KFKI-1991-32/C В. LUKÁCS Sz. BÉRCZI 1. MOLNÁR Q. PAÁL (•ditors) SYMMETRY AND TOPOLOGY IN EVOLUTION Hungarian Academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST KFKI-1991-32/C PREPRINT SYMMETRY AND TOPOLOGY IN EVOLUTION B. LUKÁCS, Sz. BÉRCZI, I. MOLNÁR, G. PAÁL (eds.) Central Research institute for Physics H-1625 Budapest 114. P.O.B. 49, Hungary T.F. Farkas: Hexa Lath Knot. (1982) The Material of the 2nd Evolution Symposium of the Evolution of Matter Subcommittee of the Natural Evolution Scientific Committee of the Hungarian Academy of Sciences, 28 29 May, 1991 HU ISSN 0368 6330 В. Lukács, Sz. Bérezi, I. Molnár, Q. Paál (eds.): Symmetry and topology In evolution. KFKI 1991 32/C ABSTRACT The Volume contains the materials of an Interdisciplinary symposium about evolution The aim of the symposium was to clear up the role of symmetry and topology at different levels of the evolutionary processes. The following topics are treated: evolution of the Universe; symmetry of elementary particles; asymmetry of Earth; symmetry and asymmetry of blomolecules; symmetry and topology of living objects; human asymmetry. Б. Лукач, С. Берци, И. Молнар, Г. Паал (ред.): Симметрия и топология в эволюции. KFKI-1991- 32/С АННОТАЦИЯ Сборник содержит материалы интердисциплинэрного симпозиума по эволюции. Целью симпозиума являлось выяснение роли симметрии и топологии на разных уровнях эволюции. Были затронуты следующие темы: эволюция Вселенной, симметрия элементар­ ных частиц, асимметрия Земли, симметрия и асимметрия биомолекул, симметрия и то­ пология живых организмов, асимметрия человека. Lukács В., Bérezi Sz., Molnár I., Paál G. (czerk.): Szimmetria ós topológia az evolúcióban. KFKI 1991 32/C KIVONAT A kötet egy evolúciós témájú Interdiszciplináris szimpózium anyagalt tartalmazza. A szimpózium célja a szimmetria és topológia szerepének tisztázása volt az evolúció különböző szintjein. A témák a következőek: az Univerzum fejlődése; az eloml részek szimmetriája; a Föld aszimmetriája, a blomolekulák szimmetriája és aszimmetriája; az élőlények szimmetriája ós topológiája, az emberi aszimmetria. HtrmooucTicM This Volum» is tu» written me tori»> of tn» symposium СШШТ "Symmetry ana Topology in evolution' held on g9-30tn Hmy 1991 in *. Luttcs: on tne Mathematics et Symmetry Brettings * Budapest under tn» aigis of tn» M»tvr»l evolution (Geonomic) sci­ 6. rati: Topology «f the Universe 9 entific Committee of tn» Hungarian Academy of Sciences, by tue B. Luttcs: rne evolution of ceamie symmetries a evolution of Hsttor Subcommittee of tn» sat a committee. Hopefully F. eioct: Chiral I ty in t»e Elementary iattractieas I* it »ill be a member of an annual series: tne previous volume was ey. Marta a Adrienné Htjtsy (with a Bemar» of E. Iilts-Almir: On 'evolution: from cosmogenesis to Biogenesis-, published as KFKI- Distortion and Secular Changes of earth's roree Fields it I99I-SO. Tue Committee is an interdiscipiinary scientific body St. aercsl: Symmetry Chant»* by Cellular Automata tn for general overview of natural sciences to develop new long- Transformations of Closed Double-Threads... г» range connections among different disciplines; tne Subcommittee Г. OlOct: ChiralIty of Organic Molecules «« started m 1999. tries to follow (or trace Pack) tne steps of B. Lüttes: On Geometrie Symmetries and Topológia* of Forms ar seif-organisation of matter from tne Beginning to (at least) tne 1. Moiaar: The Organisation of Vol vox Organisms la evolution 3i present. A'gnes Hoi Da t B. Lutics: Volvo»; a Simple Example for Unfortunately, no constructive method is known to view Spontaneous Symmetry Brett lags M 'Natural Science" as a unit; tne global picture is deduced from I. Mo lair: The Conservation of Morphogenetie Mechanisms in the separate discipl nes. Still, an overview may give answers to Evolution it old Questions. For example, on all levels of organisation the ac­ St. Béres i, в. Lutte л а г. Mointr: on symmetry and Topology of tual path nf evolution may be regarded eitner as accidental or as the Organisms In Macroevolution Ti predetermined. Such Questions nave connotations not only profes­ ss. B*rcti: symmetry and Topology...of the Fibonacci-Plant sional but pm losopmcai. reiigioiis. »c. as well, we do not want Structures 90 to leave tne grounds of strict natural science and try to show up B. Luttcs: vertebrate: individual or colony? 