Creativity involves the production of complex patterns from simple processes. What is the least initial complexity required to generate both simple patterns and bios? Four main mathematical ideas have been developed to characterize complex processes: steady states, oscillations, chaos, and noise. The empirical studies reported in Section I indicate a fifth type, which we call bios. Empirically observed patterns serve as beacons to guide the development of mathematical models for creativity. Also, the different terms in the equations must be possible to interpret in terms of actual physical aspects of natural processes (figure 13). The quantic nature of action points to discrete iterations; differential equations and calculus imply and require continuity. To study creativity, one must consider processes rather than static patterns; thus we compute recursions with varying parameters; we call them kinetic equations, as contrasted to standard dynamic recursions computed with a constant parameter. Kinetic equations generate a single time series with a succession of patterns (“mathematical development”).

The simplest and most universal process, present in both matter and empty space, is flux. By flux we refer to the spontaneous, apparently random, erratic fluctuations observed at all levels of organization. The existence of flux is a necessary consequence of the uncertainty principle that, excluding certainty, excludes absolute rest. The simplest physical process is the formation and destruction of pairs of opposite virtual particles within the limits of the action quantum. This indicates the fundamental character of opposition.

Figure 13: Simple physical processes containing basic forms reproduced at all levels of organization.

Within this symmetric flux, there are local flows of directed action. Planck’s quantum of action, inertial movement, cardiac contractions, and thinking, illustrate action at various levels of organization. Action is the flow of energy in unidirectional time. Energy and time are inseparable: the unit of change is action, not energy. Yet energy and time are distinct, and this conjunction of inseparability and distinction is exemplary of the union of complementary opposites. Energy and time are quantum conjugates. As energy is never created nor destroyed, action conveys only one dimension, time. Processes are continuous sequences made of discrete action units.

A random walk in which the time series is formed by the addition of successive random terms, models flux (random numbers) and action (conservation and additivity features). Random walks are sufficient to generate bios-like time series (including Mandala patterns when the random steps are integer). But random walks fail to generate the simple periodic patterns and shifts that accompany bios in natural processes. Further, the assumption of that changes result from independent events is often difficult to justify and impossible to prove.

Processes of interactions, i.e. relations, are far more common that isolated interactions. Entities are embedded in groups of interactions. I use the term group (a set in which every member has an inverse or opposite) to indicate that relations of opposites are central to the web of interactions. First, there are fundamental opposites such as attraction and repulsion, positive and negative charge, and, at higher levels of organization, female and male, true and false. Also, a relation shapes the interacting processes, making them complementary to each other. Complementary processes and entities interact with each other, and, conversely, mutual feedback differentiates interacting processes into complementary opposites.

Relations organize systems into a central material core, a larger energetic field, and a larger and expanding informational field (figure 14). This model of concentric spheres reminds one of the Mandala. Whitehead pointed to this pattern of organization for the solar system: a material, high-energy core (sun), an energy field (planetary system), and an unlimited radiation of information (light) into colder space. In a similar manner, a person's world comprises a material core (person), an energy field (family, friends, coworkers, community), and a larger information network (social communication) that exceeds his life tenure.

Figure 14: A system is formed by a central material core, a surrounding web of energetic interactions, and a field of communication that expands in space and time.

In this light, the idea of system’s boundaries needs to be reformulated. Space separates one solar system from another. Personal systems, such as families, have no boundary, constantly evolving through marriage, births and deaths. In organism, membranes are specialized structures that serve to channel communication. Systems do not have boundaries that separate them from others. Instead of boundaries, there are fields of interaction, and mutual feedback. Organisms, biological communities, ecological systems, continually interact with their environment. Each input contributes to determine the following action; in turn, the inputs received by a system are at least in part reactions to its previous action. This is feedback. Feedback is a universal and fundamental process in nature. Multiple and complex interactions of positive and negative feedback processes are involved in biological development --development is not simply the realization of a pre-existing blueprint. Feedback processes also obtain between persons and their interpersonal world, institutions and society. Feedback is an essential feature of natural processes, not a special process that obtains only in particular systems or organisms. Through its repetitive interactions with others, each system becomes both self-referential and co-creative.

