Fractals

P: I am a fan of the Star Wars movies. Have you seen those?

M: Yes I have, what about them?

P: Well, the computer graphics in some of those movies are done with fractals.


Fractal Animation with Terragen M: That’s amazing! Are video games also done that way?

P: Pretty much. Fractals are patterns emerged from recurring calculations. First you use a routine to calculate some outputs based on an input, then you put the outputs back to the same routine as inputs to get further outputs, and keep going. Do this many times and fractal patterns will appear.

M: Ach so, calculations done in a feedback loop fashion again. Show me some examples.

P: OK. This one is called the Mandelbrot set. It uses a formula z_n+1 = z_n² + c, where c is a complex constant and z is a complex number. If you set z₀ = 0 and c = 0.2 + 0.4i, then z₁ = 0² + (0.2+0.4i) = 0.2 + 0.4i. And z₂ = (0.2 + 0.4i)² + (0.2+0.4i) = 0.08 + 0.56i. And z₃ = (0.08+0.56i)² + (0.2+0.4i) = -0.1072 + 0.4896i, and so on. Repeat this for about 300 times and plot these z numbers on the x-y axis and you will have a Mandelbrot fractal.


Mandelbrot Set z = z² + c. M: Such fractals look fascinating, but they don’t seem to be like the realistic rendering in actual movies.

P: That is because another ingredient is needed in the recursive recipe. It’s called the Stochastic Process, which means some random probabilities are assigned to the outputs/inputs. The math is complicated, but the software coding is not. Such Stochastic fractals look uncannily real as natural objects or landscape.


Mandelbrot island by Prokofiev.
M: Even the structure of some neurons can look like fractals!


Purkinje cell by Cajal.
P: And why do you think that is so? My guess is that they both come out of recursive processes.

M: And with Stochastic randomness added in for texture?

P: Yes. Stochastic process can add naturalness into the picture. Take the growth of a plant for example. The recursive part is that a bud can branch out into leaves, the leaves branch out into stems, and the stems branch out into buds again, and the cycle repeats. The Stochastic part is that some stems will grow into shady confined spaces, and some into sunny open places. This development is a probabilistic randomness that affects how the plant will shape up later. The sunny open spaces allow the stems and leaves to grow bigger, whereas the shady confined spaces stunt their growth.


Fractal weeds.
M: I have heard that fractals are related to the Chaos Theory. What is that about?

P: I think the Chaos Theory comes from the unpredictable behaviors of some fractal formulas. A small change in a formula’s constant or variable can result in vastly different output/input values that follow later. There is a Chinese expression for this: 差若毫厘 缪以千里 - separation of a thousand miles originates in a difference of a millimeter. That expression is about paying due diligence at the beginning. If an astronomer misses setting a telescope’s angle by a tiny degree, s/he may be looking at a different star.

Logistic Map, Bifurcation Diagram of x = r x (1 - x).
See the plot of a recursive formula x = r x (1 - x) that maps variations of constant r vs. steady-state values of variable x. At r = 3, a small increase of r creates a fork of 2 final values of x. This happens again when r is near 3.45, and again at 3.55. The compound forkings turn x into myriads of possible subsequent values. This kind of dynamics is called bifurcation (furca = fork). The branches make a prediction of the final value of x basically impossible. It is like trying to predict which way a drop of water will flow at a river's bifurcation junction. The name chaos theory simple means that small changes and bifurcations make things unpredictable.

M: This bears a remarkable resemblance to the diversity trends in natural evolution. There are hundreds of different bird species and leaf shapes. Fractal bifurcation or chaos theory sounds like a very good explanation for that. In cybernetics there are positive feedback loops that can also amplify small differences into runaway divergence, which can also explain biodiversity. Since the two theories share a similar mechanism - fractals with recursive iterations and cybernetics with repetitive feedback loops - so likely their outcomes also share an analogous trend.

P: I like this kind of correlation. There is one interesting study on bifurcation by Ilya Prigogine. He won the 1977 Nobel Chemistry prize for his dissipative structures, described in a layman's book called Order out of Chaos: Man's new dialogue with nature. Something very paradoxical is observed about order and chaos in nature. When water is near its boiling point, it is bifurcating between bubbles and liquids before it turns into steam and escapes. From the outside the boiling water appears chaotic by the raging bubbles. But inside the bubbles, the water molecules are moving oddly in uniform directions. Kind of like soldiers obeying commander’s orders while a battle is raging. On the other hand, when water is in room temperature in a container, it appears very calm and orderly on the outside, with uniform clarity and stillness. But on the inside, the water molecules are zigzagging chaotically in a typical Brownian motion fashion. This is like a chess player studying the pieces on a board. He appears still. But in his head neurons are firing like firecrackers on the 4th of July. Anyway, I think I have some ideas about your cybernetic loops now.

