Louis H. Kauffman and Francisco J. Varela- Form Dynamics

8/3/2019 Louis H. Kauffman and Francisco J. Varela- Form Dynamics

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J. SocitJlBioi. Struct. 19803, 171-206

Form dynamicsLouis H. Kauffman

Department 0/Math ema tia , Un iv er si ty o f I lli no is at Ot iazgo C irc le ,Chicago, /11inois 6 06 80 , U SAand

Francisco J. VarelaUnivers idad de Ch il e, Facultad Cienci1s, Qzsilla 653, Sant iago , Ch il e

This paper is an exposition and extension of ideas begun in the work of G. Spencer-Brown (Laws of Form). We discuss the relations between Conn and process,distinction and indication by the use of simple mathematical models. These modelsdistill the essence of the ideas. They embody and articulate many concepts thatcould not otherwise be brought into view. The key to the approach is the use ofimaginary Boolean values. These are the formal analogs of complex numbers -processes seen as timeless forms, then indicated (sell-referentially) and re-enteredinto the discourse that engendered them. While the discussion in this paper is quiteabstract, the ideas and models apply to a wide range of phenomena in mathematics,physics,linguistics, perception and thought.

1. IntroductionOUf theme is best indicated by the following experiment, and by a correspondingpassage from the beautiful work, Cymatics by Hans Jenny (1974):

0140-1750/80/020171 +36 $ 02.00/0 1980 Academic Press Inc. (London) Limited

Sprinkle sand over the surface of a metal plate; draw a violin bow carefully along theplate boundary. The sand particles will toss about in a rapid dance, swarming and forminga characteristic pattern on the plate surface. This pattern is at once both form and process:individual grains of sand play continually in and out, while the general shape is maintaineddynamically in response to the bowing vibration."Since the various aspects of these phenomena are due to vibration, we are confrontedwith a spectrum which reveals patterned figurate formations at one pole and kinetic-dynamicprocesses at the other, the whole being generated and sustained by its essential periodicity.These aspects, however, are not separate entities but are derived from the vibrationalphenomenon in which they appear in their unitariness ... The three fields - the periodicas the fundamental field with the two poles of figure and dynamics inevitably appear asone. They are inconceivable without each other ... nothing can be abstracted without

the whole ceasing to exist. We cannot therefore label them one, two, three, but can onlysay that we have a morphology and a dynamics generated by vibrations, or more broadlyby periodicity, but that all these exist together in true unitariness .. . . It is therefore warrantable to speak of a basic or primal phenomenon which exhibitsthis threefold mode of appearance."Here, Jenny has allowed himself to speak generally about a wealth ofexperience and concrete experimentation with the effects of vibration on


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172 L. H . K auffman and F . J. Varektvarious media ranging from sand on a vibrating plate to fluids in three-dimensional space.These are poetic ideas, metaphoric notions, and yet they have reflectionsin all fields from the wave/particle duality of quantum physics and the formsproduced in concrete media by physical vibration, to the oscillations anddistinctions that we make at every moment of our lives.Our object is to find and explore a language that expresses these ideas andis sensitive to them. Such a language should be internally consistent andexternally meaningful. The calculus of indications created by G. Spencer-Brown in his book Laws of Form is admirably suited for the project. Thepurpose of this paper is to extend Brown's language to exhibit how a richworld of periodicities. waveforms and interference phenomena is inherentin the simple act of distinction .:There is nothing new about the idea that an entire universe of fOnTIScomesinto being with the making of one distinction. In mathematics this idea isreflected by the use of the binary system, and more centrally by theconstruction of the natural numbers from the empty set via the operationof forming a collection. We wish to focus on these same themes, but ouraims are more fundamental. For example, we would give attention to theconstruction of the empty set itself. This set, r J > : : { } . is obtained by bracketingor framing nothing. Notationally, the frame is given by the brackets. Thesebrackets indicate a distinction on the planar space upon which the bracketsare written. The open space of the plane is construed as an indication ofnothing (that is, the absence of set members).Of course, these are remarks about the notation, about the specific choiceof frame. In point of fact, we conceptualize the empty set by first framingnothing and then throwing away the frame! That is, we require that themathematics be independent of the vagaries of notation. Nevertheless, it issignificant that in the beginning of set theory or the beginning of Booleanalgebra, the very process that the mathematics proposes to discuss is inevitablymirrored in its own language and notation. This mirroring quality of thelanguage is essential for our understanding. We believe that recognition of thispoint leads to a number of fruitful avenues, and we hope to. illustrate theseavenues in the paper.Our use of Spencer-Brown's notation also reflects these issues. His calculusof indications is based on the sign "l . Our comments about the empty setapply equally well to this mark I which indicates a division of the plane.This notation is actually very close to the notation of Boolean algebra. Forexample. one standard notation that corresponds to aJ is a(see Fig. 1)..

BrownianQ 1

abQlbiil51iil511 c

Booleanii

ab'iibaDf a 1 iJ c

Fig. 1


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Form dynamics 173These is nothing obscure or esoteric about our notational system. Typo-graphically, it is very similar to standard notation. Furthermore, it allowsparenthesis free expressions by generating its own divisions. Most importantly,it focuses on the fact that the algebra it supports can be interpreted as speakingabout the distinctions engendered by its own typography.It has often been pointed out that the formal structure of G. Spencer-Brown's work, and a fortiori of the present work, is equivalent to some formof Boolean algebras or switching automata. Although, in a strict formalistsense, this is correct, this line reasoning misses an essential point: A changeof context and style (notation) may reveal the intuitive and conceptual under-pinnings of a field, in a way that other, formally equivalent systems, do not .. A simple example is Roman and Arabic nwnerals.The relations between the formal structure of Boolean and indicationalalgebras are quite obvious, and were discussed by Spencer-Brown in hi s original

work (1972, Appendix 2; see also Varela, 1975). The point is that one maysee Boolean values (true or false) or switching algebras (on or off) as particularcases of a more fundamental ground which is the act of indication. A two-state abstraction is seen rooted in a specific cognitive act. From this startingpoint, an entirely new vision of traditional Boolean formalism can be obtained.Furthermore, the actual significance of some of the postulates in these algebrasis made transparent in a way that was hitherto impossible. A good example ofthis occurs in what is here called a transposition algebra, to be discussed later on.We wish to mention that the algebras here called brownian are formallyisomorphic to De Morgan algebras. As there is a large literature on such non-standard Boolean algebras, we use Kauffman (197 8b) as an entry point, ratherthan including an extensive list of references. Thus, if we make little mentionin this paper to the vast literature on Boolean algebras and algebraic logic, itis not because we wish to ignore these sources, but rather because we wish toemphasize the change in context.In order to accomplish this program, we first present an extension ofSpencer-Brawn's algebra that is capable of handling periodic indicationalforms. We then discuss how oscillations relate naturally to the re-entry offorms, that is, to their self-referential quality. tThe following is the outline of the paper.

Section 2 recalls Spencer-Brown's calculus of. indications and discussesre-entry and oscillations in a general way. SectionS shows how to make analgebraic construction for elementary waveforms. (This construction has alsobeen discussed in Kauffman (1978) where it is used to construct De Morganalgebras from Boolean algebras.) In sections 4 and 5 we develop an algebraanalogue to, Spencer-Brawn'S primary algebra, In Boolean terms it correspondsto dropping the law of the excluded middle. This allows wave-form models; wecall such algebras brownian algebras. Sections 5 and 6 develop more models

tThe present paper grew out of our interest inthe work 0f Spencer-Brown on indicatio oal mathematics,and its relevance Cor systems theory and mathematical Coundations. In the cases of 30cial and biok>gical5)'stems, the complementarity of pattern/dynamics is quite apparent. I t points directly to an examinationof recursive, self"l'cferential dynamics generating ctlherent autonomous unities, For this background an dfor the motivations from systems theory the reader should consult the following papers: Goguen & Varela(1978, 1979), Kauffman (197&, b), Varela .t. Goguen (1978), Varela (1975, 1978a, b). Wadsworth(l976). We need not explicitly retake this background and motivation in the present paper, as its themesof independel!.t interest.


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174 L. H . K auffm an and F . J. Varelafor brownian algebras, discuss an algebra of periodic sequences of varyingperiods, and present a way in which waveforms can interfere with each other.

Sections 7 and 8 examine the relation between waveforms and recursions ore-entry of forms. Section 7 discusses some periodic properties of iteration(or indicational operators. In section 8 we examine recursion in a broadersense, and establish some relations between waveforms and fixed points(This construction for recursion and fixed points is more fully given elsewhere:Goguen & Varela, 1979; and Varela & Gguen, 1978)Section 9 discusses the relations between geometrical and indicationalforms. Section lOis a summary.

