16 Apr 2009

Pythagoras' Natural Computers: The Wild Cosmic Computation of Melodies and Flowers



We previously looked at how we might characterize the formal principles of a wild logic. We ask now what wild computation would be. Does the wilderness compute numbers? If so, by what means? And what properties characterize it?

I would like to stand natural computation against mechanical computation, for comparison. My broader aim is to see how Deleuze's ideas can contribute to artificial intelligence theory.

Recall that a simple program for a Turing Machine would allow us to automatically compute the natural numbers.

Does nature exhibit anything similar? I propose two possibilities: wave motion's harmonic overtones and the golden ratio. The Pythagoreans were involved in discovering both of these phenomena.

The first principle we need to consider is wave superposition. When wave peaks cross, they add their amplitude. [Click on image for enlargement. Image credits provided at the end. Image 1]



[The source of this image also animates the superposition. Another great animation can be found here]

Prof. Walter Lewin of MIT explains this phenomenon.



If the peaks and troughs line-up, the waves add to each other. When they do not, they interfere and subtract from each other. Most times their allignments are irregular, which creates wilder patterns. [2]



Now consider if we have a wave at one frequency. Then we superpose to it a wave at twice the frequency. That means, for every two repetitions of the faster frequency, it lines-up with each single cycle of the larger frequency. The single and doubled frequencies are shown in purple.[3]



The blue wave-form is their synthesis. We see that it too is regular, but a bit more complex. If we were to take a string at one length, and play it with another string of half the length, we would obtain such a harmonic as above. This was Pythagoras' harmonic demonstration. Prof Lewin describes his discovery.



Pythagoras was able to calculate the harmonic ratios this way. His calculator was a musical instrument, the monochord. [4]



The monochord has one string. You change the pitch by moving the bridge mechanism. Wherever you stop the bridge, that terminates the length of the string. Shorter strings vibrate at higher rates. So shorter strings make higher pitches.

The monochord is an analog computer, like a slide rule. The bridge can find itself anywhere along a continuum of points across the string. So theoretically there are an infinity of possible string lengths, and hence an infinity of possible pitches.

Now, when two strings are of the same length, they produce tones of the same frequency. So if you play them together, the waves superpose and reinforce each other. Pythagoras showed this by playing two monochords with the strings fully open. If we gradually move the bridge toward the end, we can hear the two wave patterns slowly come to match each other. [The video below synthesizes pure sine waves].



The image on the screen displays the waves' synthesized form. You see that at first there is no pattern. Then they reinforce each other, but because they are off, they cancel at regular intervals, called beats. Finally they come together, and there is one solid wave form.

Now we will move the bridge toward the center of one monochord, while letting the other one continue playing at its full length. We want to see where we obtain another strong point of reinforcement. As we near the middle, we will experience that 'beating' phenomenon, but it will not be so pronounced.



Because the smaller string is half the larger one, it makes two waves by the time the larger string finishes one. This causes the waves to reinforce each other once every repetition of the larger cycle, as we saw in this image. This is the octave.



What we find is that the next strongest reinforcement is at one third the length, the next at one fourth the length, the following one at one fifth, and so on. Let's listen to the next three harmonies. The first will occur when the smaller string is at one third the longer one's length. The following one is at one fourth. And the final one is at one fifth.



We find that these harmonic ratios continue along the natural number series. [Click on image for enlargement. Image 5]



Here is a colored version of the image often associated with Nicomachus the Pythagorean's Manual of Harmony. It displays the ratio divisions along the monochord's string. [For more on the harmonic divisions, see Friedrich Nietzsche's lecture on the Pythagoreans] [6]



There is a legend that Pythagoras discovered these ratios first by hearing metal-smith hammers clanging on anvils. He noted the harmonic relations of the different sized hammers. Then he went to a "canon" instrument with strings whose tension was varied by means of different sized weights. [7], [8]





Each successive harmony on the string is another natural number. Hence wave harmonics compute the natural numbers. If we have a vibrating string, we do not need to measure where its midpoint is using a ruler. We can find it by moving the bridge until we reach the octave. The string thereby computes "1 + 1 = 2". The sound harmonics themselves are a precise measuring tool and computation device, sort of like a slide rule.

Although Pythagoras may not have been aware, the vibration of the one whole string already contains in it smaller wave-forms following the natural numbers. Waves are fractals. There is no pure sine wave in nature. Whenever there is a wave-form of a certain frequency, it is a composite wave that is made-up of smaller waves at higher frequencies, but at lower amplitudes. When we see a string vibrating, we see something like this [9]:



It is a blur, but with noticeable regularities in it.

We know that the string moves up-and-down like a jump-rope [10]:



This makes the basic frequency. But traveling within this wave are smaller waves, and they follow the natural number series [11].



And again, all together they make a form that looks something like this, over a period of time:



[See this site for an animation that shows how the wave-forms synthesize on the string.] Here Prof. Lewin explains the discrete values of the "natural frequencies." The subdivisions within a string "go on to infinity."



