee geomeies (Buy 1993). These foms wee exensively used y Felix Candela, and Le Cousie s aaion o hem is demonsaed y Philips Pavillion. 2.2 SPECIAL FEATURES The mos impoan feaue of uled sufaes in espe o design is hei simpliiy of onsuion elaive o hei siking shape. Buy (1993) emphasizes hei ailiy o failiae onsuion: Any of hese sufaes an e simply enough desied as he poins of oigin and eminaion of an appopiae nume of saigh lines. Fuhemoe, he omponens of a lage whole an e aved independenly off-sie and hen e assemled o fom he final omposiion. 3. Developmen of he paamei epesenaion The mos familia way of onsuing he sufaes in CAD pakages is using lines ha join oesponding poins eween wo algeai uves. In mahemaial ems, howeve, he mos ommon way of desiing he

4 4 E. PROUSALIDOU, S. HANNA sufaes is using a saigh line moving along a uve wih he dieion given y anohe uve. A sufae is alled a uled sufae, if i has a C 2 -paameizaion of he following kind (Kühnel 22): f(u, v) = (u) + v X(u) (1) whee is a (diffeeniale, u no neessaily egula) uve and X is a veo field along whih vanishes nowhee. The v-lines (wih onsan u) ae Eulidean lines in spae. They ae alled geneaos o ulings of he sufae and an e inuiively undesood as he vaious posiions of he geneaing line L, he movemen of whih depends on one paamee. The onsuion of a sufae equies wo paameeised uves, i.e. he ase uve and he dieo uve. The ase uve (o dieix) of he sufae is he uve along whih uns he saigh line. The line exends in he dieion given y he dieo uve d, whih may e visualised as a veo field (a given sequene of uni veos ha vaies oninuously) on. Mahemaial equaions fo he wo uves an e expessed as posiion veos. The paamei funion of he helix expessed in veo fom is: [aos(), sin(), ] (2) Simple ansfomaions of he helix esul in a se of he uves needed o geneae uled sufaes. If he z veo omponen is suppessed he uve is degeneaed o a ile. If he x veo omponen is suppessed he uve is degeneaed o he paamei sine funion. If he x and y veo omponens ae suppessed he uve is degeneaed o a line. HELIX [ aos(), sin(), ] CIRCLE [ aos(), sin(), ] SINE [, sin(), ] LINE [,, ] TABLE 1. Cuves epesened as Veos of igonomei funions.

5 A PARAMETRIC REPRESENTATION OF RULED SURFACES 5 LINE CIRCLE HELIX SINE LINE CIRCLE HELIX SINE LINE CIRCLE HELIX SINE LINE CIRCLE HELIX SINE TABLE 2. Ruled Sufaes

6 6 E. PROUSALIDOU, S. HANNA The ominaion of hese fou uves as ase and dieo espeively geneaes sixeen ypes of uled sufaes. The oaion of one uve aound he axes geneaes hiy wo exa sufaes (sixeen fo eah oaion). Eah sufae of a suue an e expessed using six paamees: hee fo he ase (a,, ) and hee fo he dieo uve (a d, d, d ). The plane and ohe sufaes equie wo uves along diffeen axes. If he ase uve is defined as menioned aove, he dieo uve should e defined eihe as [ d sin(), d, a d os()] (3) o [ d, a d os(), d sin()] (4) The poedue ha was followed was he addiion of an exa value in he x omponen of he dieo veo so ha i eomes [a d os()+ d d, d sin(), d ] (5) whee paamee d d is applied fo oaing one uve efoe geneaing he sufae. The sufaes geneaed in his way ae pesened in Tale 2. The ase uve is onsanly oiened along he Z axis while he dieo uve is oiened along axes X, Y and Z. Among hese foy eigh sufaes he plane, he one, he ylinde, he hypeoloid and he helioid ae inluded, in addiion o he ommonly used sinusoidal sufae. When all n sufaes ha fom an ahieual omposiion ae indexed, a 13 n maix an epesen a uilding. a a a a n n n a a a a d1 d 2 d 3 dn d1 d 2 d 3 dn d1 d 2 d 3 dn d d d d d1 d 2 d dn x1 x2 x3 x n y1 y2 y3 yn z1 z2 z3 zn x1 x2 x3 xn y1 y2 y3 yn z1 z2 z3 zn Whee: α,, : paamees of ase uve [ aos(), sin(), ] and α d, d, d, d d : paamees of dieo uve [ aos()+ d, sin(), ] x : anslaion of sufae aound X axis y : anslaion of sufae aound Y axis