Feedback Loops

Feedback Loop, by GliderMaven
M: A feedback loop is a loop with high rates of transformation in its circulating matters. Some loops are not feedback loops because their velocities of transformation are slow. That is due to the presence of circulant reservoirs that reduce change. The water cycle is an example of non-feedback loop. It has oceans, lakes, and polar ice caps acting as reservoirs. Maybe less than 10% of those reservoirs undergo transformation in a thousand years via evaporation or melting. A feedback loop like the human life cycle has a much higher rate of transformations, starting from the cells of an egg and a sperm. In the movie Little Buddha, the Tibetan Buddhist teacher Lama Dorje says to the little boy Jesse that in 100 years everybody will be dead - that is impermanence. And even if some people do live longer than 100 years, the human life cycle still goes through 100% transformation in, say, 200 years.

P: That is a sobering thought. I wonder that since the water reservoirs will also transform 100% eventually, what difference does it make whether it takes 200 years or 200,000 years to complete a transformation? Why name a loop specially as feedback loop?

M: The distinction has one advantage. We can see evolution taking place much quicker in feedback loops, since evolution is about patterns of change. For example, diversity and specialization are two hallmarks of evolution. Why are there diversity like the hundreds of fish species, or specialization like the tail that they all have for swimming? These two hallmarks can be explained in terms of feedback and transformation when we get to the point of cybernetic loops, or feedback loops with governor and sensor.


Evolution of man and technology.
P: Speaking of feedback loops, I just remember the Fibonacci numbers and fractals in mathematics. How about the rock cycle and oxygen cycle? Are they feedback loops in your terminology?

M: Rock cycles are not feedback loops because the Earth’s crust is a gigantic reservoir. The oxygen cycle may be either way. I don’t know what percentage of oxygen in the atmosphere turns into carbon dioxide when living creatures breathe, or the percentage of carbon dioxide changing back to oxygen when plants photosynthesize. If the percentage is small then it is not a feedback loop. If large, then it is. When I look at the changes occurring in humanity, I see not only physical transformations, but also mental and social transformations like beliefs, feelings, wishes, relationships. They all change so very much over time. Perhaps language and culture and genes are the reservoirs in the human’s life journey. I don’t know. I am just thinking out loud. Anyway, why do feedback loops remind you of Fibonacci numbers and fractals?

P: I just read about the Fibonacci sequence and fractals the other day that they are closely related to natural growth patterns and formations. The thought of them comes up because in computer programming there are iterative loops and recursive procedures. These programs can generate Fibonacci numbers and fractal patterns in a nice, elegant way.


Goldener Schnitt Fichtenzapfen by Wolfgangbeyer at de.wikipedia
M: Iterative loops and recursive procedures? I do not speak computer lingoes. I use computers only to read emails or surf the web. That’s all. The computers are often buggy and I hate that.

P: Ah yes, computer problems. Who does not have them? Anyway, the Fibonacci numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21,... follow a simple rule. Each number is the sum of the two previous numbers. 5 = 2 + 3,  8 = 3 + 5, and so on. Because of this rule a repeat or iterative loop can reproduce them easily. A recursive function can also do it and that is a hacker’s art. The feedback part is that when a number is generated, the output is fed back as an input to generate the next number.


Fibonacci Blocks
M: Neat. Like you say, the sequence of Fibonacci numbers corresponds to many natural growth patterns. What’s more, someone has noticed that the ratio of two adjacent numbers later in the sequence becomes close to 1.618. That number is called the golden ratio, and it has an important implication about natural growth.

P: Which is what, beauty? I have seen the ruin of a Greek Parthenon that was said to be built with the golden ratio somewhere. It does look grand and nice.

M: The golden ratio dictates that a population growth, following the Fibonacci sequence, is geometric in nature. The numbers in each generation is about 1.6 times of the preceding one, making the population increase ever more sharply rising. Geometric growth is what we have in the world’s human population today, which some call the population explosion. What is not said is that cancer, the abnormal growth of body tissues, is a disease where some cells are also growing geometrically in numbers.


Normal and Cancer Cell Divisions by Dorineau.
P: Huh? Are you saying cancer is natural like the Fibonacci sequence, or that human population growth is like a cancer? No, cancer is not natural. But it is kind of true the way you put it. That is strange. Populations of natural species do not grow geometrically in this world. I know rabbits can breed like crazy. But you don’t see rabbits everywhere in ever increasing numbers. Predators like coyotes and bobcats eat them.