=

2. Recalling the calculus of indicationsThe calculus of indications (Spencer-Brown. 1969. 1972) is based on onesymbol, L, the mark. It can be viewed as an abbreviation of 0 , and hencemakes, by cleaving it, a distinction upon the plane in which it is written. Inthe context of the calculus of indications, the mark is taken to be the nameof the outside part of a distinction in an arbitrary domain or indicationalspace or as an instruction to cross the boundary of this distinction.This gives rise to two forms of equation:

A I. Form of Condensation ..., ..., = "lA2. Form of Cancellation - = , =

Here the blank indicates the un-marked state. These equations have variousinterpretations. Thus Al is indeed the form of condensation. where two thingsare realized to be identical in form. The two marks in Al may be seen differentlyby regarding the left mark as making a distinction in the plane, while the righmark is a token or name for the outside part of this distinction. When we sethe mark that makes the distinction as itself indicating the outer space, thenthe two uses condense giving "l "l = I.Similarly, form A2 may be interpreted by the sentence: To cross from themarked state is to enter the unmarked state. Here the outer mark is interpretedas an instruction to cross from the state indicated within it. The outer markis an operator in this interpretation. Operating on itself, it cancels itself.Before proceeding into calculation, it should be remarked that the readerof this page is himself or herself, a mark distinguishing a space. Thus thiscalculus is self-referential in the broadest sense. In fact, one of the essentialfeatures of this approach is that the scribe and the forms of description arereflections of one another. We become individuals by making distinctions;The distinctions we make reveal (and sometimes conceal) who we really are

These initial forms of equation lead to a calculus of expressions that Spencer-Brown calls the primary arithmetic. In this system one considers arrangementsof marks such as


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Fo rm dynam ic s 17 5An expression like e is regarded as a clear distinction if for each mark in theexpression there is no ambiguity about which marks it contains and whichmarks contain it. In this notation, this is made clear by examining the horizontaloverhang of each mark. If 0 were used instead of -, ,then a well-formedexpression would simply be any finite disjoint collection of rectangles in theplane. Thus we would write

For two expressions e and f. we write e =f if there is a finite sequence ofsteps of type Al (condensation) or A2 (cancellation) leading from oneexpression to the other. Thus

.'. e = I {A2}

;;

i l = J l l (A2);= = J l l (A2)

The following facts can be shown.(a) If e is any wen-formed expression, then e can be obtained by a sequenceof steps from one of ...., or(b) There is no sequence of steps leading from'" to


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176 L. H. Kauffman and F. J. VarehHence each expression is equivalent to either the marked or the unmarkestate, and this will be referred to as its value. Two methods of evaluation a

worth noting. The first method is in the form of calculation as indicated abovone looks into the deepest space of the expression, where there are marks thdo not contain other marks. At such places condensation or cancellation mbe applied. In the second method, one regards the deepest space as sendinsignals of value up through the expression to be combined into a globvaluation. To do this, let m stand for the marked state, and n for the unmarkestate. Thus mm = m, mn = nm = m, nn = n, and nil = n, T il = m. Now uthese labels as signals as in the following example:

Here = 1 i l 1 has the value n = .' This procedure starts from thdeepest space and labels those values that are unambiguous until a value fthe whole expression emerges. Here is one more example:

II l = ] f l= II IIIIJIJ~-----=::.. i;lm

Inlmn l J n ~m------=::.. i;lmHence e = I.

It should be noted that these methods are quite compatible. They reflethe dual nature of an expression as (self) operator or operand. Viewed as aoperator, the expression filters its own inner signals, creating a pattern owaveform that culminates in its evaluation. Even at this level, the relation tthe Cymatics metaphor of flgure/vibraticn/dynamics (that is, to the descriptivpair figure/dynamics that may represent the polarities of vibration) is quiapparent: The geometric form of the expression represents the figurcalculational steps are an elementary dynamic; signals of value moving througthe expression represent a kind of periodic vibration.This relation of form and dynamics is even more apparent if we alloan expression to contain a variable that vibrates (i,e. changes in timeConsidere = ail

If a = n then e~ m l 1 1 l n l m


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Form dynamics 177If a = m then e~ ni l n l m l nIf a varies from n to m periodically, then the signals nand m will forma moving pattern analogous to the moving light patterns on a sign in TimesSquare.Taking another point of view, we may regard the signals as moving outwardlike ripples on a pond so that a time-like vibration by a yields a pattern ofthe form

aThat is, we may suppose that each time a = m, a mark appears so that iftime is represented as t = 1,2,3,4, ... thena = { m todd

n t evenand the outward expression grows in the pattern:

1 ....

Viewed in space we would see something like

= f = iJ

where the deepest space is now indeterminate due to its vibration. Herethe formis maintained by the vibration (or growth) at its center. Since the deepestspace is indeterminate, calculation has abated. Form and dynamic have becomeone with the vibration. Nevertheless it must be noted that part of the vibrationhas been remembered as the external spatial pattern of the fonn. This patternis maintained (by the central vibration) against the dynamical pressure towardsimplification (via calculation).Viewed entirely spatially. this temporal form becomes an in/inite'expressionconsisting a descending sequence of marks. As such, its interior repeats itself.


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17 8 L. H . Kauffman and F . J. VarelaThe form becomes identical to part of itself. Thus, for f as given abovf ""fi ""Ll (see Varela, 1975). This description, f = p , where f re-enteitself, can thus be seen as a self-reference. This is the spatial context.Tem~oral1y, we may view f = J I as a prescription for recursive actiof___..:. J 1 . Thus "l__. , - - - - - - 1 . " - - " . . . and this regenerates the waveformThus vibration yields self-referential spatial form, while the associated recursidynamic to the self-reference unfolds the vibration (once again) into a temporaoscillation. We shall deal with these relationships of recursion and re-entryforms to oscillation in greater detail later in the paper.It is now appropriate to consider Spencer-Brown's algebra for fmiexpressions: Various algebraic patterns are seen to be true in the calculusindications. Thus at = a. Q 1 5 1 c = liCl 5 C 1 and a t a l = for any fmiforms a, h. c. This leads to an algebra with initial equations and a numbeof consequences. At the end of this section we have given a table consistinof these initials and consequences, forming the primary algebra. Formallthe primary algebra is an axiomatic form of Boolean algebra written in thnotation of the calculus of indications. We feel that there is a significaconceptual gain in placing Boolean algebra within this wide indicational contexIt is also desirable to deal with infmite forms and/or waveforrns algebraicallAn apparent paradox seems to emerge: Suppose that the primal)' algebraapplicable to forms such as f = fl.Then

f=f=ffl=Jfil=Hence f = ji => f = . How are we to interpret this? The simplest waout is to realize that in Spencer-Brown's calculus of indications we have thmarked state which is purely spatial having no temporal component , and thunmarked state connecting everything else. Thus f, being vibratory, has beecast into the unmarked state. It has been so cast by the form of positionjjI p i = . If we wish to articulate temporal forms we can do so by limitinthe cancellation given by the form of position. We shall show how to do thin the next section. The result is an algebra where there are many 'selinterference' terms of the form P f P I . Cancelling all such terms yields thprimary algebra. Leaving them gives an algebra capable of caring for vibratorforms and self-reference.Index for primary algebraInitialsJ1. m P l =12. p rl qTll = P I q il r

Consequences01. a l l = a02. Qlilb = alb03. la = "l04. iilbl a = a05. a a = a

(reflection)(generation)(integration)(occultation)(iteration)

(position)(transposition)


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Form dynamics 17906. Ql511albl = a07. ~ = QCl 5 i C 108. Ql5rl Crll = al5 1 C li aJ rl0 9. 5 lrilaJfli XlrlYl rll = flab! rxyl

(extension)(echelon)(modified transposition)(crosstransposition)

The primary algebra is complete with respect to the primary arithmetic(calculus of indications). That is, a = { 3 is a consequence of 11 and J2 if andonly if a = { 3 is a theorem about the arithmetic (see Spencer-Brown, 1972,p. SO).

3. A waveform arithmeticConsider again 1= ji. By taking successive replacement of 1in its equivalentform we have a sequencef.p,fil ; f i i I , . . .Now, if we let 1 take the two possible initial values 1nd 11 , and applycondensation, we get two sequencesI, " I,= 0 , I, = n , I,

and= n , I,= n , I,= n , - " = n , . . .These two sequences can be looked at as the successive values of two basicwaveforms

i =j =

This temporal interpretation of re-entering expressions which can take novalues in the primary arithmetic, was first proposed by Spencer-Brown himselfin Chapter 11 of Laws 01 Form. He calls these temporal values imaginaryBoolean values. A few years later, in the preface to the American edition ofhis book he says:What we do in Chapter 11, i s to extend the concept (of imaginary numbers) to Booleanalgebras ... The implications of this, in the fields of logic, philosophy, mathematics, andeven physics, are profound.What is fascinating about the imaginary Boolean values, once we admit them, is the lightthey apparently shed on our concepts of matter and time. It is, I guess, in the nature of us

all to wonder why the universe appears just the way it does. Why, for example, does itnot appear more symmetrical? (Spencer-Brown, 1972, p_ xi).Actually, Spencer ...Brown did not develop the formal side of these ideasextensively in Chapter II; he pointed into a rich direction, but did not exhibit

specific constructions. On the other hand, his book is a courageous attemptto deal with these issues, beginning as it does in the simple. undefmable soilof distinctions and moving upward into formal realms, We now present a specific


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180 L. H . K au ffm an and F . I. Vareladevelopmerr' of imaginary values. We hope that what is seen from our workmay help to explain the importance of Spencer-Brown's ideas.

A s a first step toward making temporal expressions well-defined objects,note that in the example provided above 7 1 = i if we interpret"" as ordinaryinversion plus a half-period shift. That is , suppose we have a periodic patternx = . . . a b a b a b a b then we defineXl = 5 lQ151aJ51Ql51 iil .

Thus if ithen

..il= i.Note also that ij = Isince at any given time either i or j is marked.