7 A PARAMETRIC REPRESENTATION OF RULED SURFACES 7 z : anslaion of sufae aound Z axis x : oaion of sufae aound Z axis y : oaion of sufae aound Z axis z : oaion of sufae aound Z axis 3.1 HYPERBOLIC PARABOLOID The aove saegy epesens many of he egula ypes of uled sufaes, u he onsuion of he ommonly used hypeoli paaoloid equies he use of anohe ype of uve, he paaola, whih an no e podued y a linea ansfomaion of he helix. Is funion is expessed y he posiion veo [,, 2 ] (6) and hus equies one addiional, quadai paamee e d. The pogam was modified y adjusing he geneal funion fo he dieo uve: [a d os()+ d d, d sin(), d +e d 2 ] (7) To epesen his exa lass of sufaes his single exa paamee was added o he veo s omponens. 4. Applying he epesenaion The epesenaion was implemened in a Poessing pogam eaed o daw muliple sufaes. This eeives a seies of paamees fo eah sufae as inpu y he use and daws an assemly of sufaes ha epesen he uilding. Speifially, fo eah sufae he pogam eeives he paamees a,, and a d, d, d, d d as inpu and exeues he algoihm as follows: A eah sep: 1. inemen paamee y a onsan nume 2. alulae ase and dieo uve veos fo value and daw a poin wih oodinaes defined y eah veo 3. join onseuive poins o daw wo uves 4. alulae slope of dieo uve y suaing suessive values fo eah veo omponen. 5. nomalise slope veo 6. daw a line of fixed lengh saing fom he poin on he ase uve and has is dieion defined y he slope of he dieo uve, i.e a uling. This ieaive poess is epeaed fo evey poin of he ase uve. A new line is geneaed fo eah value of paamee. The assemly of hose lines ompose he sufae. In ode o daw he assemly of many sufaes as a unified whole he ode was modified o inlude he sufae as a lass, so ha

8 E. PROUSALIDOU, S. HANNA muliple insanes an e eaed simulaneously. As long as he values of he paamees fo eah sufae vay, he insanes eaed display a wide vaiey of foms.

use. These allow peise onol of he elaion eween sufaes. One he sufaes ae assemled ogehe, hei ineseions an e ompued ased on he popeies of he ulings. 4.

El Pa Conol Towe o Mies Seagam Building. The values of he paamees ae pesened in 13 n maies, wih he exepion of he hypeoli paaoloid assemly.

8 8 E. PROUSALIDOU, S. HANNA muliple insanes an e eaed simulaneously. As long as he values of he paamees fo eah sufae vay, he insanes eaed display a wide vaiey of foms. The pogam employs six moe vaiales o define he ansfomaion maies ( x, y, z, x, y, z ) and some addiional gloal vaiales suh as he inemen of value, and he line lengh, whih an eihe e fixed o modified y he use. These allow peise onol of he elaion eween sufaes. One he sufaes ae assemled ogehe, hei ineseions an e ompued ased on he popeies of he ulings. 4.1 REPRESENTATION OF REAL BUILDINGS The pefomane and effiieny of he sysem was esed y onsuing onise, paamei desipions fo a seies of exising suues, anging in fom fom Calaava s Bodegas Ysios and Bofill s El Pa Conol Towe o Mies Seagam Building. The values of he paamees ae pesened in 13 n maies, wih he exepion of he hypeoli paaoloid assemly. The maies vay in size: hey ae 13 n whee n is he nume of sufaes. Aive paamees of eah sufae ake a eal o inege nume value, ohewise hey ae se o. The affi onol owe of El Pa omines ylindial and onial sufaes wih a hypeoloid. Values of he maix ae pesened in moe deail in Tale 3. The effe of hanging a single value in he maix is demonsaed in Figue 2. Figue 1. Taffi Conol Towe-El Pa Figue 2. Vaiaions of fom y hanging a value in he maix