Table 4. Mathematical developments generated by feedback

A_tis the time series, f(A_t) is the function of A_tthat produces change, and g_tis the gain of this feedback.

Generator: A_t+1= A_t+ g_t * f(A_t)

Patterns generated

f(A_t)

Initial fixed point

Terminal

attractor

B tree of bifurcations T braid of trifurcations

Chaos

Bios

Multiple logons

A_t* h- A_t(logistic)

*

¥

B

x

A_t* h– A_t-1

*

¥

T

x

sin A_t(process)

À

¥

B

x

x

sin A_t-1

À

¥

T and B

x

x

sin A_t-j, for j >4 and g = 1

x

sin A_tfor g_t = A_t

x

A_t* sin A_t(bipolar logistic)

À

~0

B

x

x

**

A_t*sin A_t-1

À

~0

T

x

x

**

h-A_t* sin A_t

À

h

B

x

**

h –A_t-1* sin A_t

À

h

B

x

**

h-A_t* sin A_t-1

À

h

T

x

**

A_t+ h –A_t-1* sin A_t-1

À

h

T

x

**

A_t * h-A_t* sinA_t

À

~0

B

x

x

**

* The logistic map, computed with a constant gain, has a fixed point.

** Ascending sequence of logons if the initial value A₁< terminal attractor h. Descending logons if A1_t> h.

In natural systems, feedback is bipolar and diverse [François, 1997; Sabelli, in press]. Given the enormous diversity of natural environments, any action evokes some synergic responses and some antagonistic ones, one or the other predominating at different times. This bipolarity of natural feedback contrasts to purely positive and purely negative feedback mechanisms that have found such a wide range of applications in engineering. The creation of organization requires a combination of positive and negative feedback. If positive feedback processes would predominate, there will be no check to exponential growth in which plants become invading weeds, animals become pests, beliefs become self-fulfilling prophecies, and ideas are enthroned by bandwagon effects. Conversely, if negative feedback would predominate absolutely, there would be little change, and no evolution.

Mathematical feedback

Table 4 summarizes the results obtained with some of the models for coexisting positive and negative feedback. It includes logistic-like equations in which positive feedback predominates at low gain and negative feedback at high gain, process equations in which positive and negative changes occur at all gains, and a combination of the two (bipolar logistic equations). In all three classes, there can be feedback to the immediately preceding action A_t-1, to a past action A_t-k(delay equations), or to change in action A_t– A_t-1. Delay may be expected in feedback processes. The difference A_t– A_t-1 is the physical substrate of information. For instance, sensation is a function of a change in the stimulus rather than on the intensity of the stimulus. Recursions involving difference A_t– A_t-1 generate trifurcation and braids in their way to chaos and bios, instead of the usual cascade of bifurcations. Note in table 4 three routes to chaos and bios: (1) linear, (2) bifurcation, and (3) trifurcation.

Biotic development by bipolar feedback

Bipolar feedback is embodied in recursions in which change is a trigonometric function of the preceding term [Kauffman and Sabelli, 1998; Sabelli and Kauffman, 1999; Sabelli, 1999]. In the kinetic process equation

A_t+1 = A_t + k * t * sin A_t

the feedback gain g is a function of time ( g_t= k * t), This recursion models action, opposition, and transformation, three universal components of natural processes. Iteration models the sequence of discrete and unidirectional actions; the sine function models the interaction of coexisting opposites; the gain g = k * t represents the intensity of these opposite interactions; and the sequence of actions A_t forms and transforms patterns. Each new action A_t+1 represents a continuation of the previous action A_t at time t, plus a change that results from positive (augmenting, synergistic) and negative (decreasing, antagonistic) responses to A_tdepending on the sign of the sine function. This feedback is bipolar and diverse, spanning the range from plus to minus g through the continuity of the trigonometric function.

