1 • IMtlM" Hco_ E=~t:~'!. .. "'...-.",·Yv_ /sTRUCTURAL STABILITY and MORPHOGENESIS An Outline of a General Theory of Models Tranllated from the French edition, as updated by the author, by D. H. FOWlER U....... ".04 w ....... + Witt. 0 ForOJWOfd by C. H. WADDINGTON 197.5 (!) W. A. BENJAMIN, INC. ,,""once;;! &ook Program Reading, Mauochvsetfs I: 'I) _,...... O""MiIo .O,~c;;.,· 5,..,. lolyo J r If ! "• f• Q f iffi Enlll'" £d,,;on. 1m Orilon,l", pubh5l1ed in 19'12. • 5tdN1i1. Itruclut. t1 r'IMIfphas~oe b~ d'uroe Ihlone ghw!:,,1e des modiO'le!o bv W. A. IIMlalnln, 11><:., ',,;h.-naQ Book Pros.,m, Lib ...., or Cell lr"' Cltiliollinl! In Publication Data Them , Ren ~. 1923 - Structural stablll ty and morphogensis . 1. Btology--Mathematlcal. lIIOdels . 2 . Morphogenesis-­ Mathematical. models. 3. Topology-. I . Title. QH323 .5.T4813 574 .4'01'514 74-8829 ISBN 0-8053 -~76 -9 ISBN 0-8053 - S~?77 - 7 (pbk.) American M.,hftT1.,IQI Society (MOS) Subje<l Oassifoc"ion SdI~....., (1970); 'J2o\101Si'A2!l Copy.lllht elW5 by W. A. 1IM~IT"n . Inc. Published 1,"",1",,_sly'" CMid• • AllIt_l "JlHsned res,1t! , .... ed. mN... , I • , permIssion~::'~~:t.:~ oft""" ';:;~~~~~~~pYbllsl!e. , ~~~~~ BooIt Pros •• m, RNdinl. ~nKhuseltJ ()'867. U.S.A. END PAPERS; The hyperbolic: umbillc In hydrody... mIa : Waves . t Plum ItJllnd, M8Uachvselts. f't1oto courtesy 01 J. GLH;klnheirner. CONTENTS , FOREWORD by C. 1-1 . Waddington • • . • • • • • • • FOREWORD to the angmal French edItIO-n. by C. II . Waddington . • • • • • • • PREFACE(tran,lated from the French edition) · . xx iii TRANSLATOR'S NOTE . ... C HA PTER I. INTRODUCTION J.I The Program . 1 A. T he succession 01 form . 1 B. Sclcnce. and the IndeterminiSm of phenomena 1 1.1 TlwlheoryojmodeiJ. 2 A. Formal models • 2 B. Continuous mooels ) !.J A h"'Qt1COllhllCJSCflhlcul dlgrtJSlon • 4 A. Q uahlali\t or quantilali\t . 4 B. The lihaoo,.. of h,story . • , C. An CJl lenSlOn of oLir basic mtulIIon • 6 l.f Tltt' NHlSlnKl/()If of II trIOdel . • 7 A. The catastrophe set . • 7 B. The mde~ndence of the substrate • 8 C. BiolOgIcal and Inert forms • • • 8 O. Conclusion . , • • Appt>ntl/Jl: Tilt tI(),lo ~ oj un oIJ}«1 10 \ '\ CHAPTER 2. FOR)'I AND STRUCTLRAL STABILITY . 12 1.1 1M lIudy ojjorms • • • · · • · 12 A. The usual sense of form or shape. 12 8. The space of forms • • JJ c. Structural sUiblIJty. • • 14 D. Nonform forms. • • • 14 E. Geometrical forms. • • 1.1 Structural stability and sr/",lIIfir OOSfrC<I!Wn . " A. The conditIons of scientihc cxpcnment. " B. The quantum objecllon . • • • " c. IsomorphiC processes. • • • " D. The nature of empirical functions " "- Regular points of a proccss . "18 11 Strurtural stability and motkls CHAPTER 3. STRUcrURALSTABI LITY IN MATHEMATICS 21" J. J Th~ gen~ral problem. 21 A. Continuous families and bl furcanon. 21 B. Algebraic geomeuy . 21 C, Geometrical analysis. 23 D. Differentiallopology . 23 E. DifferentIal equauons 25 F. FunctIOnal analYSIs and partial differential eqlJauons. 28 J.1 AlgdJro u1l(/ morplwst'fWsis, • . • • 29 A. An ex.ample of bifurcatIOn. 29 B. The universal unfoldlllg of a SlIIgulanly of flnlle c(Xhmension . 31 C. An example; the ulII~ersal unfoldlllg of y - r . 