M: That’s right. That is an example of cybernetic loops in nature that regulate population growths. Although geometric growth is natural, and cancer is that, we don’t see cancerous growth everywhere precisely because there are cybernetic loops that cut down population explosion into population balance, or ecological balance. But back to cancer. I mean to say that we need to look at cancer and its treatment differently.

P: Differently how?

M: The common notion of cancer is that it is caused by carcinogens or genetic defects. Somehow these factors make normal cells multiply like rabbits and become lumpy tumors. I suspect that normal cells can multiply like rabbits by themselves without the help of carcinogens. What the carcinogens are doing is probably damaging the regulatory mechanism of the cybernetic circuits the cells are in, resulting in some growth to become uninhibited and run away like a freight train. I have heard some cancer patients choose to forgo chemotherapy or radiation therapy that emphasize on killing the cancer cells. Instead they turn to organic whole-foods lifestyle. Some of them report that they have gotten rid of cancer successfully. I guess it is because the organic whole foods nourish the body parts damaged by carcinogens and bring them back to health. Then the balance of cybernetic regulations is restored, and cancerous growth is under control and become normal cells once again.

P: That sounds like a possibility. But are we still talking about the mind or memory?

M: Well, kind of. We are building up some contexts for talking about memory, kind of like laying out perspective lines for a drawing. Will you continue about fractals? I want to know how they relate to feedback and nature?


Fractal Broccoli by Jon Sullivan - http://pdphoto.org/PictureDetail.php?mat=pdef&pg=8232

Loops

When they meet again Pedro does not tell Mary his latest dream, for fear that it may ruin their friendship. Mary is excited to talk about her idea of memory in terms of feedback loops so she has not noticed Pedro's dreamy look.


Racetrack Playa, Where Rocks Are Moved By The Wind, by Jon Sullivan, PD Photo.org
Mary: OK, loops first. You know that on Earth everything moves. Something like a mountain may look stationary, but in time it will move to somewhere else and become something else. All things move. When they move back to where they started from, then that is a loop. In nature we can find abundant examples of loops, cycles, or circuits.

Pedro: Alright, I can imagine that everything moves. But how do you know the movements are always cyclic? Can't they be random?

M: Maybe, but let's focus on the cyclic ones first, for they account for the majority of natural events. The water cycle has been taught in science class for centuries. That's a classic example of H₂O molecules moving and transforming in a looping pathway. Actually, water pathways are complex, running in small loops within a larger loop. But they recycle nevertheless, with the help of solar energy and gravity.


Water Cycle, By William Waterway
P: Come to think of it, I have heard of rock cycle, oxygen cycle, life cycle, and whole bunch of other cycles. Can there be a thought cycle, like an idea going from one mind to another and then coming back? Wait, are you saying that memory is like a recycling of… of... sensations in the brain?


Oxygen Cycle
M: There's more to memories than just recycling of sensations or thoughts. But looping pathways have something to do with memory, our memory anyway, not the computer memory. In the brain, there are billions of neurons connected to each other. Neural impulses travel from neurons to neurons. Where do they end up? My guess is that they don't stop in some brain regions and rest forever. There are looping pathways in the neural networks for these impulses to go around and around.

P: If that were so, wouldn't the same memory be recurring forever? Nobody's memory is like that, except maybe some people with dementia who keep saying the same things over and over. Oh, I mean fanatics, not dementia patients.

M: Little children often do the same thing repeatedly. They pick up something and put it in the mouth as if they remember to do so. Adults have their routines and habits. People with dementia may ask where their socks are every 20 minutes. Of course, compulsive behaviors of obsessive people are noted for their repeating patterns. Dieters are caught in the cycles of fasting and binging. Alcoholics or smokers need their fixes like a clockwork. The serious cases are the ones who want to quit, to leave the cycles, but cannot. Anyway, memory is not as simple as that because natural loops are more than cyclic movements. It is that and transformation along the way.

P: Transformations in natural loops? Do you have some examples?

M: Well, in water cycles the water can become ice, or vapor, or fog, or snow, and back to water. Or it can combine with something else and becomes bodily fluids when it is absorbed by living creatures. It can be sap, juice, sweat, odor, blood, pee, mucus. Once transformed, the shape, size, boundary, behavior, and name of the object are all changed. In modern language we talk about these watery things as if they are unique and separate from others. But in the context of loops, they are transformations of a single circulating object.