91 the available most coherent picture as a starting point of any E. Labor: On the Concept of Shape or Form in ... and Biology 9T deduction. Looking at tne neighbouring levels sometimes the E. Ltbos: Enumeration of Metwort Structures of Various Symmetry causes or сопзеяиепсез can be recognised. Types tor AS an example: we are chiral (can tell our right hand from Si. Béren: Platonic-Archimedean Cellular Automata... Hi our left in tne vast majority of cases); tne neutrino always can A'gnes Hoiba, B. Lutics » 1. Hoinir: Asymmetry in the Development too. and tne electron sometimes. is there any connection among of the C. Elegáns and its Evolutionary Implications lit these fact*? The premature answer can be avoided by recognising E. Ltbos: Attractors and Knots Hi that tne chimpanzee is not chiral. L. Eordos: symmetry and Asymmetry in Evolution of vertebrates itt The lecturers of tne symposium are experts of tneir own A'gnes Hoiba i B. Luttcs: Hominid Cerebral Lateral nation as fields; tne Organising Committee (tne editors) took tne responsi­ spontaneous Symmetry Brett Ing 130 bility of selecting tne actual persons, but tne texts are the lecturers' sovereign works. The Organisers tried to establish overlaps and links between lectures, and sometimes tney insert their comments in tne Gothic italics of tms text, with a signa­ ture £. Tne lecturers are not responsible for anything in eotnic itaiics. we nope that this Volume will promote some further interdis­ ciplinary work. The editors would like to thank tne Theory De­ partment of tne Particle and Nuclear Institute of tne Central Re­ search institute for Physics (кгт RHKI) for background and tech­ nical help. The pub 11cat ion was partly supported by the OTKA fund •Developmental constraints on tne evolution of ecological Spe­ cialisation'. editors (and Organising Committee) 0. Lukacs (president), Si. Bérezi, I. no mar, G. Pas I evolution of natter Subcommittee of tne Natural evolution scientific Committee of tne Hungarian Academy of Sciences OB THE MATHEMATICS OF STNSTIT BBEAE1B6S if the symmetry breaking is caused by an external influence then generally it it proportional to the influence (and then one B. Lukacs can compare the acting forces). If the breaking is spontaneous, Central Research institute for Physic*. H-1523 Эр. 11». Pf. «9., this is not necessary at all. Budapest, Hungary «. AB EXAMPLE FOB BOB-SPOBTABEOUB STMHETIT BBEAEIBO: OBLATEBESS Received on tetn July, 1991 A rotating fluid body is often an oblate spheroid (although other solutions of the hydrostatic equations exist as well 12)). For small angular momentum Я one can get simple approximate for­ Syametry ana Topology in evolution. e<t- a. Lukacs i al. кт-1991-зг mulae as follow: The rotation results in an inertial "force", including its •potential" one gets ABSTBACT V(r.e) 5 -GH/r • XeT'lln'e (Z) Deviations -from symmetries ere discussed. We distinguish two Then the equipotentlal surfaces (of which one Is the surface of cases: i) symmetric laws * asymmetric external disturbances lead the fluid [3)1 are given by roots of a cubic equation. Again for to approximately symmetric actual states; ii) symmetric laws snan <г they can be approximated in a transparent way. Consider a without other influences lead to states asymmetric in an extent value v0<o. Then rc = r„|e): determined by the laws themselves via spontaneous symmetry Ьгеак- re = (7GM/vem*xi«JHa)'/<ve'))iin'ej (Ji ing. Then the extent of asymmetry or symmetry breaking is the ratio of tne two terms in the bracket II for, say, *>i/f. Substituting terrestrial surface data, the ratio is 0.0017, rather moderate. o. lmooucrioa Tr.ls Paper is devoted to the mathematics of the different 5. OH SPONTANEOUS ЗТНИЕТВТ BREAKIH6 kinds of symmetry breaking, since the present Volume deals with Again instead of a precise mathematical definition here we symmetries, our coals are limited: we do not want to go into fine adopt the following transparent one: details of definitions, but try to demonstrate that a symmetry THE SY1.1ETRY BREAKING IS SPONTANEOUS IF THE LAWS ARE SYMMETRIC may be broken for different reasons and to show how the extent in STILL THE MOST NATURAL CONFIGURATION (SAY 0ROUN0 STATE) FOR THEM which it is broken can be quantified. IS ASYMMETRIC. Tne simplest example is mechanical equilibrium in a potential t. OB SUBSUMES V(x):V(-x>. The potential is even, i.e. mirror-symmetric: this instead of a complete definition let us use the following will be the symmetry. The most natural configuration is now tne one: equitibrlum one; it that is not unique, then the ground state A SYMMETRY IS A TRANSFORMATION LEAVING SOMETHING INVARIANT. among them. Here come the details. Example: consider a sphere. By rotating it remains the same. The condition for equilibrium is x:x ; (dv/dx) : о («) For a precise definition of geometrical symmetries see an­ 0 0 other paper in this Volume Щ. For simplicity we restrict ourselves to polynomials, such -.hat V(0) s 0 2. OB ВЖЕАК1В6 V(ia>) z »m (S) If there is no symmetry at alt. nobody speaks about broken V(-X) s VIX) symmetry, we speak about а broxen symmetry if we are going up with the degree of the polynomial - 1) the situation is almost symmetric; »o 2) tu» situation ought to be symmetric still it is not. The symmetry can be broken The only possibility compatible witr (5) l* t) by external asymmetric influences; V0(X) x О (») 21 spontaneously. Then (4) holds everywhere; the ground state is degenerate.