32 D. The generallhoory of the unlH'!'Jal unfoldlll8 JJ E. The case of a function . 34 CHAI'TER 4. KI NEMAT IC OF FORMS: CATASTROPH ES 38 <.1 Spa/lUI p,or~ssn • • • • • • 38 A. The morphology of a proces' 38 8. AHraclol'$ . · · • • • • • • 38 C. The partition 11110 bulllS • • • <.1 MUlht'nrul'eal modt'ls fly "BulaT prornsd. "40 A Static models • 40 8. Metabolic models . • • • • • 40 c. Evolution of fields . • • • • 41 COIIU1l11 vii O. Equlv al ~nc~ of mod~ls 41 E. Iwmorpluc procems . 42 4.J ClJrQStrophu . 42 A. Ordinary catastroph~ paints 42 B. Essential catast roph~ painu 4. 4 "'OI1'hoger/~fIC fi~/ds (WOCiUfed "'illl/«ol cu/oJlropilu. "44 A. StatIC modell . 44 B. Stable IlnglilantlC'S of wave fronll C. Metabolic models. "46 4," PnfimJlIQ'Y clanifjrallQn af cOltutrophc 47 A. The domain of ulstcnce and basm of an a\tractor . 47 B. Conflict and bifllrcation catastrophes 47 ' .6 Tlwmody1Wmicol coupling. A. Mu;:rocanomc;al entropy. "4' B. The interacuon of t",osystelT1$ 49 C. The approach to the equilibrium state with ihermodynamlcallnleraction . • 50 D. Polam:ed dynamIcs . E. The pseudogroup of local equivalences of a field " Th~ " 4. 7 ndllcttl field. • A. The defmitlon of the redllced field . " 8 . The !Ioelf-mterachon of a fIeld: th~ ~~oluhon " of th~ reduced fIeld . " CHAPTER 5. ELEMENTARY CATASTROPHES ON R4 ASSOCIATED WITH CONFLICTS OF REGIMES . 55 "" Fitlds of grudlenl dynmllJO and Ihe assocIated Sialic- madid 55 A. The competItion bet"'«n local regimes. 55 B. Muwel1's con~enuon . 56 ".1 Tht algebraic SI* afpoilll smgulunlies af 0 /HJltnfjal fimCIIQrI . 51 A. The cat.astrophe set . 51 B. Blfurcahon strata. 57 C. A study of lsolatttl smglliar painu: coran" 58 D. The resldllal slngulanty. 59 ,., Ca/(Ulrophes of caronk O/~. • • 60 A. Strata of codimension lero . • • ,\ ". Strata of codimension one • • • • ,\ I. Strata of fold Iype • • ,\ 2. ConflIct llrat• . • " vi ii COOIUIIIS C. Strata of codimension two. 62 L The transversal in1t:rseclion of two fold strata 62 2. Strata of cusp points and the Riemann.H ugoniOl catastrophe. 62 3. Connict strata . 6l 4. The transversal intersectIon of a fold stratum and a connict stratum . 64 D. Strata of codimension th ree 64 1. The swallow's tail 64 2. Transition strata. 67 3. Connict strata. 68 E. Strata of codimension four: the butterfly 68 j.4 Eft'men/Ilry mlC15lrophes oj c(H"llnk /Wo . 73 A. UmbiliCi. 73 B. ClassificatIOn of umhilics . 74 C. The morphology of umbllics 7S I. The hyperbolic umbillc: the crCSt of the wave. 7S The elliptic umblhc: the haIr 78 3. A remark on the terminology. 81 4. The parabolic umbilic: the mushroom 81 D. A general remark on bIfurcation catastrophes 90 j.j The morphology oj breakers OJ j.6 Al/roclOrs oj 0 metabolic field. % CHAPTER 6. GENERA L MORPI-I OLOGY. 101 6.1 Themoinrypes ojjormJotldlheir'.'I."ollllion . 101 A. Static and metabolic models . 101 B. Competition of aUTaCtors of a HamiltOnian dynamic. 102 C. Creation of a new phase; generalized catastrophes. 103 1. Lump catastrophes. 103 2. Bubble catastrophes. 103 3. Laminar and filament catastrophes. 103 4. Catastrophes with a spatial parameter. 