P: Yes, I see now that in rock cycles it is the same. A rock can become rock, sand, molten lava, dust, gemstone, ore, soil, or even minerals that are taken as vitamins by people. It stays or circulates in the body for a while. When the body dies it returns to earth and continues on in the rock cycle.


Rock Cycle, By ZeWrestler
M: Although there are rock cycles, it is hard to notice transformations of rocks. They seem to be the same rocks every time we look at them. That is because they take so much longer to change. In life cycles it takes less time. Still, many of us think that the notion of self, the I, is something static and separate from others. We do not see the I change as transformations along a life cycle. That’s due partly to how we use our language, and partly to observations from our own perspectives. Take the question 'who am I' for example. If we ask someone who is 7 years old, 14 years old, 21, 28, 35, 42..., who is happy, sad, living in a big family, in a single-parent family, with or without children, parents, spouse, jobs, teacher, friends, enemy, idols, master, followers, then the different answers will show a pattern of transformation. As a person goes through different stages of life, his body and his notion of self changes. His views of the world, his interests, his sufferings, all change. In the Buddhist language this is impermanence. But I prefer the notion of transformations along the pathways of a complex loop.


Das Stufenalter der Frau by F. Leiber.
P: Yeah? What is the difference?

M: Well, with terminology like loops we can explain the expression ‘the more things change, the more they stay the same’. How can something be the same if it keeps changing? In loops it can. It is moving through the same loops while transforming inside the loops. If we take this a step further, we should update our language to reflect this. Change all nouns from a static name to a rolling movie. 'I' for instance. Change I to something like I-life-cycle or I-life-story or the-Watsons-life-loops or the-Shakyamuni-clans-cycles. We can spin it with Latin or Greek to lend it an air of authority. How about Mary-Cyclus or Pedro-Kyklos?

P: Okay, okay, that is a little too far fetched for me. Maybe because the Earth is spinning and revolving around the Sun day in and day out, and the Moon is orbiting around the Earth, that everything in nature goes around in loops. What about the feedback loops that you mentioned? Is there something else?

Anxiety

Pedro has been thinking about Mary since he finds out she is single. That night he has a vivid dream. Waking up the next morning he writes it down hastily while it is still fresh in memory. Later during the day he notices that the dream is full of anxiety. Has the anxiety caused stiff muscles in his neck and shoulders? Or is it the other way around, or both? He reads his dream story again...

I am in an open space with a group of people. We are supposed to pair off into couples and compete in the game of Amazing Race. Right away I look for a woman partner who is physically fit. But they are choosing others who are as athletic as themselves. I walk around and see a small woman alone. She comes over to me and asks me if I know what the word two-moon means in Chinese. I say it means friend. She laughs, and I grab her hand to join the others in the race. Many people are there before but now I see no one. The place is empty except for a few deer foraging in the distant. My woman partner is very tall now and have big breasts. She looks down at me and I let go of her hand. There’s no one competing in the Amazing Race anymore.


Hashiguchi Goyo - Woman after the Bath (Yokujo no Onna) - Walters 95881
We walk to a hot spring pool where some people are bathing in nude. She takes off her clothes and joins those in the pool. I start to take off the armor suit I wear for the race. After some time I still have a few layers of clothing on me. So I sit down by the pool to find an easier way to do this. A cafe nearby puts out a sign that says lunch is served. Everyone is getting up and walking towards it. I go over there also and look into the windows to see what is on the menu. I see Mary in there cooking. She is talking to several male co-workers and laughing. My heart twinges with insecurity. How beautiful she is yet she doesn't see me. I sit down outside to wait until she is off work so I can talk to her alone.

Then I see Mary walking out of the cafe down toward a creek. I call out her name and she turns around. I ask her where she is going. She says she is going to check out the creek. ‘Do you want to have an adventure with me?’, she asks. I’m happy as a clam hearing that. Then a Buddhist monk comes by and starts talking to her in earnest. I politely stand by to let them finish the conversation. Mary seems to have lost interest in me and keeps on talking with the monk. So I turn away and walk toward the creek by myself. In the shallow creek there are many rocks protruding out. I walk on them from one to another in the middle of the creek, looking at the clear running water and little darting fishes. There's nobody around, but I see in several places signs that someone have been there before. I stop and look at a pile of stacking rocks standing 5 pieces tall vertically…


Reading it Pedro feels none of the anxiety experienced during the dream. But he knows it’s still buried in him somewhere somehow. He wonders how he can melt away this inner tension?

An Outline

During the weeks when Pedro looks into the design of computers, he notices the Internet is similar to interconnected neuron cells of a brain. When Mary brings up definitions of brain and mind he finds them oddly fitting.