Now the sequence ... ababab . . . can be looked at in a different way,somewhat similar to the construction of the complex numbers from the realnumbers. That is, we think of the ordered pair (a,b) as representing theessential features of the sequence. This shift in perspective permits a moredetailed algebraic treatment of the notions described above.3.1. Definition. Let B be any algebra (or arithmetic) satisfying the initials forthe primary algebra. Let B = {(a ,b) la ,b EB}and define (i) (a ,b ~ = (D1, iii) (inversion plus shift)

(ii) (a,b)(c,d) = (ac,bd). a,b),(c,d) EA )Let i = n. = = n )and j = ( = = n , ' l ) . Make the identification B Cb via a = (ap) foraEB.The smallest collection of such pairs representing waveforms is. in fact, one

containing no variables, but only the constant elements ' l , ~ .i, j . Call itthe waveform arithmetic V. By the previous defmition we see that:3.2. Proposition. Let P= 1,11 denote the primary arithmetic.Then V = p ,A s required, the basic waveforms i,j arise out of the static forms of theprimary arithmetic. They also return to it by the relation 1= ii.Next, note the following:


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Fo rm dynam ic s 18 13.3. Proposition. InIi occultation and transposition are valid,and i = il, = = Jl, ii =LProof. We shall verify occultation and leave the rest of the proof for the reader.Let X = (a,b), Y== (c ,d ). We must show that X l y l X = X in 1 3 :~ X = (Ol.al)(c,d)1 (a,b) (3.1 (i))=(Dlc,Old)! (a,b) (3.1 (ii))= (Q l d l ,me! )(a,b) (i)=(Qld!a,blclb) (ii)= (a,b) (occultation inB):.FYI X=X.This proposition suggests that in the waveform arithmetic and the relatedalgebras!J the relevant initials may be occultation and transposition. The nextsection develops this idea.3.4. Remark. Note that (as in 3.3) if X= (a,b), Y = (c,d) then XI Y = (0 1 c,al d)in lJ . It is interesting to consider this in the light of Spencer-Brawn'sinterpretation of implication (see Spencer-Brown. 1972, p. 114). Ininterpreting the primary algebra for logic, Spencer-Brown lets - '1 stand for T(true) and stand for F (false). Then a -+ b becomes a b. Thus in X Y resolvesinto two implications inB: b -+ c and a -+ b :

X: ... a bob a b ...t ~ ~ ~ ~ ~ ~ ~Y: ... cdc d cd ...Thus, if X - Yin!J is taken to be XI Y we see that implication in / : J has atemporal component. Each term of the Y-series is 'implied' by the previousterm of the X -series, Further study of this remark promises to be veryinteresting.

reveal

4. Brownian algebrasLet a,b, ... , p,q, ... be a coUection of variables, and let expressions involvingthe cross, "l ,and these variables be defined as in the primary algebra. That is,whenever A)J are expressions, then A I , HI, A I B, ABl ,AB, are also expressions.The variables themselves are expressions, as are ...,and (blank). Take thefollowing two initials as valid;Initial J . Occultation

II P I q Ip = p conceal

Initial 2.12

TranspositionPrIlfrl l = P l q J l r collect

distributeCall any algebra satisfying 1I and 12 a brownian algebra. As in the primaryalgebra, the cross operates on whole expressions and the juxtaposition


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182 L. H . K auffman and F . J. Varelaoperation (continence) a,b .... ab also operates on any finite collection oexpressions. Since it simply expresses the distinctness of these expressionsviewed together as a whole, continence has no particular order properties. Inmore orthodox algebraic systems this would be expressed explicitly byintroducing initials for commutativity and associativity. Here it is not so muchthat commutativity and associativity are tacitly assumed, but that we arearticulating a level at which they do not yet exist! Service is paid to this pointby ignoring commutativity and associativity in all subsequent demonstrations.Since the cat is already out of the bag. we must make do with tacit conventionsto substitute for true simplicity. We now derive some forms of equations validin these algebras.

augmentDemonstration 1 a = 1i l (CI)=, (II).Consequence 4. Echelon

(e4) ii l b l c l = Q C l ~ break

Consequence 1.el.Reflectiona n = a reflect

reflectDemonstration.==::----,a n =a\ilil a l l

' " a n a l atl a I I= = III a n a= a n a'" a

Consequence 2. IterationC2. a a = a

(11)(II)(12)(11)(11 ).

iteratereiterate

Demonstration. a a = a n a=aConsequence 3. Integration

(e3) "l a =1

(CI)(II).

reduce_... . . . .

make


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Form dynamics 183Demonstration

iii b l c I = iii z J i l J c I (Cl )= aa D l e l l l (12)= s a 5 1 c I (CI)

Consequence 5. Combinationres) iii , 1 i5 l ,I = ao 1 , 1 combine_..

-e--'

splitDemonstration Q 1 , 1 0 1 r I = Q 1 rl O J , 1 1 1 (Cl)

= Q 1 I z ) 1 I l r I (I2)= 7 i l i I rl ten

The next two consequences are special cases of e5 and e4 respectively. Wearticulate them because they are related to corresponding forms in the primaryalgebra.Consequence 6. Catalysis

(e6) Qlj J bl = QI b l b 7 ) ] 1 release

distribute

dissolveDemonstration Apply CS.

Note that if b O J I = (blank), as in the primary algebra, then e6 becomes acrossed fonn of generation (see the index for the primary algebra in section 2).Consequence 7 . Tension

(C7) Q J b l ill 7 ) ] 1 = U l bmll atone. . . . . ...-attend

Demonstration. Applye4If b 'O n = (blank), then tension becomes Q J b l iil ljiJ = Q 1 I = a, the formof extension in the primary algebra.

Consequence 8. Modified transposition(C8) aJ D r 1 C r I I = Q l m C l i aJ ii collect

Demonstrationiii D r I C r J I = Q l Drl C r l i i t rcn

= a J m C l I , 1 1 (12)= iii 0 1 E ll aJ r 1 l (C4).


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184 L. H. Kauffman and F . J. VarelaConsequence 9. Distribution

(C9) Pl qll T'iS1 = ;m P S I q ; : 1 Q s J shum ecutDemonstration

Pl Q l I " f1 s U = p rfTIl q fI S i l l (12)= # I P s I i r 1 f s i a (12)= prj ps t q;:l Q s I (Cl)

Consequence 10. Modal crosstransposition(CIO) regulate

Demons fralionaxil ~ a Ql Z 1 1 1

= X T Ilil Y l l a l Zl I I Q l ti l nil= Xl] Y1l n l an xnl til Ql

(C8)1 1 1 1 (C8)

(C6, Cl)

accommodatea x l i ill F l I a Ql Z l l l = ' l iXl Ql Y[ ill a l XYZI

= a x l Ql Y I Q] a l(We thank Mark Kauderer for this demonstration.)

It is worth comparing the consequences in the brownian algebra with thcorresponding results (see index at end of section 2) in the primary algebr01,02,03.04,05 and 08 are valid in both algebras. Catalysis is the browniaimage of generation; tension corresponds to extension in the primary algebrmodel crosstransposition corresponds to crosstransposition. In each of thescases cancellation of terms of.the form P I P I yields a corresponding resultthe primary algebra.By adopting the initials occultation and transposition we have constructedan algebra closely reljted to the primary algebra that automatically avoids thcancellation of P I p . A s we have seen. these self-interference terms Pl pare important for handling waveforms. For these purposes they should barticulated rather than ignored.By taking the initials (II) and (12) we have done no violence to SpencerBrown's original grounds, but have simply adjusted our sights to perceivpatterns already inherent in the form.It is also worthwhile to compare this change of language (the primary algebrversus brownian algebra) to other more complex linguistic situations. Foexample, in Benjamin Wharfs studies of Hopi Indian languages (see Wharf1956) he finds language structures that allow a view of the world that iprobably closer inspirit to the wavefonn algebra than to the primary algebr


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Form dynamia 18 5(which keeps the all or none quality of Aristotelian logic). To change viewpointit is not enough to simply throwaway a rule (say throwaway P P i l = ) . Thisabandonment must be embedded in a positive framework (such as thebrownian algebra) that consistently holds open the possibilities hoped for. Thuswe have in the pair Primary algebra/brownian algebra a prototype for manyissues involving change of perspective and shift of language.

S. Completeness and structure of brownian algebrasWe have already seen that the arithmetic V generated by "l , 1 1 , i, j with l l = it]l = j and ij = -, satisfies 11 (occultation) and 12 (transposition). Hence V is amodel for a brownian algebra. All of our consequences hold for expressions inV. In fact, the next theorem shows that brownian algebra is complete withrespect to this arithmetic ..5.1. Theorem. Let Ci and / 3 be two algebraic expressions. Then Ci = t J is aconsequence of II and 12 if and only if Ci = f 3 is true in the arithmetic V.

The proof of this result requires some preliminary work as outlined below.The first result we need is an algebraic reduction of form (similar to Theorems14 and 15 inLaws of Form).5.2. Proposition. Let Ci be any expression in the brownian algebra. Then Ci canbe reduced to an expression involving no more than four appearances of a givenvariable. More precisely, suppose that X is a variable in a. Then there areexpressions A,B,C,D involving no appearance of X so that

e x = AX! Bii1CX X U D .Proof. First note that, by using echelon, any expression is equivalent to anexpression no more than two crosses deep. Hence we find that0: = x a l l bll ... Xanl bnl XCii Xdl1 . . X cml xdmll X e1 ! . . . X e p l fwhere aI, ... ,an,b1 , ... ,bn,c1 , .. , cm,d1 .. " dm ,el,' . " epf areexpressions in which X does not appear.

Note that XiiI b l = XI b l aJ bl by combination (C5). Similarly. xa Xdt =XXI d l X Cl d l . Thus the proposition follows at once from these facts andrepeated applications of C5.The next result involves evaluating an expression at. "l, 1 1 , t.t. That is, weshall have an algebraic expression a: = a:(X) involving a variable X. Theexpression e x may be viewed as a function on V = { "l , 1 1 , i, j} that gives avalue in V when all of its variables are replaced by elements of V. Similarly,0:(1), o(ll), a(i) and o:(j) are also functions on V. (Here X is replaced by -', 'll, iand j respectively). In the next proposition the symbol = refers to equality offunctions on V.5.3. Proposition. Let a(X) =XA1 XI B I X XI ci D where A.B,C,D areexpressions involving no appearance of the variable X. The following equalitiesof functions on V are valid :

0 : ( 1 1 ) = A I D , a(l) = B I Dcx(;) a: (j) = A I m D D~~I=D.