The reason we note these constituent overtones is because in this way, the string does not just compute the natural numbers, it visually displays them as well, just like slide-rules, abacuses, mechanical computers and electronic ones too. And because each sub-wave is a discretely different numerical value, 1, 2, 3, 4 and so on, the string displays its computation in digital. Here he explains how the harmonics go on infinitely. This is important, because that means in a very short amount of time, the string computes all the natural numbers perhaps to infinity, which would be impossible for any classical digital computer.



Now he will explain that the instrument strings do not just play one frequency, but also its higher harmonic frequencies as well.



Here he will use a device to indicate those higher harmonics.


But still we do not know precisely what are the higher overtones and how strong they are. We want a way to determine that in every wave there are certainly many more that follow the natural numbers. This will assure us that in fact waves do compute the natural numbers, and perhaps even infinitely many of them in a small amount of time.

The means for analyzing-out the constituent wave forms is a method called Fourier analysis. I will leave it for mathematicians to explain how you do it. I cannot. But we just need the results. We want to see with our own eyes the countless natural numbers that nature and the cosmos are calculating in so many places at once.

I found the animations and explanation at Peter Ceperley's site to be very helpful for grasping the basics of Fourier analysis. [the other parts of the site are wonderful too.] But we will follow the essential parts of Professor Lewin's lecture so that we can grasp enough to see the constituent waves.

Here he explains that the string's up-and-down motion is also its back-and-forth. This will lead us into his animation.



[See this page by Peter Ceperley for a helpful animation showing this wave phenomenon.]

We will now see how Fourier analysis allows us to determine the series of smaller frequencies and the amplitudes that superpose to make the larger wave form. In this case we begin with a triangle wave and find the curvy sinusoidal waves that make it up. You will see how waves are always complexes of smaller constituent waves, even though their synthesis only indicates them implicitly. The red wave is the actual wave. The blue waves are all the wave motions who synthesize together to make the red wave.



The peak of the red wave does not come to a triangular point. That is because we would need to include the full infinity of constituent blue waves, each one smaller then the prior, that together will fill-out the full wave-form.

Now Lewin will explain the Fourier analysis for a sound wave that we might find in the air, and not just as a mathematical abstraction. The analysis will show how the wave contains not only its full frequency, but also the first constituent harmonic (at double the frequency), the second (at triple) and so on.



Here he uses a device that can perform the analysis, although not to a great extent.



Below we can see displayed an analysis for a guitar playing a tone at around 300 hertz. Notice the spiking at 600, 900, 1200, and so on. Each spike is another multiple of the original wave. That means there is the fundamental wave, and moving through it are waves of half the size, a third, and so on. So the one guitar wave computed the natural numbers. We here are able to detect it calculating up to 10, but it is conceivable that the string computes them on to infinity.[12]



We investigate Fourier analysis because it shows that indeed waves are already computational engines. They compute natural numbers. The analysis just pulls out the implicit computations, some of which we can see just by looking at the string's motion. Trained musicians in fact can distinctly hear many of the successive overtones in the series.

We only need small moments of the vibration to deduce the numerical ratios that the wave is computing. The series of component waves is infinite. So in a matter of just moments, a wave computes and displays countless (perhaps infinite) natural numbers. This is something that a digital machine could never accomplish, because it can only complete tasks within finite numbers of steps. Even if it is shown that the string's computations are limited, and that digital computers can do better, nonetheless, it is quite remarkable that nature herself is constantly computing natural numbers to begin with. Now consider also how waves are ubiquitous in nature and the cosmos. Some reduce all things to waves or vibrations. The universe is sublimely great. Every moment countless galaxies are computing numbers. The world around us is a staggeringly sophisticated and extensive computer.



I propose another possibility for natural computation: the golden ratio. We can extend this to other naturally occurring irrational numbers like pi, but I begin with phi, the golden number.

The Pythagoreans are credited with discovering the golden ratio (Livio 35). They found it in the proportions of a pentagram.

I am interested in finding natural things that are like calculators in way that is similar to how an abacus or slide rule is a calculating device. The monochord string was one option. Now I propose a flower or other plant whose growth pattern follows the golden ratio.

Euclid defines the golden ratio in the third definition of the fourth book of his Elements:
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.



We see that we are dealing with an analogy. But normally analogies have four components:
a is to b as c is to d
This also formulates a proportion.

Just a ratio is one value taken in relation to another value:
a/b
A proportion is one ratio taken in relation to another ratio:
a/b = c/d
The golden ratio seems like it can be expressed as a proportion:
a+b / a = a/b
We see that we only have two terms. So on the one hand it is just a ratio; there are only really two terms that are being related. However, they are related in such a way that the relation itself self-relates. What we see is that the microcosm (a/b) is proportional to the macrocosm (a+b / a). The way that the whole relates to its larger part is the same as the way its larger part relates to the smaller one. What interests us here are not the terms, but the relations between them. We obtained this harmony with a middle term, a, the larger part. The larger part acts both as the smaller to one value, and the larger to another. But that relation in both cases is proportional. The golden ratio is magical because it is a self-proportional proportion.