The amplitude and complexity of the patterns in the single time series generated by this bipolar feedback increases with the magnitude of the feedback gain (figure 15). For g < 2, the equation converges to an odd multiple of À (e.g. p for initial values between 0 and 2p). At g > 2, it generates symmetric opposites that diverge as the gain increases. Following the initial bifurcation, there is shift in each branch of the series that looks as a bifurcation in which only one outcome is visible (unifurcation). A cascade of period-doubling bifurcations generates 2^Nperiods, followed by period 2 + chaos, and then by chaos with intrachaotic periodicities, among which period 6 and period 4 are prominent. When g < 4.604, A_t expands both positively and negatively, generating aperiodic biotic patterns resembling those observed with cardiac data. This is bios, an apparently erratic pattern of much larger magnitude than the preceding chaos. Process chaos is negatively autocorrelated, while bios is positively autocorrelated (Pearson’s coefficient for 1 lag +0.99); unpredictability obtains at the boundary between chaos and bios.

Bios is terminates with a flight toward positive or negative infinity (infinitation) when the gain equals an even multiple of À (full rotation) and at few other critical values. As g increases further, new biotic series emerge, and further infinitations follow –a mathematical metaphor for death and renewal, essential features of living processes. Also, bios converges to bioperiods 2 when the gain equals odd multiples of À (half rotation inverting the sign of the sine function). Bios generates a new pair of opposites from which a new biotic phase emerges.

Figure 15: Time series generated by the kinetic process equation. Values of A_t(y axis) as the gain increases with time (x axis). Upper chart presents the steady state, bifurcation, unifurcation, cascade of bifurcations and chaos that obtain at low gain (1.5 to 4.5). The lower chart shows the time series from gain 1 to 7 with a larger scale in the y axis that allows one to see the biotic phase, and two infinitations; the initial phases are not discernable at this scale. This development starts from any initial value, no matter how small. If the cosine function is employed instead of the sine function, this development starts even from zero. The gains at which these various phenomena occur are dependent on the rate of change of g. The numerical values mentioned in the text obtain when g is kept constant.

Figure 16: Distinguishing characteristics of deterministic patterns generated by bipolar feedback. Note the sensitivity to initial conditions of periodic patterns generated by bipolar feedback. Bold indicates series globally sensitive to initial conditions

Bios, bioperiods, and infinitations, albeit widely different, share significant characteristics (figure 16), including global sensitivity to initial conditions absent in chaos (figure 17). Also, biotic trajectories are extremely sensitive to changes in the rate of change of the gain.

Figure 17: Bios and infinitation show global sensitivity to initial conditions while chaos does not. Time series (N =90000) generated by the process equation at constant gain 4.1 (chaos), 4.61 (bios) and 2 p (infinitation). Initial Value: left = 1.000; right side 1.001, except for chaos, in which an initial value of 5 is used to highlight the difference between locally sensitive chaos and globally sensitive bios.

If the gain first increases and then decreases, the time series evolves from an initial steady state to greater complexity, and then devolves back to simpler patterns. We call these sequences “lifeforms” (figure 18). A most remarkable feature of bios is revealed by these kinetic equations. If the series evolves up to chaos, a subsequent decrease in the gain leads back to the initial steady state. In contrast, if the series reaches the biotic regime, a subsequent decrease in gain does not lead back to the starting point. One is tempted to relate this mathematical irreversibility of bios, absent in chaos, to the irreversibility of physical processes.

Figure 18: “Lifeforms” generated by a process equation computed with a gain that periodically increases and decreases (sinusoidal line). If the gain surpasses the threshold for bios, there is no complete reversibility to the initial value. In contrast, if the time series reaches only chaos, there is reversibility.

Complexity increases with the amplitude of the time series up to a limit, and then decreases (infinitations, bioperiods), consistent with Hegel-Engels’ law relating quantity and quality. The difference DA_tbetween successive terms of the series equals the product of the gain g and the sine function ranging from 1 to –1. When the gain is low, the range of A is smaller than the amplitude of the difference DA_t. When the range of A_treaches 2p, the time series expands into bios, becoming much larger than DA_t(figure 19). This disproportion between long duration change and moment-to-moment differences distinguishes bios form chaos. As a result, the transition from chaos to bios represents a bent in tridimensional space (figure 20).

Figure 19: Time graph of the gain 0.0005 * t, the time series A_t (black lines) and the time series of the difference between consecutive terms of the series A_t - A_t-1(white lines) as the gain is increased by 0.0005 at each iteration.