103 D, Superposition of catastrophes. lOS E. Models for a generali7.ed catastrophe; change of phase . 105 F. The formaliWlion of a generalized catal;trophe . 106 6.1 ThO' gt'Omelr), of 0 c(mpling. lOS A. Mean field . 109 , " B. Examples of mean fields aSSOCiated with an elementary catastrophe • · 110 1- The fold . • · 110 2. The cusp. • · ItO C. Mean fields associated with a coupling. • · 11 2 D. Mean fitld. scale. and catastrophes • · 113 6,3 Scmomic nwdel5 . · 11 3 A. Definition of a chrcod . · I 14 B. A subchrcod of a chrcod · 115 C. The family tree of chrrods · I 16 D. Condllio na l chrtOds li nd levels of organization · I 16 E. Ellamplcs of semantic modcl~ . · I 17 I . The diffusion of a drop of ink in water . • 11 7 2. Feynman graphs in the theory of elementary particles . · 11 7 F. The analysis of a semantic model. · liS O. The dynamical analYSIS of the chrrods of a Slatic model . · 1IS Appl'ndi~: splrol n"bula/' . · 119 CHAPTER 7. THE DYNAM IC OF FORMS • 124 • 7.1 Modeu oj m« h(m;cJ . · 124 A. Limitations of classical and quuntum mOOcls . · 124 B. Determinism. · 125 7.1 In/ormQIIQI' and topoJog'('Qf complexi/)'. · 126 A. The present usc of the idea of information • · 126 B. The rdative nature of complexity. • · 127 C. Topological complexity of a form • · 127 D. The choice of a ground form . • · 12S E. Complexity in a product space • · 129 7.3 II1/orm<lliQI1, SigflijiCQI1«, al1d strrN:tural slability. • 130 A. Free interoction . • · 130 B. The entropy of a Form . • · 133 C, Competition of resonances . • · 134 D. Information and probability _ 134 7. 4 EI1f.'rgy and JpUllul complexily . • 13S A. The spectrum . • 135 B. S\I,lrm-Uooville theory in several dimensions . · 136 C. Agmg of a dynamll:al system and the evolullon of a field towanl equllibnum . · '38 7.S Formal dynamiC'S. · '38 A. The origin of formal dynamics . · 139 B. Phenomena of memory and faclhtatlon • , .. 141 C. Canalizallon of equilibrium . · 142 D. Threshold stabilization . · 143 E. Threshold stabilizauon Ind the theory of games. · 143 F. Other formll aspectS of a coupling; cOOlllg · ' 44 7.6 Form and mjOf"nIlJl'Qil . · , Appendix /. ConU""OIl()n of e"ne~ (md fhe firSI law of Ihtrmodynamics . · '46 Appendix 1: TtJpOIogirol complexity oj a dynamic . · 147 Appendix J: Infimlt rompltxity oj 8«1"'tlri('al /omu . · '48 The symmrtry priM"ipft . · '48 TIlt Jp«t and jo,"u of a fUMlion $plKt . · 149 CIIAPTER 8. BIOLOGY AND TOPOLOOY . • 151 8.1 Tht lopoJogicalllJfH!C'l of biologicllf morphogtntsis · 1S 1 8.1 Form In bIology; Ihe idea oj (l phtnolYpe · 1S2 A. The 'patill form . .. · 152 B. The global form .. ... • · '" 8.J MoI«ular broJogy and nKNJIMge"nclS • • · '54 A. The madequacy of biochemIstry • • · '54 B. Morphology and bIochemIstry. · '56 8.4 InformUlion In bioluK>' . • 157 Apptndix; Vitalism and reductionISm • ISS CHAPTER 9. LOCAL MODELS IN EMBRYOLOGY · 16 1 9.1 T1w writty oj forol m«hamsms of "wrphlJgellfiis in lHoiogy . 16 1 9.1 ThtprntnlullO/tojlMmodcl . 162 9.J A discUSJlon oj hiJIINlc/,I,hwrtts . · 161 A. DeHlopmcnt of mosaic type • 167 B. Gradient theories . • · '68 9.4 Models in pflml/il:c epigcnclJ oj amphibia . · 169 9.5 \lodeb J« tM pnmilil'c sITeok · 113 COfIIMII " A Gastrulation m buds .