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186 L . H . K auffman and F . J. VlUelaProof. The first two equations are obvious. For the last note, since 71 = i , a(i)lA 1 T B 1 Tn D = i A I B l ~ n D (using cs twice). Similarly, rx(j) = i A 1 m t i l l DHence (letting E = ill HI C l i ) we havecdO I a (j)1! = ; ; ; ; ; ; ; ; f f , i i O " . ! = D ' I - - jE 1 ! ! !! P " "" D. . . .

= fill J E l l 1 D= f j E I D= j E I D

(12)rcr, en

= D(li = "I)(II) .. a (0 1 ~l

= xiI l l X I B l x x I I iIl B 1 C l i D= x A l D l l 'I HI D I x X I D ! A 1 B l C I 1 5 1 1

(CIO,C1)(12) .

Finally, a(l) aU) = iE 1 D j E I D= 71 7 1 EI D= il l E I D= E I D

: . C i . (i) a (j) = A I HI ['I D .This completes the proof of the proposition.Proof of Theorem 5.1. We are given two algebraic expressions a and fj such thaCi. = / 3 can be proved as a theorem about the arithmetic V. This means that a=as functions on V. We wish to show that under these conditions a = fis demonstrable from the initials II and 12. The proof will proceed byinduction on the total number N of variables in the two expressions.If N =0, then a = D and ( j : ; ; ; D' where D and D' are constants. By hypothesis,D = D' and there is nothing to prove. Note that the only algebraic constants are"Iand (blank).Thus we assume that N ) 0 and that the theorem is true for a smaller N. LeX be a variable appearing in one or both of the expressions Ci.,/3. By Proposition5.2 we can assume that Ci. = X A 1 X I BI xXi C I D and {j = XA 'I XI B 'l X XI c I D'where A.B,C,D,A',B',C',D' are expressions involving no appearance of X.By the evaluations of Proposition 5.3 and the hypotheses of this theorem ithen follows that the following formulas are demonstrable: - .

(e2, csi(7l = i, T I = j)(ij = "l)

(i) AI D = A"I D'(ii) 1 J l D = S '1 D'(iii) D = D'(iv) Al B 1D D = A"'l If'1 Cl D'.

We now apply modal crosstransposition (C 10) to demonstrate Ci. = ~ :a = XAl XI B I XXI. ~ ! . D

Now substitute, using equations (i) - (iv), reverse steps, and conclude thaC i. = { : j . This completes the induction step and the proof of Theorem I.


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Fo rm dynamic s 187Some technical comments are in order here. We have actually proved thatany free brownian algebra is complete with respect to V. That is, given a set S,

one can form an algebra B(S) by regarding the elements of S as variables withno special relations. We then form a set of expressions (S) by tne followingrules:(l) s is an expression for each sES.(2) I and (blank) are expressions.(3) If X and Yare expressions, then X I , Y I ,and XY are expressions.The initials I I and 12 generate an equivalence relation - on (S)' We let B(S)::::(S)/- and say a: = / 3 if a: - P for a, P E (S).The collection of equivalenceclasses, B(S), will be called the free brownian algebra on the set S.Note that V itself is not a free algebra. The relations 1 1 = t, I I = j and ij : : : : " lare not consequences of 11 and 12. If S = {iJ} , then V may be regarded as theresult of placing extra relations on B(S). We shall not formalize this notion

now. but shall return to it in section 8 when algebraic structures for self-reference are discussed.We can now examine not only the initials and consequences of a givenbrownian algebra, but also the relations between algebras. The structurepreserving maps between the objects of a given class are at least as important asthe objects themselves. Thus we now give the definition of homomorphismsbetween brownian algebras.SA. Definition. Let B, B' be brownian algebras. A homomorphism h: B ~ B' isa set-mapping such that

h( ) =h(l) =I

and hex y) ::::h(x) h (y)h(Xl) = h(x)1for all elements x, y E B.

It is well-known that in a free algebra B(S) any homomorphism is determinedby its values h(s) for sES. In the case at hand, a homomorphism between abrownian algebra B and the waveform algebra V. amounts to assigning to each ..variable sES a value in the waveform arithmetic. Using this language, Theorem5.3 may be reformulated as follows: .5.5. Theorem. Let B(S) be a free brownian algebra on the set S.Then for a, ~EB(S), e x . = ~ if and only if h(a:) = h(m for every homomorphism h: B(S) -+ V.This theorem, in tum, has another reformulation that places B(S) inside alarger waveform algebra. We first need the notion of a cartesian product ofalgebras (not to be confused with the ......onstruction of section 3).5.6. Definition. Let Band B' be brownian algebras. Then the product algebraB x B' is defined by taking the cartesian product of the underlying sets anddefining operations by

(i) (a,b) I = (lil 7 7 1 ) (inversion but no shift)(U) (a,b )(e,d) = (ae bd),


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18 8 L. H . Kauffman and F . 1 . VarehSimilarly, if A is an indexing set and we have algebras Ba, aEA then we canform the product of all of these and denote it by n Ba.

aEARemark. A wave interpretation for this product construction will be givenmorfully in the next section. However, suppose that B = B ' = R where R is anotherbrownian algebra. Then aEE means that a = (x,y) where (XJl) connotesperiodic pattern ... xyxy . .. . Thus

a = xyxyxy ...b= zwzwzw ..and ab = . . . (xz)(yw)(xz)(yw) EB

However, we may choose to view ... xzywxzyw as a pattern of period 4The element (ap) E B x B formalizes this notion. In general B x ... x B(k factors) can be interpreted as an algebra of patterns of period 2k.5.7. Theorem. Let 8(S) be a free brownian algebra on the set S.Let A ={ h:B(S-+ V) be the set of homomorphisms of B(S} to the waveform arithmetic VLet Vh denote (a copy 00 V, corresponding to each homomorphism hEAThen there is an injective homomorphism

(I: B(S)___'" n Vh .hEAProof. Defme (I by (l(x) = n hex) for each x EB(S}. (Note that h: B(S) -+ Vh.hEASince x = yin B(S) if and only if hex) = h(y) for all hE A , we have x = y if andonly if (l(x) = (I(y). Hence (I is injective. .

In fact, Theorem 5.7 is true for arbitrary brownian algebras. The proof of thimore general version follows from further reformulations and the use of deeperesults about De Morgan algebras. For references, and a discussion of this poinseeKauffman (1978b).From our point of view, this result is quite significant, since it shows thaany brownian algebra may be seen as a subalgebra of a wave-form algebran Vh

hEAThe latter is generated entirely by self-reflexive elements, that is, by solutionofx = Xl.Thus the wave forms associated with the simple re-entering form x = Xl=Lstand at the base of all our considerations. The 'real' logical or indicationavalues such as -, are seen as combinations of synchronized waveforms (ij = I)This principle remains true in the general context of all algebras satisfyinoccultation and transposition.In this regard it isworth noting that the self-reflexive elements of a brownianalgebra are irreducible. That is:5.8. Proposition. Let B be a brownian algebra containing X such that XI = X . IX = YZ with 1 '1 = YandZl = Z, then X = Y= Z.Proof; Note that the proposition will follow ifwe show that AB = AC => B = CwhenAl = A,Bl =B, CI=C.(X=YZ=>YX=YZ~X=Zetc ... )Toprovethis, we proceed as follows:


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Fo rm dynam ic s 18 9B=T:::i1B

= BAI B= C A l B=MB= eBI A B 1 I= CB A ' Z 1= BlAiI C= BA I C= CAl C= DAI C

.. B =C This completes the proof.

(11)(B =Bl)(CA =AC=AB=BA)(C=(J,A = . 4 1 )(12)(AB = AC)(12)(Al = A,HI = B)(BA = CAl([1 = C)01)

6. Varieties of wavefonns and interference phenomenaIn the algebra ! J , associated to a brownian algebra B, we find period twosequences of elements from B. Thus for a ,b E B we have the correspondence(a,b) ~ ... ababab . " . So far, however, we have encountered only waveformsof period 2, namely those of the waveform arithmetic V.There is no reason tostick to patterns of period 2 (high frequency) in the context developed so far.Let us now discuss explicit constructions for waveforms of arbitrary period.6.1 Definition. Let p be an even positive integer and let k = p/2 so that p = 2kLet B be a given'brownian algebra. Define Sp (B) to be the set of sequences inBof period p, That is

Sp(B)={a={an} ,nEZ Ian EBandan+p =an for all n} .(Z denotes the set of integers.) This collection of sequences can be transformedinto an algebra by extending the operations of continence and crossing in thefan owing way:(0 ab = {(abln} , (ab)n = anbn for n E Z

and a,b ESp (B).(ii) Ql = t (linn1, (al)n = an.k , k =p/2

for n E Z anda E Sp(B).Thus crossing is accomplished by combining ordinary inversion for B with ahalf-period shift (whence p must be even). This extends the previous use ofcrossing for the A construction.By way of illustration, consider a sequence of period 4, and the special caseB = P (the primary arithmetic); that is, consider S4(P). Take one suchsequence a,


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1 90 L . H . Kauffman and F . J. Varela

a =

~ = . . . . . . _ . _ . _ _ . n . . n . . n . . .,. ~ a =

1=

Thus iil a = a.It is easy to verify that occultation and transposition hold in Sp(B). Thu

