How long is a piece of rope?
Everything has a chaotic element within, it's just for the observer to discover and perhaps use it to advantage. This never occurred to any of us when we were building a lean-to shelter for the kids on a piece of abandoned land. To craft the sides of the structure it first required that a rope be stretched between two trees, then successive layers of branches and long canes would then be leaned onto this support to provide the classic structure. However, there was one problem, the two trees that were most favored were just a bit too far apart to permit tying between them the single piece of rope that we carried. How to accomplish the impossible?
While others were searching for another pair of trees or looking for a piece of rope that could be joined to the one we had, Don came up with a solution that would have made Beloit Mandelbrot(1) proud. Seeing the rope not as a single length of twisted sisal fibers, but as three separate strands twisted together to give a proper size and strength, he simply unwound the three to yield not only the length needed but an additional piece as well.
The rope "strands" tied between the two trees met our requirements and the lean-to was constructed in short order.
Now you ask what has this to do with Mandelbrot and chaos. The evolving concept that not everything is as it appears to be is just another way of saying that organization gives way to chaos and chaos gives way to organization. There is benefit in looking for the dark underside of reality.
Consider this particular piece of rope. Let's say we take a one foot section of it and look at what it actually represents. It can be defined in three dimensions ^� length, and diameter which has two dimensions, across and up-down. But how short or long can be made this one foot piece of rope? You can shorten it by doubling it back on itself and if the rope is sufficiently small in diameter, it can be made very small. In fact taken to a limit, you could say that the rope's length could approach zero.
So the rope can be made to be essentially zero in length, but how long can it be made to be? We can separate the three strands as Don did, and we find that the three together are longer than three times the length of the rope because now it is no longer twisted but the individual strands can be made straight and when laid end to end may actually be over four times as long as the original rope. If one takes an individual strand and separates it into the original sisal fibers and rewind them into a smaller strand, we can create a "rope" of much greater length. (The individual fibers, if polypropylene or the like, are spun as a single strand and then chopped to a specified length and when combined into a "tow" are the beginnings of thread, rope or sheets (if you're a sailor.) From the manufacturer's view, the diameter of the strand can be varied dependent upon the ultimate requirements and that diameter can be made quite small as example, of the fineness of a silk worm's cocoon or a spider's web. Taking this to an extreme, and then using the goldbeater's techniques, it could be made extremely thin and flat. If cut into narrow widths, and the flat pieces then joined together, the length becomes almost infinitely long. Now our, so called, rope has dimensions much different from that originally viewed. But what is the actual length dependent upon?
For this, we need to look at the rope itself. We have previously defined it as having three dimensions, usually referred to as x, y, and z so that it can be visually and graphically portrayed. But as Donald discovered, by unwinding the rope we have revealed a fourth dimension. You can call it density if you like. This actually represents the amount of space that is actually filled by the rope itself plus the free space surrounding the individual strands. Imagine if you will, the solid component could be separated according to Donald's method, then fluffed up so that the volume is equal to that of the original rope. Certainly the strength would be less, but the apparent size and shape of the rope and its dimensions would appear to be unchanged. We have substituted space for substance. How much can you continue with this extension (and is there some practical value other than extending the rope between two trees)? The answer provides an interesting study in the science of packaging.
To prevent damage to goods that are placed in boxes, bags, portmanteau, and such, you might use sand or some other dense material to prevent the goods from being banged about. Or since a lightweight material would be a better choice, you may choose to use excelsior which was used until very recent times. With introduction of plastics, the era of "popcorn" arrived, (some bright souls have used nature's own pop corn in a similar fashion) as a packaging material. Here you surround your goods with air pockets held in place by a thin film of polystyrene or the like. This may be carried to an extreme where plastic pillows filled with air or some other gas is used to protect the goods. Moving companies choose a different approach, they separated glassware and other materials by placing them in sectioned boxes, a practice used by those who transport bottled liquids as well. However, whatever the appearance of the packaging, the concept is the same. Protect the goods by preventing them from either banging together or against the case itself.
Mother nature improvised a second approach. She chooses to suspend a fragile item in a liquid. Consider the egg. The "shell" is a part of the package and may be either hard or pliable (as example; chicken or turtle eggs). The albumin contained within is a suspending media, which by its viscosity buffers the embryo from moving rapidly through the media and impacting against the shell.
Let's go back to our rope again and separate it into the separate strands. We discover that each is in-turn made up of shorter fragments which we called "tow" or fibers dependent upon the material from which our rope is made. If man-made, tow will be the better term, as the extruded fibers as they are formed are cut into short sections which makes it possible to handle them to make thread, rope and such. In the case of nature's fibers, as example, cotton, flax, sisal, etc., the length is whatever nature intended and man accepted. Regardless, the fibers are twisted together so that friction between fibers in the twist holds them together imparting strength. Untwist the strand and it loses the property that was desired. Nevertheless, as Donald did; the "rope" retains some useful purpose.
In the case of the fibers, having a twist gave strength, in the goldbeater's leaf, the metal regardless of how thin it is beaten, will retain some strength, not much, but some. Look at the fiber, whether natural or man-made. It is a chemical, an organic molecule that has a characteristic structure. However, there is a limit to the size of the molecule and only by having a number of them together do you have something that can yield a fiber. In fact there is some reason to compare the thin gold film with the plastic or natural fiber. The gold atoms have an affinity for one another which permits the beater to practice his art. In the case of the cotton fiber or a polyethylene one, the presence of a great number of molecules of the same type associate together and by exclusion of the environment in which they exist, they appear together, but like the gold leaf, they can be made to separate. Rayon is a man-made fiber that is reformed from cellulose (the name for organic materials of wood or cotton origin). By dissolving the organic fiber in a solvent and then "spinning" it into another, a fiber forms which has properties not like those of its parent. Spinning is the process of causing the liquid polymer to pass through an orifice or die and having the polymer harden to a product. Spiders do this all the time yielding a fiber which is formed into a web or a "bungee."
Imagine if you will, the rope which we started out with, by the magic of chemistry caused to be reshaped into a single fiber through an orifice so small that it would permit exit of a fiber a single molecule in size, the rope would be of a size approaching a finite limit but much longer than originally envisioned. However, there is yet another "extension" of our rope. If as in the case of gold, individual molecules could be shown to have an affinity for each other, then they could be spaced apart with a suitable gap between each individual molecule. Instead of forming a polymer, then the individual glucose molecules (for that is what the cotton is composed of is) could be strung out in a line. But they need not touch, only be close enough so that they have an attraction between each other. Picture them as like a group of magnets spaced so that they retain their position but still give a structure to the "fiber." Rather than magnets, they molecules may be attracted by forces called "van der Waals."
How much further can you stretch the rope? There is a limit which can be calculated.
By the way, when you have the atomic forces holding together the individual molecules you now go from a weak disorganized structure to one that has greater strength than ever envisioned in the simple rope. This is the second teaching of the mathematicians and physicists: At some point, the unexpected happens, instead of having less and less strength, at some point there is a transition and great strength is found. That's chaos theory, but it's not chaos - it's organization. And that's what Donald's efforts with the rope teach; organization can give rise to chaos and then organization can be created from chaos.
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Beloit Mandelbrot - Chaos, Making a New Science, James Gleick, Penguin Books, New York, 1987.
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