9 A PARAMETRIC REPRESENTATION OF RULED SURFACES 9 ROW BASE CURVE paamees ig. funion DIRECTOR CURVE paamees ig. funion RULED SURFACE TRANSLATION axis unis 1,17,17,,,1,, [,17os,17sin] [,,1] - ile line - 2,2,2,,,1,,2 [,2os,2sin] [,,1] Z ile line 2 3,6,6 3,,,,,36 [,6os,6sin] [3,,] Z ile line 36 4,3,3,,2,2,,5 [,3os,3sin] [2os,2sin,] Z ile ile 5 5,2,2,,,1,, [,2os,2sin] [,,1] - ile line - 6,3,3,1,1,6,,65 [,3os,3sin] [os,sin,6] Z ile helix 65 7,3,3,1,1,-6,,65 [,3os,3sin] [os,sin,-6] Z ile helix 65 8,53,53 3,,,,,41 [,53os,53sin] [3,,] Z ile line 41 TABLE 3. Values of El Pa-maix in deail

longiudinal axis. 5 5 4 2 2 2 2 2 21 1 1 1 1 15 2 6 5 5 5 5 7 55 55 41 41 5 9 41 41 55 55 25 425 Figue 3.

45 17 3 45 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 5 4 24 24 24 24 24 24 24 17 17 17 17 157 145 95 45 5 55 15 155 1 24 1 12 11 12 12 12 18 11

10 1 E. PROUSALIDOU, S. HANNA In he ase of Bodegas Ysios, he load eaing walls ae a sinusoidal shape and he oof is a uled sufae wave, whih omines onave and onvex sufaes as i evolves along he longiudinal axis Figue 3. Bodegas Ysios Palae of Assemly is a moe omplex omposiion of diffeen sufaes: hypeoloid, planes, ylindial sufae in addiion o a pyamid PI / 6 PI / 6 PI / 6 PI / 6 Figue 4. Palae of Assemly Los Mananiales povides an illusaive sample of an assemly of hypeoli paaoloids oaed aound a enal poin. I was seleed o es he exended vesion of he pogam PI / 4 2 * PI / 4 3* PI / 4 4 * PI / 4 5* PI / 4 6* PI / 4 7 * PI / 4 Figue 5. Los Mananiales

A PARAMETRIC REPRESENTATION OF RULED SURFACES 11 The Seagam Building is omposed of plana sufaes. I was seleed fo a ompaison o he epesenaion poposed y L.Mah.

11 A PARAMETRIC REPRESENTATION OF RULED SURFACES 11 The Seagam Building is omposed of plana sufaes. I was seleed fo a ompaison o he epesenaion poposed y L.Mah Figue 6. Seagam Building 5. Disussion 5.1 GENERAL CHARACTERISTICS OF THE SYSTEM The paamei epesenaion povides signifian infomaion in he maix of values. Looking a he maix, infomaion an e insanly aquied aou he ype of sufaes ha ompose he suue (plana o uvilinea), he nume of diffeen ypes, vaious heigh levels, exisene of epeaed elemens, elaion eween he loaions of elemens e. Also some values, possily signifying ommon feaues, an e diely ead. By expessing a suue as a seies of values ha oespond o a speial onfiguaion of elemens, he designe an insanly eae muliple vaiaions of fom pesened as hee-dimensional models. The use ineas y aleing values. The model an e modified a limiless amoun of imes while always mainaining elaions eween pas. Emedding ules and onsains in he epesenaion, in he fom of paamees ansfoms he ohewise passive epesenaion ino an aive one (Kalay 24). I has een oseved fom he esuls ha small hanges in values an lead o unexpeed hanges of fom. Thus expeimening wih andom values an lead o ineesing eaions ha may povoke he designe s eaiviy. The use of a limied nume of vaiales endes he epesenaion highly flexile. Elemens an e assemled in an unlimied nume of ways. Moeove i is effiien in ems of speed and simpliiy. Emedding ules and elaions eween elemens of he sysem makes i vey easy o undesand, manipulae and use. Wih no speial effo a hee-dimensional model an e podued simply y eneing a nume of values.