Sp(B) is a brownian algebra. The next result gives a more precise idea of thstructure of this algebra. by showing that it is indistinguishable from tuples oforms of period 2.6.2. Proposition. Let B be a brownian algebra. Then we have the followinisomorphism of algebras: s,(B) ~ Uk I i I

1=1where 1 3 1 (1 = I, ... ,k) denotes (a copy of) E corresponding to each of thintegers 1 through k, and k = p/2.Proof. Let Sp(B) = {(a,/3) 1 0: = (aI, ... a,d, {j = (;31, ... ,/3d and e x ; ,{jj Efor I~ , j ~ k ) . Define operations in Sp (B) as follows: (ex 13)(a' 'p)ala'l,' .. /Xk,cx. 'k), (j31 1 3'1 , ... ,f3kf3',t,(ex'p) = ( jf tl , . .. . P k l ) . (0 :- ;1 , ... ,~ ).Then. S p (B) is a brownian algebra. and we may map Sp(B) to Sp(B) by thfunctlOn~: Sp(B)~Sp(B)whereh (a) =a ... ,ak), (ak+l .... ,ap. Thisclearly an Isomorphism. On the other hand. we have an isomorphism:

g: s ; , (B)--:---+ Uk B Igiven by g(a'p) = al , /31) , (al.f32), ... ,(at .13k. Hence the composition g oilthe desired isomorphism.When p = 2 this result shows that SdB) = = E . as expected. WhenB "" P, thprimary arithmetic, then,

Sp(P) = = Uk VI1= 1(where V, is a copy of V). For the example of a ES4 (P) discussed above, onobtains S4 (P) = = V x V and a -E + (n,I). (11.In = (I, j).By combining two sequences in the same indicational space. we produce ainterference between the waveforms they represent. This is what is involved ithe extension of crossing in (ii) above. Interference follows quite naturally fo

waveforms of the same period (but arbitrary frequency). We are left with thquestion of interference of waveforms of widely different period, so that wmay handle the resulting patterns in an adequate way. One approach, presentedbelow, is to concentrate on the least possible period resulting from thinterference,Let lcm (p,q) denote the least common multiple of the integers PIl. For evenintegers pI/. the least common multiple is also even, and we may definemapping II: Sp(B) x Sq(B) Sr{B) where r = lcm (P II) and II [(a,b) ] n =


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Form dynamics 191anbn. Thus two sequences of different period interfere to form a new sequencewhose least period divides the least common multiple of the initial periods. Bychoosing to look at interference in this way we are simply stressing the highfrequency components of the interference pattern. We are also insisting on arule for assigning period to the interference pattern that will work wellalgebraically. Least common multiple has this property, aswe shall see.What are the difficulties in constructing an algebra of sequences of varyingperiods? The first point can be seen in the waveform arithmetic V. There ti =Iwhere i andj have period 2. If we must assign a period toI,hen 2 = Iem (2,2)seems to be the natural choice! We might say that the mark, I,n V resonateswith period 2. This leads to no confusion since everything in V (or l J ) hasperiod 2. In the more general situation Imay have to be regarded differentlydepending upon the assigned period. Consequently integration in the formIa = Iwill fail. It will hold only if we disregard the period of a. Since themarked state represents the observer, it is not surprising that we should find aspectrum of marked states, each corresponding to a different resonantcondition. In order to capture this aspect, we define a generalized brownianalgebra as foUows.6.3. Definition. A generalized brownian algebra is an algebra satisfying theinitials(i) aa = a (iteration)(ii) all = a (reflection)Note that such a generalized algebra has nothing more than exteriordescriptions of calling(I-,= I)nd crossing (1) = (blank.

6.4. Definition. Let B bea brownian algebra and let S(B) denote the set ofperiodic sequences in B with even (assigned) period. If a, b E S(B), then Viewrite a = b when an = bn for all n , and p(a) = pCb) (p(a) = the period assignedto a). Operations are defined as follows:(i) (ab)n = anbn, p(ab) = lcm (p(a),p(b.(ii) (cil)n = an -k , k = p(a )/2

p(al ) = p(a).Since lcmilcm (x .Y),z) = Icm (x, fcm (y , I for any integers x.Y,z, the operation(i) is associative. Associativity becomes explicit at this point.Another way of putting our observation about resonant marked states is tonote that we may assign any period p to the empty sequence 4 J = . . . , , , , . . . ES(B). Then ~ = . " I, I, I, I, ... also has period p. Thus we cannot write1= ... I, I, I, ... without noting its period.6.4. Lemma. S(B) is a generalized brownian algebra.The proof is immediate.In order to understand the structure of S(B) we now give conditions underwhich occultation and transposition hold.

6.5. Theorem: Let ab s: E S(B) be members of the sequence algebra for abrownian algebra B. Then(i) aJ b l a =a if and only ifp(b) dividesp(a);(ii) aJ 0 1 1 c = Q C I D c l l whenever the largest power of 2 dividingN is the samefor N= pea), N = pCb) and N = p(c).


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192 L. H. Kauffman and F. J. VarehThe following lemma will be used in the proof.I6.6. Lemma. Let k(a) =lp(a) for any a E S(B). Then(i) p(b) Ip(a) (x Iy means x dividesy).. pea) I(k(ab) +k (a.(ii) If pea) = 2aN, pCb) = 2aM when Nand M are odd,then p(a) I(k(ab) + k(a and p(b) I(k(ab) + k(b)) .

Proof of lemma. The proof is omitted.Proof of theorem. If QJ b l a =a then p(Q1 b l a) = pea). But p(a l b l a ) = lcm (p(apCb~and:. pea) = lcm (p(a), p(b and this implies that pCb) Ip(a).Conversely, suppose that pCb) Ip{a). Then for any n, an . .t (ab)-k(a) = an sincek(ab) + k(a) is a multiple of p(a).Hence

(aI b l a)n = F a l b l )n an= (til b)n oJ: (ab)1 an= an-J: (ab)-J:(a)1 bn'J:(ab)1 all= Q , ; l bn-Ic (ab)1 an= an (occultation in B).

This proves (i).To prove (ii) first note that since lcm (/em (x,y),z) = lcm (lem (x,z), lcm(y,z for any integers x,y,z, p(QI ' i i i e) = p(tiCl ill }or any a,b,e _E S(B}Thus we must show that Qj 'O U e and Qc1 D c! are identical term by term. Ware given that p(a} = 2aN, pCb) = 2aM, pee) = 2aR where N,M and R are oddThen ( Q J 1 j j j e ) 1 1 = (aI 01 )l1ok (ab)1 en

= an-Hab)-.t(a)1 bn-J:(4b)-Hb)11 en= a, ; l 6;;l en

Since pea) I k(ab) + k{a) and pCb) I k(ab) + k(b} (by Lemme 6.6.) .On the 0ther hand,(QC] m )n = (ilcl 6C l ) 1 : 1 -J: (abd

= (aeln -k (abe)J: (ae)1 (bc)lI-k (abe)-J: (be)!= (oe)nl (bcf,J

Sincep(oe) Ik(abe) + k(ae) and p(be) Ik{abe) + k(be) (by Lemma 6;6) .= an enl bn e n l= ;,;] bJ en= (aI til e)n

This completes the proof of the theorem.Remark. If we let S(B) denote the set of elements a E S(B) such that pea)2aM for Modd,then transposition is satisfied ins


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Form dynamics 193iteration, reflection and transposition a transposition algebra. We see that foreach a, SZ(B) is a transposition algebra. By (i) of 6.5 it is not a brownian since~ a = a if and only if p(b) I p(a). It is easy to verify that echelon,combination, catalysis, tension, modified transposition and modal cross-transposition are valid in any transposition algebra. Occultation is not aconsequence, as the models SJ(B) show.Remark. It is worth noting exactly how close a transposition algebra comes tobeing a brownian algebra. If we include the initial " " 1 l = I,then we can getoccultation as a consequence:ifbI a = ~ a = aa1 ti l a l l = iii b I a D = -=rt;R a = 1 1 a = a .Thus"" becomes relativized to the period of the sequence that it interacts within a generalized brownian algebra. We can assert that "lb = . . . , for any b only byallowing the frequency of ...,to change with b. Nevertheless, the rules I' = . . . ,and 'll = (blank) still hold. Even if I is seen to be resonating at a givenfrequency, it still calls and cancels itself.

7. Constructing wavefonnsIn dealing with waveforms, we have so far assumed that there are sequences ofelements from an algebra B. The relationship between the sequences and theunderlying algebra has remained mysterious. We now show that the operationsof the algebra B itself are capable of generating oscillations, by the simpleexpedient of recursion. That is, given an algebra B and an algebraic operationT:B -+ B, we consider the iterates T" = I, T' = T, T " = ToT, ... ,m+1 = moT.If there is an integer p such that 'jT'+P = 1 '" for all n, then T can be used toproduce sequences of period p,For example, let T(x) = X l . Then P(x) = x and yn+z= T" for all n, Tproduces the sequence: x, Xl, x ,X l, x ,X l, ... In this case we have an algebraicversion of the sequence. That is, if xEB, then a = (x , X l ) and { 3 = (Xl , x) belong to!J and represent two phase-shifted versions:a: xXlxX lxX l .1 3 : X1xX lxXlx .

We are given T :B -+ B and obtain the corresponding mapping t:!J ~ ! Jdefined by the same formula. Note thatt(a) = 4 1 = (x ,xl)J =


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194 L . H . K au ffmDl! and F. J. Varela7.2. Theorem. Let B be primary, and T:B _. B an algebraic mapping. Let t bethe corresponding mapping on!J. Then there exists a z E ! 3 such that ttz, = Z. Infact, we may take Z = (T(x),P (x) or Z = iT ? (x),T(x for any xEB.Proof. To see that Z = (T(x),P (x)) is a fixed point fort, rust note that for anyZ = (a,(j) E13, tiz: = a L I b2i c

= (aa,oI3)1 b(13l ,03 X c= (Of3l, G O O ) (bi lL b J j i l )c= ( a m bel l i c . a Ciimc ) .