Transition into bios represents a change in dimensionality. Note in figure 20 that bios develops at an orthogonal plane to chaos.

Figure 20: Phase space portrait of the time series generated by the kinetic process equation.

Pointing to numerical counterparts of opposition, 2^N and À^N repeatedly appear as significant outcomes and gains --2 is the simplest opposition while 2p represents a full circle of opposites. At g = 2, the time series splits into period 2; at g = 2², period 12 develops; at g = 2³, there is infinitation. Prominent periods are 6 = 2*3 and 4 during chaos, and period 2 and period 4 bioperiodicities during bios. À is the initial fixed point for initial values between 0 and 2À; even multiples of À separate basins of initial values that converge to odd multiples of À, and unifurcation occurs when A_t= 1.5 À and 0.5 À. Bios emerges when A_tranges from 0 to 2p. As gain, À produces unifurcation; odd multiples of À produce period 2, and even multiples of À produce infinitation. These results point to an intriguing relation between unifurcation and bioperiodicity, and infinitation and convergence (which looks like an infinitation for vary small values of gain). Two abstracts the pairing of opposites; p describes the numerical relation between a pair of diametric opposites and the circumference formed by an infinite number of oppositions. We are tempted by no lesser men than Pythagoras, Galileo, Gödel and Jung to think that numbers represent form, just as they represent quantity and order.

When the previous action A_tserves as the feedback gain (figure 21), the time series generated by bipolar feedback evolves directly from a steady state to chaos. This illustrates a road to chaos other than bifurcation. Also, chaos does not develop when the cosine function replaces the sine in the feedback term. This is one example of the many asymmetries between these complementary opposites.

Figure 21: Process equations computed with gain = A_t. For the sine equation, chaos develops for almost all initial values, except for 1.5 p, 3.5p, etc. For the cosine equation, the series converges to 0.5 p for small initial values, and infinitates at critical initial values (e.g. between 4.9 and 6.9).

Symmetry and asymmetry

Opposition represents a measure of symmetry, but it does not exclude a degree of asymmetry. To investigate the dependence of creative processes on the relative magnitude of opposites, the process equation is modified to At+1 = At + g * (q + sinAt), where q allows one to introduce asymmetry in the bipolar feedback. Since the sine function varies between 1 and -1, total asymmetry obtains when q = 1 or q = -1. At high gain, biotic patterns occur when opposites are symmetric (q = 0), and steady states or divergence obtain when opposites are absolutely asymmetric (q = 1 or -1). Between these two extremes the biotic pattern acquires a trend, merging seamlessly with divergence. We call these trended biotic patterns “parabiotic” (figure 22). A small asymmetry reduces slightly the gain required to generate bios. In the same manner, at low gains, small asymmetry increases and large asymmetry decreases the complexity of pattern (figure 23). Thus, complexity increases with the intensity and symmetry of opposites (see figure 6), albeit there is nonlinearity (small asymmetry and moderate intensity are most creative). Kinetic symmetry is more creative than dynamic symmetry. Alternating a positive and a negative asymmetry (e.g. q is positive for odd iterations and negative for even iterations) generates complex patterns at lower gain than the symmetric recursion with q = 0 (figure 24).

Figure 22: Biotic and parabiotic patterns generated by the process equation with gain 4.61.

Figure 23: Time series generated by the process equation at gain 2.8 (left) and 3.2 (right). In the symmetric case (extreme left, q = 0), period two is generated. As asymmetry increases (from 0 at the left to 1 at the right) we observe an increase and then a decrease in the complexity of pattern.

Figure 24; Kinetic asymmetry is most creative. The entire time series generated by the process equation is here represented using a modified “logarithmic” scale. On top, the standard time series with q = 0. Bottom: the time series computed with q = 0.1 for odd iterations and –0.1 for even iterations. Note the development of period 2, chaos, bios, infinitation, and new bios, at much lower gain.