12 12 E. PROUSALIDOU, S. HANNA 5.2 LIMITATIONS The inenion of using minimal daa esis he appliaion of he paamei epesenaion. I is no appliale o suues ha ae omposed eihe of uled sufaes swep along a fee fom uve, suh as DeCOi s Paamoph, o of fee fom sufaes, suh as he Guggenheim Museum in Bilao. 5.3 FURTHER WORK Apa fom fuhe developmen of he epesenaion o inlude fee fom uves, i an e inegaed wih 3D CAD/modelling sofwae as an exploaoy geneaive design ool. Anohe geneal developmen an e he implemenaion of onsains ha onne one sufae o he ohe suh as paallelism, pependiulaiy and angeny. If he epesenaion is used as a geneaive design mehod, onsains suh as sufae oninuiy would e useful o ensue ha all designs make sense geomeially, o o eah ojeives suh as suual sailiy. Consains an also e speified as ondiional elaions. Inopoaing ondiional expessions migh exend he inees of he mehod. The paamei epesenaion may also e implemened in seah ehniques. The poposed epesenaion would e inegaed in a genei algoihm y mapping he paamees ha desie a sufae o a genoype, he alleles of whih would oespond o he se of values. An iniial se of ules and onsains esponding o a given polem and speifiaion of finess ieia is equied, y whih a finess es would evaluae how lose he values ae o a given example o ieion. Suh a poess ould e used o find a aional uled sufae definiion esemling any given fom, as in he uilding epesenaions in seion 4, and seve as a poin of depaue fo fuhe design exploaion. Alenaively, an analysis of sola shading, suual pefomane o ohe popeies ould e pefomed on he model and he fom opimised o solve a speifi design polem. 5. Conlusions Ceaing a paamei epesenaion involves hoosing he igh paamees and esalishing onneions eween hem. In his ase i has een ased on elaions eween elemens ha ae oh simple and eonomi: he vaiaions podued y he ansfomaion of he helix. This eonomy of paamees esuls in a model ha is poweful in ha i equies vey lile numei daa o desie a lage ange of possile foms. The nume of paamees needed (he fewe he ee) and he ange of foms apale of eing epesened (he moe he ee) ae he evaluaion ieia fo he mehod. In ems of epesenaion, his minimises

13 A PARAMETRIC REPRESENTATION OF RULED SURFACES 13 edundany in he daa equied. In ems of seah (e.g. genei algoihm), i edues he dimensionaliy of he seah spae and heefoe simplifies he poess. The ih speum of podued foms, inluding plana and a signifian nume of uvilinea foms, do desie he geomey of a lage nume of uildings and eause of he ahieved paameeisaion of he vaious geneaing uves in a single model, he nume of paamees needed is edued. The moe familia mehod of joining wo ounday uves y line segmens has he advanage of allowing any feefom uled sufaes o e defined, u he majoiy of uves used in aual uildings ae aional uves defined y he poposed mehod, and fo hese he paameeisaion is minimal. Fo uildings onsising of uled sufaes, a signifian amoun of infomaion aou he fom an e enoded in he maix of values. Having a limied nume of vaiales ineases flexiiliy of epesenaion. Using sufaes omposed of saigh lines as uilding loks of he sysem may simplify sai alulaions as well as he aual onsuion. The model impliily offes indiaions of he suue and is ehaviou. Minimal enoding ased on numeial paamees ould enefi gealy fom employing onsains o links eween design elemens, somehing ha an e easily ahieved in his model eause i is mahemaially explii. Consains ae a pa of implemenaion o follow depending on how he model is o e used. The esuling implemenaion allows an ineaive manipulaion of he model y hanging paamees. Muliple vaiaions of he model an e apidly podued so ha design exploaion is pefomed y seleion of opimal vaiaions. The poedue is simple, developed fo a limied nume of paamees, use-fiendly and effiien in ems of speed. Apa fom he inegaion of he epesenaion in 3D modelling sofwae, enoding suues in his way makes possile is inegaion in evoluionay design ehniques, suh as genei algoihms. I has een demonsaed how a simple paamei sysem ased on elaions of elemens and using limied amoun of daa an e developed as a useful design ool. I an suffiienly epesen a wide vaiey of foms effiienly and wih few vaiales. Aknowledgemens The auhos would like o hank Chion Moam fo his assisane in he developmen of he Poessing pogam and his ommens.