Thus it suffices to show thatT(x) = aP (x)1 b T(x)1 c

and P(x) = aT(x)1 bTft)'l cwhen T(x) = l iXl bill c.This is a straightforward computation.

Thus the algebraic structure of 1 3 reflects the properties of periodicsequences that are generated from B. Sequences from B become algebraicfixed points in 1 3 .What we see emerging here is a beautiful harmony among oscillations,re-entering fonns,aIgebraic operations and their fixed points. So far we haveseen that the re-entry of a form to its own indicational space, as in x =il = iJgives rise to a fundamentally new arithmetic V, where we have the waveformsand j. These are fixed points for the cross " " 1 in the new algebra V. We can

attempt to generalize this situation by showing how every re-entering form wilgive rise to an oscillation: the fixed points of the operator represent the spatialview of the oscillation, while its associated sequences represent the temporalcontext. Now in order to do this, we have to be able to construct an algebrawhere infinite expressions are defined, so that we are assured that everyalgebraic expression actually has a fixed point in the same algebra, rather thanin a larger one (as 'in the waveform arithmetic where i is in V but not in P). Wprovide such a construction in the next section, and in so doing we will see thatthe correspondence between pattern (re-entry) and oscillation (sequences) wilbe partially lost. Although every re-entry form will oscillate, a given waveformcan be generated through many alternative operators when they are allowed tore-enter.The remainder of this section will be devoted to procedures for generatingsequences of arbitrary period. As we have seen, sequences of period 2 may begenerated by an operator with a single variable. In order to generate sequencesof period other than 2, itis necessary to use recursion on more than onevariable. For example, let T :P x P -+ P x P (P denotes the primary arithmetic)be defmed by T(x..v) = < Y l . XIy l ) . Then


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Form dynamic s 195T(I,") = (""1,1)T (""1, 1 ) = ( = i l , ~ )T(=il,:;l) = (I,~'l)

and hence T produces two entrained oscillations of period 3:

We say that T produces an oscillation of period 3 and dimension 2.Notice thatwe may also represent this two-dimensional waveform by the spatial pattern ofre-entry of the operation which generates it: That is, T(XV IyV) =(XV, yV) represents the spatial fixed point and in this case,

x " = ~ y " = X 9 J y " I = X 9 J J . Hence X" = 1IWe now show that it is a simple matter to determine operators T thatproduce a given wave-train,

7 .3 . Definition. Let P denote the primary arithmetic, and P" :::: :P X P X x Pthe n-fold cartesian product of P with itself. An algebraic operator TiP" -4 pnis a function

Ttx , ... IXn) = (T 1 (Xl I" ,X n), I T n (XI , ... ,xnwhere each Tk (x 1 , ,X n) is an expression in the primary algebra involvingthe variables Xl, . , Xn. We say that T is periodic if there exists an integer psuch that T P +n = 1' " for all integers n > N (p and n are non-negative integers)where N is some specified integer.Example. Let T:P2 -4 p 2 be defined by T(x,Y) = (Xl Y i l , x y t ) . Now p l =

a.b.c.d where a = (I,""), b = (I, "I), c = ( ', ') and d = ( , , "l).Itis easy to verify that T(a) = b, T(b) = c, T(c) = d and T(d) = c. Thus we saythat T has period 2 since '[ '11+ 1 = T " for n > 1 . Note that P*T since T(a)= b while P (a) = d. With this definition we immediately obtain thefollowing result:7.4. Theorem. Every algebraic operator is periodic.Proof. p n has cardinality 2n (with respect to arithmetic values). Hence for anyfixed X = (X I' Ixn) E P" , the sed 'P I (x) i n = 1,2, ... } is finite. Thusthe sequence T(x), T l (x), ... must be eventually periodic for each x. Sincethere are a finite number of such X I the least common multiple of thecorresponding periods is necessarily a period for T.7.5. Theorem. Let 1 T : P " -+ P be the projection to the first co-ordinate, 1 T ( O :I,.I~ ) = at. Let{a = [an} In = 1,2, ... ] be any periodic sequence of values


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196 L . H . K au ffm an and F . I. V arelafrom P. Let p be the least period of a. and choose n so that 2n.1 < p $;ZH.Thenthere exists an operator T : p n -+ P" of least period p, and a starting vector ofvalues x E P" so that an = 1I'(P (x for n = 1,2, ....

Thus the sequence a can be seen as the first component of an x-dimensionalentrained oscillation.Proof. We shall give an algorithm for producing the requisite operator. Thefollowing notation is convenient. Let b = (bl / 1 2 I , bn) E P", and letA(b) = AI A1 .. A where ill = 1 if b, = , and ~ = 0 if b, = "l,Regardt:,(b) as an integer expressed in the binary system. Let neb) be thecorresponding decimal integer. Let lJ(b) be the following operator:

~(b)(x."",xn) = b.(X l)b2(X 2) ... bn(xn) !where bi(x) = XI if bi = -,

x if bi = "I.Note that {I(b):Jln -+ P and {I(b)(x) = I~ = b. For computations it isoften useful to use neb) as the name of Mb).

Nowchoose b I,bl ,.. ,bp E pn so that0) b, * b, if i.* i .(li) 1I'(bk} = ak ,k = 1,2, ... ,p.This can be done since 2n-1< p ~ 2 1 ' 1 1 .

Let Tk(xi ... ,xn) = (J(bal) (J (bal) ... (J(b~) where a., ... ,~ isthe set of indices a such that kth co-ordinate of b(O:+I) is marked (we view 0:modulo p so that bp+l = bl). Finally, let Ttx, , ... xn) = (TI (x), ... ,t;(x.

It is easy to verify that T(bk) = bkt and T(bp) = b s . Thus T produces thedesired periodic sequence. This comple1tes the proof of the theorem.In order to illustrate the foregoing theorem, suppose that we wish toproduce the period 5 oscillation:

That is, al = -, > a2 = ' . a3 = I, a" = ' , as = , . Then we take n = 3andwe may choose bl ,... Ibs as in the chart Fig. 2).H~re . .Hence r, = 20 = x.Y1 z l xyzl = XZl

T2 = 1 = xy Z iITJ = = 0 5 = xy z I X I y tIThus T(x,y,z) = (rn, xy Z 1 , xyzi Xl y zH ) .


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Form dynamics 197

IIx Y Z ntb)

bl I ~ ~ 4b2 ~ ~ ~ 0b3 I ~ I 5b4 ~ = J l I Ibs ~ f =jl 2

T (x . r, z) = (rn. xyil [ xyzl X l r II [ )

Y:X :

r.:

Z:Fig. 2

In general there will be many operators T such that an = 7rTWx) for a givensequence a = { all} . We can easily generalize this lemma to embed anyentrained oscillation (of arbitrary dimension) in a higher-dimensionaloscillation without repetitions (in the sense that bi * bj for i * [) .Applyingthe same algorithm, we thereby obtain an operator T whose projected iteratesgive rise to the original entrainment.In the next section we look more closely at the formal structure of thiscategory of operators.

8. Re-entering forms and infinite expressionsIn the primary arithmetic there is no separation of operator and operand. Themark, I, takes both roles as context and viewpoint demand. We haveconstructed certain 'complex' arithmetical forms (such as i and j of thewaveform arithmetic) but these have not been constructed as operators.Rather, they are seen ssfixed points of an operator. Thus 71 = i and 7 1 = j. Inorder to return to a viewpoint wherein such fixed points are seen also asoperators, we need to re-consider the self-referential context.What is the meaning of f = iJ? We have suggested in section 2 that iJmay

be viewed as an infinite form f = ';] ~ ;:;j I I so that f is literally contained as a


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198 L. H . K auffman and F . J. Varela

proper part of its own indicational space if = ]l). When we write f = : = = 1 1 1 =Jl we are not saying that f may be calculated from 1 1 , but rather that rand 1are identical as infmite forms. Seen in this light, f operates on itself by way ofthe re-en try. This operation is the very lifeblood off = iJ giving it the stabilityexpressed by iJ = L J .I n this sense the very notion of a self-contained form is closely tied to self-reference. The extent to which a form is seen to be autonomous/self-containedis directly proportional to how we find its stabilities. We push at it here, pull atit there. It seems to react, stabilize, retain shape and intelligibility. How canthis come about? Even to a pure solipsist, the experience of relatively stable,seemingly external forms must present something of a puzzle. And yet the

simple form of L J = ;:;,I I presents a shift of perspective. Let be given onlyan operator T (say T(x) = Xl ). Allow the operator to operate on itselfinfinitely as in A = T(T(T(T(T( ... . Then T(A) = A is a formal identity.The infinite concatenation of the operator on itself gives rise to a stability withrespect to this operation. It becomes possible to say of A: it has fonn. It is anobject with a life of its own.