As biotic patterns obtain only when opposites are symmetric, the fact that cardiac beat series show a biotic pattern suggests that the opposite actions of accelerating and decelerating nerves and hormones that regulate cardiac intervals must then be largely symmetric --this is to be expected, as marked cardiac acceleration or deceleration can be lethal. Some economic processes show a parabiotic pattern that combines biological-like creativity with an asymmetric trend. Extreme trend leads to infinitation, a catastrophic acceleration that provides a mathematical metaphor for economic crises.

Delay and trifurcation

Figure 25: Top: Process equation computed with delay 1, 4, and 5, and gain 1. Bottom: Time series generated with increasing values of gain g_t. The horizontal axis also corresponds to time. The biotic phase (not shown) is similar to that generated by the kinetic process equation.

Trifurcation and period 6 occur for feedback processes involving three terms of the series, such as with delay or with iteration of the difference between consecutive terms. The introduction of delay in the process equation generates period 6 (delay 1), a braid (delay 4), and bios (delay > 4) (figure 25 top). Note that there is no chaos. The kinetic process equation with delay (figure 25 bottom)

A_t+1 = A_t + g_t * sinA_t-1

generates an initial steady state (p) that trifurcates (central value = p). A trifurcation is a bifurcation that conserves the initial fixed point. The trifurcation leads to chaos, and biotic patterns that resemble those observed in heartbeat interval series better than those generated by the process equation [Sabelli, 1999]. A prominent period 2 starts at g = p, separating two biotic phases.

Figure 26: Trifurcating equation showing an initial braid with a wall, a branching, chaos, and eventually leading to a biotic pattern that overlaps with period 6 (“colored bios”).

In the trifurcation equation (figure 26)

A_t+1 = A_t - A_t-1+ [k * t * sin(A_t)]

six strands of the trifurcation) form a beautiful braid that shows many complexities, including chaos, expansions, walls (sudden interruption of part of the braid), and multifurcations. This braid eventually expands into 6 biotic strands symmetrically distributed around 0, that approach and separate in a rhythmic fashion. There is no initial steady state. Braiding knots periodic patterns around each other. While a bifurcation cascade separates, a braid unites.

Logistic development

Computing the well-known logistic equation with an ever-increasing gain generates a time series that resembles the bifurcation diagram of the static logistic map, but lacking an initial fixed point (figure 27). Instead the time series grows exponentially up to the point of bifurcation. Logistic development has a great generality, and can be produced in several ways with bipolar feedback (figure 27), but in these cases the time series converge to an attractor rather than blow up. There are logistic-like development within the time series generated by the process equation (figure 15).

Figure 27: Logistic developments generated by the logistic equation computed with an increasing gain (top); a minimum gain of 0.45 is required. Logistic developments are also generated by bipolar feedback (lower three panels); in these cases p is the initial fixed point, and the series converge to 0.0002 in one case and to 3 p in the two other, rather than exploding.

The logistic equation requires initial values between 0 and 1, gains between 0.45 and 4, and flights into infinity after chaos. In contrast, bipolar feedback is very flexible, allowing any value or sign for the initial value and the gain parameter, and generates biotic patterns beyond chaos. A_t is the conserved action, g is the energy, and the coefficient of g is the information or opposition. In the logistic equation, the opposites A_t and 1- A_t are inversely proportional, both ranging from 0 to 1. The logistic development is thus arrested at chaos. In the process equation, sin A_t is not inversely proportional to A_t; it is positive sine for A_t less than p and negative for A_t larger than p. When either A_t or 1- A_t are added as coefficients of the gain in the process equation, the development also is arrested in the chaotic phase and converges to a point attractor.

As in the case of the process equation, computations of the logistic equation that involve delay (figure 28) or the difference between successive terms (figure 29) generate trifurcations. These two equations do have an initial fixed point, and converge to attractors. .

Figure 28: Logistic equation with delay.

Figure 29: Logistic-like equation in which the difference between consecutive terms is computed.