Mary: Those definitions come from information I gathered while researching. I have found that Chinese has a rich vocabulary for the mind. For example, thought (思), imagination (想), meaning (意), and memory (念). These 4 characters are of the type huiyi (会意), words that we can infer their meanings by the combination of the radical and the other part.

思 (心+田), si, thinking = mind + farmland. Thinking is like plants/weeds growing/decaying in a farmland. Seeds become plants that reproduce seeds again.

想 (心+相), xiang, imagining = mind + appearance. Imagination is images playing out in the mind.

意 (心+音), yi, meaning = mind + sound. Meaning (or wish) is the inner voice that resonates.

念 (心+今), nian, remembrance = mind + now. Memory is a reflection of past events in the present moment.

There are several other Chinese words like these. But taking them all on at the same time is biting off more than I can chew. So I focus my research on just one word: 念, memory, the precursor of the Western word mind. The other aspects of mind seem to be closely related to memory.


Allegory of Memory. A man writing; representing the faculty of memory. Engraving by T. Jenner (?), c. 1650.
http://wellcomeimages.org/indexplus/image/V0007658.html
Pedro: Neat. So memory is the key. A big part of computer data processing is also memory - storing, indexing, retrieving. Everything about computer memory, both software and hardware, is known. But when it comes to human memory, neuro-scientists and computer scientists are still scratching their heads. It is a big mystery how and why it works.

Mary: Is that right? I sort of expect so.

Pedro: Yeah? Why is that?

Mary: Because everything about computers, memory or otherwise, is designed. It’s created by people and materials that are isolated and arranged in a specific way. Computer scientists in general don’t socialize well. Computer circuit boards similarly will be buggy if you leave them exposed to coffee or cake or pets. The human brain, on the other hand, is evolved, not designed. Evolution comes from interactions. The brain thrives when exposed to external stimuli, and withers if isolated. Such different patterns make me think brain and computer share very little in common.

P: The Internet is like a brain. It’s also evolved, not designed. And the Internet is certainly not like a computer. So when the mind remembers something, what is going on in the brain? Have you found anything that may solve this puzzle?

M: Yes we have. And it has something to do with evolution.

P: We? Who’s we?

M: I mean other people have studied and thought about this before. I have not come up with all the information by myself. I only dig up what they leave behind and put two and two together.

P: Oh, I thought it’s your boyfriend or something.

M (smiling): No. I don’t have a boyfriend.

P (smiling also): I’m single too. Please go on with what you’re gonna say.

M: OK. In the 20th century a key mechanism of evolution process has become largely understood. That mechanism is cybernetics. Cybernetics can be a context to describe human memory. However, biological scientists are not familiar with it yet, even though cybernetics is relevant in almost everywhere in the natural world. This is probably due to history. Alfred Russel Wallace, the lone naturalist who first noticed the relationship between evolution and cybernetics, was eclipsed by Charles Darwin in terms of the popularity of their evolution theories. The word 'cybernetics' comes from Greek κυβερνητική (kybernetike), meaning "governance". It is cybernetics that guides robots to imitate actions of intelligent creatures. However, many people are worried about robotic intelligence. Remember that movie Terminator where Arnold Schwarzenegger plays the villain cyborg? Those robots built by the Cyberdyne Systems Corp. become self-aware and take over control of the world. Well, in reality cybernetics itself is not a sinister science. But playing with matches can be dangerous.

P: Sounds scary. I've seen that movie. But in terms of evolution, do you mean that robots are the next stage of human evolution, like the TV’s Bionic Man? Or was it Bionic Woman?

M: Ah yes, I remember those TV shows with Lee Majors and Lindsay Wagner. I don’t mean robots are the future. No one knows for sure where evolution will take us. The lyrics of the song Que Sera, Sera says, “Whatever will be, will be… The future’s not ours to see…” Anyway, I mean the direction of evolution can be described by cybernetics after the fact . To engineers, cybernetics is a way to control or regulate some behaviors of a system, like the robots at the assembly lines, or cruise missiles that home in on their targets, or pacemakers that help heart beat properly. To scientists, cybernetics is the study of feedback loops that have a sensor and a governor.

P: Words like cybernetics, feedback loops are Greek to me. I’m still not clear what cybernetics is and how it relates to evolution or memory. Can you give some examples?

M: OK. Let's see. I have examples that can be separated into categories: loops, feedback loops, and feedback loops with sensor/governor. Sort of like following evolutionary progression. That should lead us to memory eventually if we don’t get lost.

P: We may get lost, but we may also find something else. Maybe we will find enlightenment, ha! If we find peace and love and happiness along the way that’d be swell, too.