It is no denial of reality to suggest that what we call objects are in factnothing more than operations taken to such a limit. In fact we do suggest thisand invite the reader to consider the idea in all of its rernifications. When weinvestigate natural forms we continually come upon circularities of structurethat lend stability. Are not these precisely the same as our formal images ofself-containment and self-reference?In order to understand the main idea for allowing infinite fornts, considerthe following example (due to Spencer-Brown). Let r e x } = XaJ b l . Theniterating T we have, as expected,P(x) =xal b l a l b l = xalbl = T(x).Thus

" X a J b l = Xafb l a l b l = Xci] h I a l b l a l b l = N1 hi a l b l a t b l a [ h Iand if we allow this process to proceed indefinitely, we are led to contemplatethe infinite expression, x V = .. i i 1 b l a ! b I .This form contains a copy of itself, and thus re-enters its own indicationalspace

xV = xV a l b ! .B y going to an infinite expression, we have eliminated x as a variable andobtained a form, or spatial pattern, which embodies the operation.In other language, xV is the fIxed point of T, for

T(xV) = x Y a l b l = . . :If!bl a l h l = xV .The equation T(xV) = xV is an expression of the direct identity of theseexpressions; it is not a statement that one can be calculated from the other. In


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Form dynamics 199general, by going into a suitable structure, where infinite expressions areallowed, we are assured that every operation will have a fixed point solution,for we can form xV = T(T(T( ... ) = lim 1'",the infinite concatenation of this operator. In this universe of infinite forms weare free to express the re-entry of forms. What needs to be examined is therelation between spatial re-entry and its temporal quality: given a pattern, howdoes it vibrate?In order to explore this question we first have to construct a universe ofinfinite expressions and see how re-entry is expressed in them more precisely.We have two basic clues from the previous discussion. First, in order to form aninfinite expression we may allow it to grow step by step, and never commandthe process to come to a halt. This involves introducing the idea of an order inthe class of expressions, so that at each successive step the new expression is abetter approximation to the infinite one. In the limit, the sequence ofapproximations defmes the infinite expression. This is a process somewhatreminiscent of the idea of order, approximation and limit in calculus. (Theorder being introduced here, however, is quite different from numericalapproximation). Secondly, in order to have re-entry, it is enough to considerequations among these infinite expressions, or, in the other words, fixed pointsfor operators in this extended domain.As an example, with the operation T (x) = x al hi we may consider asequence of approximate expressions thus:

1C lal b l c Ial b l a l b l C '"where we start with an undefined exwession 1, and successively apply T.HereC denotes the order relation and x = lim 1'" (1). The reader will have ton-k>:>forgive a full exposition of the details of order and approximation for infiniteexpressions. We hope that we have given the flavor of it; the full technicaldescription may be found in Goguen & Varela (1979) and Varela & Goguen(1978).Thus we now assume that we have before us a well defined class Boo ofindicational expressions finite or infinite. Elements of Boo look just likeordinary indicational forms, except that they might grow indefinitely! CallBoothe classof continuous forms.Here is a rapid sketch of some algebraic ideas that can be handled inBoo:8.1. Definition. Let B.,..(X") be the class of continuous forms on X" = {x 1,xl, ,Xn} That is, the variables Xl,'" 'Xn may appear in the expressions ofBco(xn). A system 0/ equations in Boo(X") is a function E: xn ~ B~(XII)where Bg,,(Xn) denotes the n-fold cartesian product of Boo(xn) with itself. Weshall write xj- = E,(x) as the ith equation of E (x = (XI,xl,"" xn n. Thefollowing can be proved.8.2. Proposition. let E be a system of equations in Boo(Xn). Then there isA E EB!b(xn) which is a minimum fixed point for E, E(.AE) = AE AE is called thesolution of E over Boo(xn). In fact AE = lim 1 ' (1,1, ... ,1).


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200 L. H. K au ffm an a nd F . J. VarelThis proposition assures us that this new universe B C I O is large enough tohandle all kinds of re-entry. We have no idea how complex the equation Ecould be; perhaps it is inflnite. Thus, for the present purposes we will narrowour scope, and consider only those inflnite expressions in B o o that correspondto finite re-entry (e.g. where the re-entry can be indicated on a sheet of paper.)

8.3. Definition. A finite system of equations over Boo(xn) is a system ofequations E such that E: X"-+ Bn (X"). Here Bn (xn) designates the set ofn-tuples of primary algebraic expressions in the variables XI,xl, ... , xnIn this way we focus on the forms in B o o that arise from (finite) operationsinB.8.4. Definition. The set RB of rational expressions of dimension n is the subsetof B C I O satisfying: RB = { A E B o o I 3 Al , ... , A n E B o o and a finite systemof equationsE so that E(A) = A where A = (AI ,A l,A3 ," . ,An)} . That is,the rational expressions are components of fixed points for finite operators.They are single components of n-dimensional re-entering forms.

RB is an enchanted land for self-referential forms - every operationimmediately gets an associated value (fixed point), a re-entering formcomputed through the fixed point construction. Conversely, each rational formhas a corresponding operator. While this correspondence between operators andoperands is not one-to-one, we are certainly justified in asserting that it is amatter of point of view whether a rational expression is regarded as a value, oras a transformation. In this sense, operands (i,e. elements of RB) and operators(i.e, certain maps RB -+ RB) are interchangeable.tLet us now examine the relation of RB to the temporal context. Given A ER8 We know that there exists A = (A 1 ,Al, . , An) ER'b so that A = A Iand a finite algebraic operator T of dimension n so that T{A) = A. We knowthat A = lim 1'" (1, 1, ... , 1). Thus A has an associated sequence A(n)

t T h i s ismeant to point to the idea that formal doJDlins ca n be reflexive, that is, type-free. T h e fuDextent of this idea has been proposed a nd explored il l combinatorial logic and topology by Dana Scott(1971, 1972; tee alJo Wachworth, 1978). For further discussion on the notion of rat ional elements ofcontinuous algebIu see Gapen et al. (1977) and. Wright (1976). Obviously what we say here is veryinformal and expository, and the il l terested reader is encouraged to look at the aforementioned papers fora detailed discussion.

n-+ 00(n = 0,1,2, ... ) where A(n) =1'" (1,1, ... ,1) = Tn (I).At every finite n,each term of this sequence is finite, and thus we can reduce them algebraicallyby choosing an initial value (vector) for the indeterminate terms in V = T. Weimmediately see, by Theorem 7.4, that each sequence will be (eventually)periodic. If we change the initial value we either generate the same sequencewith a phase shift, or we find ourselves in an entirely new periodic sequence.These associated periodic sequences may be thought of as (temporal) states ofthe expression A. Thus every rational expression oscillates. To each pair (A, nsuch that T(A) = A there is a corresponding oscillation and characteristicfrequency.Thus we may say that (A, n has a dual nature of particle (TCA) = A) andwave (V, T(V), Tl (V) , ... ). Which viewpoint comes to the fore depends onour bias. If we insist on invariance of form, then the particle nature is apparent.If we allow the calculational dynamic and assign values, then the wave-nature isseen. Both viewpoints rest ultimately in the essential periodicity of the process


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Fo rm d yn am ic s 201we have considered.For example, consider the re-entering form presented in section 7:

XV =Since (being infinite) it must belong to Rs ' we know that it must be part of thefixed point of a system of equations. In fact XV = x~ J I , and this reveals thatit is, in fact, the first component of a system (xV ,yV) = ( Y V l , ;v, y v l ) . Thus thecorresponding transformation is given by T(x,y) = C Y 1 , X l y I ) . To see thewaveform that corresponds to x V we compute the sequence of approximationsto xV (these are the first co-ordinates of ]lI(l, 1) for n = 1,1,2, ... ):

_ L , : L l , : L l _ L I I ,~ :L l _ L 1 1 1 , : L l _ L I I I~ : ; J I l l 'n.Thus if1 , . then the wave is

',1-",11...and if 1 = 1hen the wave becomes

1',11',1...In each case. we have a wave train of period 3. The change in choice of initialvalue produces a phase shift.The inverse process is more complex. For a given sequence, there are manyoperators that will generate it, and therefore several elements of Rs can beassociated to it. For example, t EV can be produced by T(x) = Xl, but it alsoappears entrained with other oscillations as in Tex,y) = C Y I , Xl). Notice alsothat the re-entrant form flwill correspond to both i and j depending on how 1is evaluated. As it should be in the static world of forms, phases are irrelevant.The waveforms i and j condense to the spatial form iJ .Much remains to be explored about re-entry forms. For example, we havenot discussed the matter of algebraic structures. We feel that more work isneeded in this area, but shall end this section with an example.

Consider x = XxI . Certainly, this equation is satisfled by the Infinite formu:::J. If we allow iteration (aa = a) as an algebraic rule about infinite forms,theniJ will also satisfy the equation. The infinite forms iJ and i:i:J should beregarded as distinct, and yet algebraically they are both solutions to the sameequation. In general, we can impose algebraic initials on the class of infiniteforms after these forms have been fully constructed. In this fashion theuniverse of infinite forms becomes a brownian algebra.Note that it is too much to ask the infinite forms to be a primary algebrasince, if so then


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202 L H . Kauffman and F . J. Va/ 'elaThe fonn of position sends the entire structure into the void!Just as a self-referential expression has many possible stable states andpatterns so have we produced many viewpoints clustering around the generalself-referential form. Each view has its own coherence and different views arerelated in remarkable ways. These remarks, unfolding the self-referential form,only skim the surface. W e see that each self-referential form ha s a naturaloperator / operand (or wave/particle) duality. and furthermore, eachself-referential expression has an associated brownian algebra. We haveemphasized individual self-containing expressions because this context places usclosest to our experience. Each multiplicity of forms is seen within a widerform to which it condenses.

9. On geometrical fonnAt this stage it is easy to see the beginnings of the connections between ournotions of form and the classical views of geometry and topology. Thegeometer considers forms as topological spaces and asks for properties invariantunder various groups of geometrical transformations. In the abstract, there is adirect analogy with our invariances T{A) = A. It would be an incrediblycomplex task to translate most standard geometric notions directly into thiscontext. Nevertheless, this could in principle be done, since the structure of anaxiom system for geometry can express all of its operations as analogous formaloperations. The 'geometric' objects can then be constructed as formalconcatenations of these operators. This may give insight into the processes ofgeometry.This general view of invariance is attractive in that it embraces forms of allkinds from the abstract geometric forms to the shapes of living things that areso obviously continually reformed by recursive process such as we havediscussed.