Combining different types of feedback

The coexistence of several types of feedback that as may be expected to occur in actual processes. The bipolar logistic equation (figure 30)

A_t+1 = A_t + [A_t * k * t * sin(A_t)]

generates a sequence of logistic like developments separated by shifts –it generates multiple “levels of organization”. For initial values between 0 and 2À, the recursion converges to À, and then bifurcates repeatedly and develops a chaotic pattern interspersed with periodicities. This sequence of patterns is similar to logistic development, but as the gain is increased, the series converges asymptotically to a point attractor instead of exploding. Whereas the logistic equation requires initial values between 0 and 1, the bipolar logistic allows any initial value. With low initial values, the series converges towards, but never reaches, zero. Initial values between even multiples of À converge to the included odd multiple of À, then generate the same sequence of patterns as the logistic equation, and terminate by converging to the next lower odd multiple of À. In this way one obtains a cascade of logistic-like developments (steady state, bifurcation cascade, chaos) that we shall call “logons”. If the initial value is sufficiently high, there is an initial biotic phase.

Figure 30: A bipolar logistic equation.

Other creative patterns

Figure 31: Left Firing pattern. Right: overlap of firing and bios.

In addition to bios, bipolar feedback generates other patterns that display complexes, novelty, diversification, nonrandom complexity, and complex embedding plots. One of them, that we call “firing” (figure 31, left)by analogy to the firing of impulses by nerve cells, can be generated in many manners. A process equation in which the gain varies rapidly in a sinusoidal fashion illustrates that bios and firing can overlap in the same series (figure 31 right).

Mutual feedback and system synthesis

Figure 32: Mutual feedback between process equations. Top panels computed with slow increase in gain, lower panels computed with faster gain increase. Black points represent values and white lines represent transitions from consecutive values. Note the overlap of period 2 with a periodic (top) or a biotic (bottom) pattern in the A_tseries..

In nature, feedback is not only bipolar but also mutual. Within a system, components interact repeatedly, so the output of each is in part a response to the other. As natural entities never are isolated, mutual feedback is not a special, particular case. Mutual feedback also occurs among systems. Organisms not only adapt to the environment, but also create it, as illustrated by the generation and maintenance of atmospheric oxygen since the Precambrian. A process and its milieu are complementary opposites. Circular, mutual, or dialectic causation, is much more common than one-sided causation. Mathematical instantiations of mutual feedback generate enormously rich patterns, many of which show organic form (figure 32) that in some recursions repeat in a fractal fashion. Figure 32 presents an example of generation of organic forms in two dimensions. Mutual feedback may be expected to involve greater delay for the input received from the other process than for the changes induced by each process upon itself. Such delay introduces further complexities (figure 33).

Figure 33: Mutual feedback with delay. Top panel shows the complex braid generated by one time series A_t, and the simpler increase in the other series B_tuntil it reaches the biotic phase. The lower diagram shows A_t in greater detail. Note that its initial pattern combines chaos with period 8.

Sequential organization

Sequences of processes may also be expected to be common in systems. Sequences are creative processes that can generate complex patterns from simple origins, as illustrated by the cascade of equations in figure 34. The output of one feedback system generates the gain for the following system. In contrast, a circuit involving a number of feedback processes in which the gain of the first one is determined by the output of a later member in the system such as A_t+1= A_t + Et *sin(A_t), B_t+1= B_t +A_t *sin(B_t), C_t+1= C_t +B_t*sin(C_t), D_t+1= D_t +C_t *sin(D_t), and E_t+1= E_t +D_t*sin(E_t), generates a stable chaotic attractor.

Figure 34: A cascade of process equations: A_t+1= A_t + Et *sin(A_t), B_t+1= B_t +A_t *sin(B_t), C_t+1= C_t +B_t*sin(C_t), and D_t+1= D_t +C_t *sin(D_t). Note firing pattern in B_t

Arithmetic recursions

The models discussed above demonstrate at least three different roads to chaos and bios: growth, bifurcation, and trifurcation. Growth, bifurcation and trifurcation can be generated simple arithmetic recursions. The simplicity of arithmetic points to its fundamental nature. Gödel’s theorem implies that no simpler logic can account for arithmetic. The physical world is constituted by quantities (energy, time, information, matter). Action is a linear quantity; opposition is a two-dimensional pairing; organization is a topological form in a space of 3 or more dimensions (e.g. sphere, torus, etc).