14 14 E. PROUSALIDOU, S. HANNA Refeenes Benley, P: 1999, An Inoduion o Evoluionay Design y Compues, in P Benley (ed.), Evoluionay design y Compues, Mogan Kaufmann, San Faniso. Benley, P: 1999(a), Fom Coffee Tales o Hospials: Genei Evoluionay Design, in P Benley (ed.), Evoluionay design y Compues, Mogan Kaufmann, San Faniso. Buy, M: 1993, Expiaoy Chuh of he Sagada Família, Phaidon Pess, London. Buy, M: 23, Beween Inuiion and Poess: Paamei Design and Rapid Pooyping, in B. Kolaevi (ed.), Ahieue in he Digial Age- Design and Manufauing, Spon Pess, New Yok. Buy, M, Buy J, Dunlop, GM and Mahe A: Dawing Togehe Eulidean and Topologial Theads, pesened a SIRC 21. Coaes, P, Boughon, T and Jakson, H: 1999, Exploing Thee-Dimensional Design Wolds using Lindenmaye Sysems and Genei Pogamming, in P. Benley (ed.) Evoluionay design y Compues, Mogan Kaufmann, San Faniso. DeCOi: 2, Tehnologial Laeny: fom Auoplasi o Alloplasi, Digial Ceaiviy, 11 (3): English, EC: 25, Vladimi Shukhov and he Invenion of Hypeoloid Suues, Meopolis & Beyond: Poeedings of he 25 Suues Congess and he 25 Foensi Engineeing Symposium, New Yok. Faze, J: 1995, An Evoluionay Ahieue, Ahieual Assoiaion Puliaions London. Jakson, H: 22, Towad a Symioi Coevoluionay Appoah o Ahieue, in P Benley and W Cone (eds), Ceaive evoluionay sysems, Mogan Kaufmann, San Faniso. Kalay, YE: 24, Ahieue's New Media : Piniples, Theoies, and Mehods of Compue-Aided Design, MIT Pess, Camidge. Kühnel, W: 22, Diffeenial Geomey : Cuves - Sufaes Manifolds, AMS, Rhode Island. Mah, L: 1972, A Boolean desipion of a lass of uil foms, Camidge Univesiy Pess, Camidge. Mah, L and Seadman, P: 1971, The Geomey of he Envionmen, R.I.B.A Puliaions, London. O'Neill, B: 1966, Elemenay Diffeenial Geomey, Aademi Pess New Yok. Seadman, P: 1998, Skeh fo an aheypal uilding, Envionmen and Planning B: Planning and Design, 25h Annivesay Issue, pp Seadman, P: 21, Binay enoding of a Class of Reangula Buil-Foms, Poeedings, 3d Inenaional Spae Synax Symposium, Alana. Seadman, P and Waddoups, L: 2, A Caalogue of Buil Foms, using a Binay Repesenaion. Poeedings, 5h Inenaional Confeene on Design and Deision Suppo Sysems in Ahieue, Nijkek, The Nehelands, pp Todd, S and Laham, W: 1999, The Muaion and Gowh of A y Compues, in P Benley (ed.), Evoluionay design y Compues, Mogan Kaufmann, San Faniso. Rosenman, M and Geo, J: 1999, Evolving Designs y Geneaing Useful Complex Gene Suues, in P Benley (ed.), Evoluionay design y Compues, Mogan Kaufmann, San Faniso.

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