A well-known example of a geometrical re-entry form is given by the divisionof a golden rectangle: (Fig. 3)

Fig. 3


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Form dynamics 203Here we are given a rectangle with sides in the ratio I:~ (l+v'). Successivecuttings of squares leads to a spiraling sequence of similar rectangles. The givenrectangle is of the same form as the new rectangle obtained by adding a squareto its longer side. This essential identity of part and whole is seen again andagain in natural forms. The whole can be seen in any part, and at any scale atwhich the form is examined. Forms of this type (self-similargeometric forms)have been given much attention recently under the name of fractals (seeMandelbrot, 1977).The line between the recursion and the perceived fonn varies just as in OUfabstract wave/particle duality. When the particles are at the fore, their globalinter-relationships lead to exterior forms of description best suited to algebraand geometry, Nevertheless, the recursion and wavefonn sounds continuallybelow and throughout these structures.As a fmalexample,contemplate the complex numbers. Here is indeed themathematical expression of an enchanted realm where waveform and geometrylive joyfully together. All mathematicians know this, and yet our point of viewgives even this place a subtle shift.Reconsider Xl + 1 = O.View its solution as the happy wave-form solutionto the 'paradox' x = -xl That is, let T(x) = -xl. Whence the infiniteeigenforrn E so that T(E} ;: ;: E is given by E ;: ;: T (T(T ( ... = - ( - ( -(- tl r't' tl) and the associated wave train is ... , -1,+1,-1,+1,-1,+1,-1,+1. .Let R denote all real numbers, and let R = R x R with elements [a,b] witha,b E R. Regard [ap] as representative of the waveform ... abababab ...Make definitions:(1) [ap] + [c,d] = [a+c,b+d](2) [ap1 = [boO] (conjugation is a phase-shift)(3) .J=1 ;:;:[+1,-1] ,I = [1,1](4) [a,b] * [c,d] ;:;: [acpd](5) a[b,c] = [a,a] * [b,c] = [ab,ac]Now demand a multiplication a, 3 t-+ a J 3 so that (-/=-l)l = -I, that iscommutative, associative and distributes over addition. Show that necessarily:

a J 3 = Y2(a*{3 + li*/3 + a*lI - a*Tf).Thus the complex numbers are easily viewed as waveforms. Multiplicationbecomes a special operation involving combinations of phase shifts.

a + iJ.../--I = al + b.. . f=-l = a[l,ll + b[l,-l] = [a+b,a-b].Thus a+b.. . f=-l oscillates between a+ b and a-b.Re-introduce the well-known geometry so that a+lr,f=.l is represented as(ap) in the cartesian plane. View the oscillation as a circular orbiting of a pointat a distance Ib Ifrom a on R (see Fig. 4).


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204 L. H. Kauffman and F . J. Varela

o a+b

-I

+1

Wave form Cart.slan for" ,Fig. "

Combine the two (Fig. 5) by associating the orbit to each complex point on thunit circle in the complex plane. In the diagram each circular orbit correspondto two complex numbers a o...!=-I on the cartesian circle, +1 and -I ardegenerate circles, F-l is the large unit circle.

Fig. 5This view of the complex numbers expands the real line not to the plane but toa dancing buzzing line with an infinity of synchronized circular orbitassociated to each point.The two points of view are in mutual support. While the temporality has ithe usual case been restricted to the one circular orbit about the origin (and itprojections as sine and cosine), nevertheless it pervades the complexes fromwhatever viewwe take.Thus we are only stating the obvious, but we must repeat: the complexnumbers are to the real numbers as the brownian wave-algebra is to primaryalgebra. Resolution of paradox at the point of self-reference leads to themergence of new forms, temporal and spatial.


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Fo rm d yn am ic s 20510. Coda

We intended, in this paper, a discussion of form that began as simply aspossible. What is simplicity? The word simple derives from the Latin simplex, acombination of semel (once) and plex (fold) ..Thus to be simple is to be of onefold. This look into the roots of simplicity propels us at once into an entirecomplex of ideas surrounding the notion of form,Let that which is folded be some fabric of unspecified qualities. The foldprovides a distinction within the space of the fabric. And yet the fabric itselfremains essentially whole and undivided. It is our perception of the fold (ourmaking of it) that divides the fabric for us. The distinction is mutable - a pullon the fabric or a change in viewpoint will restore the wholeness at once. Theact of distinction, of seeing the fold as a fold in the fabric, becomes the fonn asseen. In this sense there can be no separation between distinctions and acts ofdistinguishing. Herein lies the point of utmost simplicity,Even to distinguish ourselves from the fabric is to move into a complexity ofcleavages that is far from being of one fold. In simplicity we can not distinguishourselves from the fabric. The fold is our fold. The act is the object acted upon.The distinction ismade and dissolved alternately and simultaneously. For thereis no sequential time in simplicity - only in the inevitable movements into andaway from complexity does the notion of time occur.Yet in perceiving a form we are aware of time's passage, of the succession ofthat form that is seen to be the same form. To see the form again and yet to seeit as unchanged is a periodicity, a repetition of form, An underlying periodicvibration, if you will.The periodicity may be purely temporal as in music, or almost entirelyspatial as in the ornate frieze patterns used to decorate walls and boundaries.These seemingly complementary views of periodicity merge when we realizethat the viewing or making of any boundary involves a periodic oscillation ordance across it. There may in fact be no boundary other than this musicaldance. The boundary becomes vibration at a distance. Here space and timemove into one another as our perceptions move toward the one fold.We must conclude at this point. This paper has been a combination of twothings: on the one hand we have presented a formalization for those intuitionswhich are clear to us; on the other hand we have hinted at many other possibleroutes in an infonnaJ,.alld metaphorical way. This seems to be necessary at thisstage of study of fonn dynamics.Our approach can be summarized as follows: from the basic notion ofindication and the primary algebra of indications ofG. Spencer-Brown, we havemoved into two essentially complementary directions in order to bring out thedynamic component immanent in a form. First we developed the notion of abrownian algebra, where waveforms can be represented through sequences.Second, we expanded indicational forms to infmitary indicational algebraswhere re-entry can be expressed properly. The relations between oscillationsand pattern can then be established through an analysis of algebraic operatorswhich emobody the dynamic quality of spatial forms.Even at the level of scrutiny we have achieved so far, there is great beauty inthe harmony of shapes and the vibrational quality of indicational expressions.So far, the music of these spheres seemed to have escaped our notice, except in


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206 L. H. Kauffman and F. J. Varelasome more special forms (such as the wave/particle duality in quantummechanics). What we see in the present context is that all of these periodicphenomena in matter, nature, and art seem to be fundamentally the same.They stem from the basic act of distinction, of creating a duality of this andthat. This primordial act is pregnant with time, space, pattern and their dance.

AcknowledgementsThe authors wish to thank Joseph Goguen and David Solzrnan, for manyfruitful conversations on Cyrnatics, self-reference and interpretations of thewave-form algebras.L. K. was partially supported by the National Science Foundation. F. V. waspartially supported by the Alfred P. Sloan Foundation.

ReferencesGoguen, J. A. & Varela, F. J. (l978). Systems and distinctions; duality and complementarityInt. J. Gen Systems (in press).Goguen, J. A. & Varela, F. J. (1979). Some algebraic foundations for self-referential systemprocesses. Math. Systems Theory (submitted fOT publication).Goguen, J., Thachter, R., Wagner, J. &Wright, 1. (1977). Initial algebra semantics.v. Assoc.Comput, Mach. 24, 68.Jenny, H. (1974). Cymatics, Basel: Basilius Presse.Kauffman, L. (1978a). Network synthesis and Varela's calculus. Int. J. Gen. Systems 4, 179-187.Kauffman, L. (197gb). De Morgan algebras, completeness and recursion. Proceedings of theEighth International Symposium onMultiple-Valued Logic, pp. 82-86.Mandelbrot, B. B. (1977). Fractals, Form, Chance and Dimension. San Francisco: W. H.Freeman.Scott, D. (1971) The lattice of now diagrams, in Springer Lecture Notes in Mathematics,No. 188. New York: Springer-Verlag.Scott, D. (1972). Continuous lattices. Springer Lecture NOles in Mathematics. No. 274. NewYork: Springer-Verlag .. Spencer-Brown, G. (1972). Laws of Form. New York: The Julian Press.Varela, F. 1. (1975). A calculus for self-reference.lnt. 1. Gen. Systems. 2; 5-24.Varela, F. I. (19780). On being antonomous: the lessons of natural history for systemstheory. In (G. Klir, Ed) Applied Systems Research: Recent Developments and Trends.New York: Plenum Press.Varela, F. J. (I978b). Autonomy and Cognition: On the Circularities of Natural Systems andtheir Cognitive Processes. New York: Elsevier-North Holland (in press).Varela, F. J. & Goguen, J. A. (1978). The arithmetic of closure. f.Cybernetics, 8, (4).Wadsworth, A. (1976). The reliitto'n between computational and denotational propertiesfor Scott's D DO models for the)o..-c:alculus.SIAM 1. Comput. 5,488.Wright, 1. J., Thachter, R. Wagner, E. & Goguen, J. (1976). Rational algebraic theoriesand fixed point solutions. Proc. IEEE 17th Symp. Found. Comput, Sci., Houston, pp,147-158.Wharf, B. L. (I956). Language, Thought and Reality. Cambridge, Mass.: M.I.T. Press.

Louis H. Kauffman and Francisco J. Varela- Form Dynamics

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