Quantities add and subtract. The recursion of these simple operations generates growth, bifurcation, and trifurcation. Adding recursively consecutive members of a series A_t+1 = A_{t-1 +}A_t generates the well known Fibonacci series, that grows exponentially, and also generates spiral forms. Whereas addition is commutative i.e. symmetric, subtraction displays the asymmetry of order: A - B is not equal to B – A, and hence there are two different recursions. The series generated by the recursion A_t+1 =A_t-1 - A_t grows exponentially in both the positive and negative direction --difference generates opposition. This bipolar growth models, exemplifies and perhaps even explains enantiodromia, meaning “race of opposites”, postulated by process theorists since Heraclitus, as a fundamental form of nature.

Figure 35:

The opposite equation At₊₁ = A_t - A_t-1 generates a period 6 composed of three pairs of opposite (positive and negative) numbers (figure 35). The algebraic structure of this period 6 resembles the relation between primary and secondary visual colors: there are three primary colors, such that the sum of two complementary opposites produces an identity element, and the sum of two primaries equals the complement of the third (figure 36). For instance, the complementary opposite of red is green, which is formed by mixing blue and yellow.

Figure 36: The cube of primary and secondary colors presents an interesting algebraic structure that is reproduced by other natural processes and by arithmetic recursions.

The importance of this logical structure is highlighted by the fact that classes and combinations of quarks can be described by analogy to colors (quantum chromodynamics). A similar triadic organization also applies to quark’s flavor. This similarity among processes at different levels of complexity may represent a homology, i.e. the result of a common origin in a natural pattern of organization, i.e an archetype [Sabelli et al, 1997]. Visual colors arise in the retina as a three-way split of the continuum of light wave frequencies.[1] This trifurcation constitutes from physical oneness to biological diversity, and may in fact expose a fundamental pattern of creative processes [Sabelli, 1989; Sabelli et al, 1997]. Triadic organization is fundamental because Sarkovskii [1964] has demonstrated as a theorem that in the sequence 3< 5< 7 < ¥ …3*2< 5*2< 7*2 < ¥… 3*2²< 5*2²< 7*2² < ¥ … 3*2³< 5*2³< 7*2³ < ¥ …3*2ⁿ< 5*2ⁿ< 7*2ⁿ < ¥ …2ⁿ< 2³< 2²<2<1, every period implies all that follow it. Thus period 3 implies all other periodicities, chaos, and infinitations. This is probably because period 3 represents the end of a causal chain that starts with a single state and a cascade of bifurcations. Period 3 is prominent in the logistic equation –in contrast to the extremely narrow window in which it appears in the process equation. Period 6 is prominent during the chaotic phase of the process equations, and overlaps with bios in the trifurcating equation described above. Trifurcations also obtain with variants of the process and the logistic equations that introduce delay and/or that compute the difference between successive terms (figures 25, 26, 28, 29). The period 6 generated by the simple equation At₊₁ = A_t - A_t-1 indicates how opposition (represented by subtraction) is sufficient to generate color-like triads.

Colors also form cyclic forms, from the period 6 of the arithmetic recursion to the color wheel in which all the colors of the spectrum merge seamlessly with each other. This points again to the union of opposites by rotation, which we have first encountered in complement plots and Mandala archetypes. Combining addition and subtraction

S_t+1 = S_t + k * C_t

and C_t+1 = C_t – k * S_t

generates a circle or an expanding spiral (figure 37), forms that, together with the linear addition implicit in action, generates the bipolar feedback of the process equation.

Figure 37: Addition and subtraction generate the sinusoidal forms used by the process equation to embody bipolar feedback at a constant or increasing gain.

In summary, these mathematical experiments lead us to the following conclusions and hypotheses:

Patterns resembling natural processes can be deterministically generated by bipolar feedback.
Recursions involving differences, the physical substrate of information, generate trifurcations and color-like structure.
Systems of mutual feedback generate organic forms.
Fundamental features of these processes can be generated by simple arithmetic relations. This is consistent with the fact that energy, time, etc have a quantitative aspect.

[1] This is not intended as a theory of color vision. The term color is used in physics to refer to a property of quarks. In the same manner, the term color is used here in an abstract sense, as a metaphor to indicate a type of organization.

Patterns generated