Attention’s Fractal Patterns
Seeing the beautiful vegetal patterns generated by Mandelbrot’s famous equation, many have made the claim that fractals reveal the secret of Life in terms of self-replicating patterns. I agree. However, there are also significant differences between the two systems, i.e. mathematical and living, which should be acknowledged — not ignored.
Life has her own mathematics, which is based in the Living Algorithm. Both Mandelbrot’s Fractal Equation and Life’s Living Algorithm are Reflexive Equations, i.e. based in feedback loops. Due to this affinity, they share a similar logic. We have shown how this so-called Fractal Logic is more appropriate for living systems than is the absolutist set-based logic of Matter. In particular, fractalized boundary lines disable the absolute truths that many attribute to verbal constructs.
However, always another ‘however’, we must remember that the relationship between living systems and fractals is metaphorical in nature, as are all abstractions. As such the fit between model and reality is not perfect, never is. Let’s examine the mathematics of the two equations to see where the fractal metaphor breaks down with regards Life. In so doing, we will discover where the fractal metaphor falls short when it comes to living systems.
Let us compare and contrast Mandelbrot’s Fractal Equation with Attention’s Living Algorithm. Although they are both Reflexive (recursive) Equations, the first is deterministic and static, while the second is open and dynamic. Quite a difference! Let’s check out the details and the significance.
Mandelbrot’s Reflexive Equations (the ones that generated the fractal patterns) are closed — no external input. Despite being closed to the outside, small differences in the initial conditions, i.e. the constant, can still yield diametrically opposed results — limit or no limit — the equivalent of distinct destination or lost — directed or aimless. No precise lines identify which points are inside and outside of the destination set.
Despite the complexity of the results (no demarcation lines), Mandelbrot’s fractal equations are still deterministic. The initial conditions always yield the exact same result. With no external input, there is no variation in the output.
Even more complex is the reflexive equation that Attention employs to interact with data streams — the Living Algorithm (LA). While Mandelbrot’s fractal equations are closed, the LA is open to external input. With each iteration of the LA’s computational process, new information enters the system. As an open, reflexive equation, the LA is not deterministic. Due to this freedom, the LA’s mathematical system is qualitatively different from the absolute determinism of Material Equations or Mandelbrot’s Fractal Equations.
Material Equations & Fractal Equations = Deterministic
Living Algorithm = Not Deterministic
Yet the LA defines the Living Realm of Attention. What does this freedom mean? If the mathematics can’t predict results, what’s the point? Not a specialist in material events, the LA instead identifies processes and their features. This is appropriate as even Life’s identity is based in process, rather than content, as a subsequent section illustrates.
Rather than illuminating a process, Mandelbrot’s equation reveals the results of an event, i.e. which constant is chosen for the iterative sequence. On the most basic level, the result is binary — limit or no limit (determinate or indeterminate). On a secondary level, the answer can also have a speed, the speed at which it approaches the limit. Some sequences are faster and others are slower. This variation accounts for the beautiful colors that are associated with the fractal patterns.
Gorgeous but so what? Not sure. I don’t know if anyone has uncovered any real utility to these static fractal patterns. Certainly don’t hear much about it in the literature.
In contrast, the Living Algorithm reveals two fractal patterns that dominate the rhythms of Attention. Rather than revealing static, deterministic results, as do the other two functions, the LA’s computational iterations identify two dynamic processes that are replicated on the very small time frames, e.g. the blinking of an eye, to the very large, e.g. an entire lifetime of mastery — one’s life work or even the Pulse of a Chinese dynasty.
The Pulse is one of the fractal patterns that we are referencing. Reflecting our Attention span, the Pulse is everywhere that there are living systems. It has some distinct characteristics that shape the form of our awareness, e.g. a distinct beginning and end; and a peak of awareness that is negatively impacted by interruptions — disproportionately so.
The other mathematical pattern that replicates on the large and small levels is what we have chosen to call the Triple Pulse. This fractal process is related to our need for sleep, rest and variation in a theme.
Natural selection has chosen to take advantage of both of these mathematical processes. Appropriate biological systems evolved to exploit the potentials of the LA’s mathematical system, i.e. Data Stream Dynamics (DSD). For example, both Posner’s Attention Model and Dement’s Opponent Process Model for the biology of sleep have a strong metaphoric relationship with the mathematical processes of DSD. Both of these models are also intimately related to a complex of biological systems in many species.