Suppose we just cut the line in half. Then the whole relates to one of its parts as 2:1, but the one part relates to the other part as 1:1. Here the ratio does not find itself within itself. There is only one very precise division that produces the golden ratio. So the golden ratio is not so easy to calculate. It is a very sensitive determination. In fact, it is so sensitive that it cannot be determined precisely, at least with digits. Like all irrational numbers, when we try to display the digits of its decimals, we continually carry and carry to the next lower digit place, never arriving upon the last one.

Certain flowers and plants compute this ratio in their growth pattern. They shoot-out new branches one-by-one as the stalk grows upward. Some plants send out branches at golden ratios. Somehow the plant just naturally computes and displays a number that no digital computer could ever calculate.

If we were to divide a circle's circumference into goldenly divided parts, we would obtain the following proportion. [14]



We can approximate the angle. [15]



Many plants shoot-out new limbs each time at the golden angle. [16]



From 1 to 2 is the golden angle. From 2 to 3 is also the golden angle. 3 to 4 as well, and so on. After a while we obtain an interesting and pleasant formation. [17]



We can see how it calculates the golden ratio and displays it to us. [18]



We can also see that the golden 'phyllotaxis' in this case produces two sets of spirals going opposite directions. [19], [20]





Some other plants display the golden spirals more prominently. [21], [22]





Now, one might be unimpressed: nature only computes and displays the golden ratio in select species of plants. But recall the last time you poured cream into hot coffee. It spiraled. Your coffee displayed the golden ratio. That spiral can be found in hurricanes. In fact, our galaxy is such a spiral. [23]



Mathematician Benoit Mandelbrot is even said to have calculated that all the galaxies in the universe are arranged in such a spiral form. Spirals within spirals within spirals. Calculators upon calculators, all throughout the cosmos.

Someone else might object that in none of these examples does the phenomenon display the ratio precisely, because on some very small level at least, it will be off by a little bit. I respond in two ways.

1) Perhaps if we averaged every 'imprecise' manifestation of the golden ratio throughout the cosmos over the course of its eternity, we would have a precise calculation of the golden ratio.

2) What is remarkable in the very least is how these natural computers are tending to display the ratio precisely, or seemingly trying to. Now also consider the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, and so on. We add the prior number to any given number to obtain the following number. If we make a ratio between any two neighboring numbers, we obtain an approximation for the golden ratio. As the numbers get bigger, the approximation becomes more accurate. Many things in nature proceed according to the Fibonacci pattern. The pattern is tending toward an absolutely precise rendition of the golden ratio, as it goes on to infinity. But that tendency is there from the start. So things in nature that display the Fibonacci sequence and proceed developmentally in accordance with that pattern also exhibit the tendency toward an absolutely precise rendition of the golden ratio.


We examine natural computation to compare it with automated artificial computation using digital mathematics. If at all nature computes, that is remarkable enough, given that we consider computation to be an artificial human invention. But what is more remarkable is that

a) natural computers might be more computationally powerful than artificial digital ones, because natural ones seem to compute infinitely complex numbers in just instants when it would take a digital computer an eternity, and

b) natural computations make-up very much of the dynamics of the cosmos and nature, so much so that it lends evidence to the Pythagorean claim that all is number and Becoming is calculation. Consider also how many things such as bubbles tend toward a spherical form. We also spoke of sinusoidal waves that make-up every actual complex wave in nature. These formations involve circular geometries, which means a bubble for example calculates pi. But pi is also an irrational number that cannot be computed digitally. And yet, so much in the cosmos tends toward spherical shapes.

Lastly, I would like to address the question of whether natural computation is "wild" or not. Wild computations would be ones that follow a deterministic pattern but that are marginally thrown-off their deterministic track by natural (and not mathematical) random interferences. We said that natural computers are precise in their tendencies. In their actualities, they might always be wild. Nature computes irrational numbers. Such numbers cannot be computed and displayed digitally, because there is something about them that always defies determination. In other words, nature is a computer that defies determinism. Nature and the cosmos at heart are wild computers.




Livio, Mario. The Golden Ratio: The Story of Phi, the World's most Astonishing Number. New York: Broadway Books, 2002.

Peter Ceperley's wave site table of contents:

Wonderful animations and explanations at the University of Salford site:

Video from:

Images from:
[1]

[2]

[3]

[4], [5]
Guthrie, Kenneth Sylvan. The Pythagorean sourcebook and library : an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Grand Rapids (Mich.): Phanes, 1987. ISBN: 0-933999-51-8

[6]

[7],

[8]

[9], [10], [11]
Jones, George Thaddeus. Music Theory. New York: Harper & Row, 1974.

[12]

[13]

[14]

[15]

[16]

[17], [19], [20]

[18]

[21]

[22]

[23]


Posted by Corry Shores at 4/16/2009 02:27:00 pm
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