Economy and Extravagance

The work of a curious fellow

The article that follows is on a topic that I have wondered about. I would appreciate any feedback that you might be able to provide. Especially errors in concept or calculation. Please send an email to if you would care to comment.

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Exploring Nature's spending habits...

Nature has a reputation for following the path of least effort when free to exercise her own preference... But wait! When would Nature not be free to follow her preference? Perhaps when confronted with willful living organisms? (see why is there life) There is a popular way of thinking that considers the activity of mankind as somehow unnatural. I am not sure this is justified. We humans are clearly part of the natural universe. Why then would our activities be beyond the scope of Nature? I suspect that it is quite presumptuous of us to assume that our activities are somehow extra-natural, disrupting Nature's plan. Being presently (and possibly temporarily) at the top of the food chain, some people idealize the non-human part of the natural world as kind of a conflict-free place of great harmony. Clearly they have not been paying close attention to the world around them. I suppose what they really mean by that observation is that humans can find peace and tranquility in nature - true as long as we are warm, well fed and generally prosperous. People in this circumstance get quite irate when people not in this circumstance disturb their illusion of a peaceful tranquil nature in an attempt to achieve the warm, well fed and generally prosperous state. But that is not what I set out to write about. We are considering the origins of life on this planet.

As I was saying, Nature manifests a remarkable economy of effort in running the inanimate parts universe. Take a look at an essay from another thread of my observations, where I discuss well behaved matter. In that essay I introduce the idea that inanimate matter follows very simple rules as it makes its way through the universe. Animate matter on the other hand seems to be decidedly unruly. This difference in behavior may contribute to the impression that nature is one thing and humanity something else entirely. As I observed in the prior essay in this thread, maintaining life requires constant effort. Such a thing seems to be an wild extravagance on Nature's part when compared with the way the inanimate part of the universe operates. Perhaps there is a hidden economy in this apparent extravagance.

I wonder if there is something about the rules by which the universe is governed, the laws of Nature as these rules are sometime called, that favors life arising from the inanimate matter making up most of the mass of the universe. This might reduce the huge improbability we find of chance alone producing life. I had better confess right here that demonstrating life flows naturally from the rules by which the universe is guided is well beyond my understanding. The best I can hope for is to introduce a way of looking at the rules governing inanimate matter that may be new to some folks and observe that in this way of thinking about things, the gap between inanimate and animate matter may not be as wide as it appears at first glance.

I need to lay some groundwork to reach this new point of view on the behavior of inanimate matter. Below are links to a few pages from my online course in the mathematics of chaos. These pages should fill in the necessary background to get a grip on the concept of an attractor, particularly the attractors that arise in the case of a simple vibrating strip of metal. Please at least skim through the pages in the table below. Detailed study, including playing with the animated illustrations might take a few hours.

Introduction to Order a closer look at chaos This will be a useful reference for understanding any of the pages which follow.
Numbers, Functions and Graphs This is a fundamental introduction to the ideas named.
Extending Graphing Concepts Here we examine some alternatives to the customary way in which functions are graphed.
Iteration and Attractors This material may be new to folks who have not delved into chaos theory previously.
Dynamical Systems Background Here we begin to explore the motion of inanimate matter in response to the laws of Nature.
The Simple Pendulum This page introduces the idea of a phase space representation of the state of a pendulum.
Pushing The Pendulum Here we examine the phase space representation of a lossy driven pendulum.
Duffing's Mechanical Oscillator On this page we use a driven mechanical oscillator to explore some irregular but periodic motion.
Periodic Attractors This page explores a system with multiple periodic attractors.
Chaotic Attractors Here we see examples of the simple laws of nature producing results so complex as to be unpredictable.
First Orbit Strands The chaotic motion of the vibrating strip in the Duffing mechanical oscillator (DMO)though totally unpredictable, when viewed in an appropriate way in phase space reveals an amazing degree of order hidden in the chaos of the motion. The image at the left is the (blue) track of the state of the vibrating strip through the space, called phase space, with dimensions of: strip position (x), strip velocity (x') and time (represented by the circle perpendicular to x and x'). This track comprises about one and one-half strands of the full orbit, starting on the (x, x') plane and moving generally counterclockwise around the time axis circle. Notice that on completing one strand, the point where the track pierces the (x, x') plane is marked with a red spot.
At the right is an image of multiple strands of the phase space orbit of the vibrating strip. These strands never intersect each other no matter how many of them we accumulate as time passes. This is one of the characteristics of chaotic motion. The region of phase space where the orbit strands gather is the attractor for the system, which is the DMO in this case. If I interfere with the vibrating strip by poking it so that its state moves off its attractor, in a strand or so the motion returns to the attractor, but from the then on the track, on the attractor, has no necessary relationship to what it would have been in the absence of the poke that I gave it. Notice that strands in this bundle are also marked where they pierce the (x, x') plane. It is difficult to appreciate the intricate structure of the attractor on which these strands lie from looking at the bundle of strands itself. Bundle of Orbit Strands
Orbit Cross Section To hide some of the obscuring detail in the strands themselves, we can just look at that region of the (x, x') plane where the red spots mark the intersection of the strands with that plane, as shown at the left. This effectively shows the cross section of the attractor as cut by the (x, x') plane. If we cut the attractor at different locations on the circular time-axis we would find that the attractor cross section changes, stretching and twisting in such a way that when we get back to the (x, x') plane the structure shown at the left is restored. Of course the structure shown at left is an incomplete cross section. It is made up of dots where a finite number of strands intersect the (x, x') plane. We would have to run the oscillator forever to get the complete picture. I have not run the model that long, but even in the fairly brief run shown here it is clear that the attractor is a complicated and intricate structure. It is an infinitely folded and twisted two-dimensional surface occupying no volume because the surface has no thickness, yet taking up space with an efficiency somewhere between two and three dimensions, maybe about 2.2 or so. Being of fractional dimension a surface of this sort is called a fractal.

One important aspect of attractors is that they exist for all sorts of dynamical systems, not just simple mechanical systems. Dynamical systems analysis is now applied in the fields of aerodynamics, biology, chemistry, ecology, engineering, medicine, process controls and sociology; and probably many others that have escape my my notice, or my recall at the moment. It seems that in spite of Nature's preference for simple rules and minimum action she is a secret lover of complexity. Who would have guessed that from Newton's laws of motion, familiar to every high school physics student, would arise an "object" as complex as the attractor we have seen here. It is true that this object resides in a space one level of abstraction below the ordinary space of length, width and height that we live in but its influence is clearly demonstrated in the motion of the metal strip in our "real" space. Wouldn't it be interesting if Nature's laws of chemistry, more complicated and less well understood than Nature's laws of motion as written down by Newton over three centuries ago, gave rise to an attractor whose influence was demonstrated in the existence of life on Earth. The amount of information contained in DNA huge but so is that contained in the chaotic motion of our vibrating strip.

So where do we stand with all those perhapses I noted back near the end of the essay on What is Life. It is wildly improbable that in a universe this young, life could have been assembled purely by chance. It is also improbable, if not wildly so, that evolutionary forces drove inanimate matter to become animate. I also suggested that perhaps the universe is arranged in a way, not obvious to us now, that favors the rise of lifeforms from inanimate matter. Clearly I have not proved that this is so but I have suggested a possible mechanism, the attractor, to improve on raw chance in bringing forth life. That leaves the last perhaps in my list. Maybe as was the case in assembling the jigsaw puzzle, the application of intelligence accomplishes what chance alone could not do in thousands of our universe's lifetimes. I will pursue this notion in the next essay.

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