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[NOTE: this was started one afternoon on a random thought/inspiration. The intention was to take a topic I had thought a little about but not come to a solid conclusion on and see where it went. Well I accomplished that goal just find and I at least think the result is worth reading. The problem is that the original intention also included finishing it before I stood back up, and instead it’s taken me weeks of random work on it as it kept getting bigger and bigger. So if it seems a little disconnected, I’m sorry, I tried to keep the continuity up but when you’re trying to just follow your train of thought where it goes you’ll find that even if you left it in New Brunswick last week, there’s no reason it wont be visiting Zimbabwe this week.]

In one second a photon could travel around the Earth 7.5 times, at 2.998*10^8 meters per second. One light year is how far that photon could travel in a year, so that means that the circumference of the Earth is 1/(7.5*3600*24*365) light years, or 4.22797*10^-9 (0.00000000422797) light years. The Sun is 2*10^-5 (0.00002) light years from the Earth. The next nearest star, Alpha Centauri, is 4.2 light years away. The center of our galaxy, the Milky Way, is approximately 3.*10^4 (30,000) light years from us. The entire width of the galaxy is about 1.0*10^5 (100,000) light years. The next nearest galaxy, the Andromeda Galaxy, is 2.0*10^6 (2,000,000) light years away. The estimate for the size of our universe is about 1.5*10^10 (15,000,000,000) light years wide.

That’s a lot of numbers right there, and when they’re that big (or that small) you begin to lose track of their size. Can you actually picture something that’s 3.5*10^18 (3,500,000,000,000,000,000) times the circumference of our planet? Of course no one can really get that picture in their head, at least not a meaningful picture, really all you get is the idea that it’s really, really big. Which is of course, true, but not all that helpful when it comes to understanding it. The question is, is the size of the number important? To be honest, I don’t know, I’m writing this without really knowing where I’m going in the hopes that I’ll arrive somewhere meaningful, even if the numbers aren’t.

Well let’s start off with another example. Let’s take the number pi and see what we can get by it. Of course pi stands for an incredibly complex (qualitatively, not quantitatively) number, it’s irrational, meaning that it’s digits continue on into infinity and can not be written as a fraction (where as the number 0.333333333 on into infinity can be written as 1/3 and is thus, a rational number).

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3.
1415926535   8979323846   2643383279   5028841971   6939937510
5820974944   5923078164   0628620899   8628034825   3421170679
8214808651   3282306647   0938446095   5058223172   5359408128
4811174502   8410270193   8521105559   6446229489   5493038196
4428810975   6659334461   2847564823   3786783165   2712019091
4564856692   3460348610   4543266482   1339360726   0249141273
7245870066   0631558817   4881520920   9628292540   9171536436
7892590360   0113305305   4882046652   1384146951   9415116094
3305727036   5759591953   0921861173   8193261179   3105118548
0744623799   6274956735   1885752724   8912279381   8301194912

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That’s the first 500 digits of the number, it’s been calculated by a Japanese firm to 5.02*10^11 (502,000,000,000) digits with no pattern found. Of course, for most people, having all those digits is preposterous. When doing calculations you may need to know it around to 3 digits, sometimes a little more, but 500 digits of accuracy is beyond the scope of any theoretical need.

So to a physicist, or whoever is using the number, all that’s interesting is the scale of the number. 3.14159 will undoubtedly suffice for most people’s calculations throughout their entire lives, and decently accurate, quick calculations can made just using 3 as pi (though don’t tell me that 3 equals pi! ).

There’s been extensive studies of the number pi, attempts to find a pattern (all of which have failed), faster ways of calculating the number, none of these things has to do with the fact that the number is approximately 3.14159, it has to do with it’s unending complexity. Certainly to these people, complexity is far more important than magnitude, giving a number’s magnitude (pi would be on the 10^0 magnitude on a normal scale of measure). Of course some people’s interests don’t prove overall interests. After all, we know mathematicians are nuts right? So we’ll take pi as an example of a number which is defined by it’s complexity but has most of its use in magnitude. We’re getting closer to arriving somewhere, but we’re still not there yet, wherever there is.

So let’s move out, back again to a solar example and compare the Earth to the Sun. The Sun, of course, is far, far larger than the Earth; it could hold about 1.3*10^6 (1,300,000) Earths in its volume. Yet how interesting is the Sun? What does it mean to us and what do we study it for? Well the Sun is a hot bed of scientific inquiry though no probes can actually be sent to it, they would be destroyed by the extreme heat long before they gave us meaningful data, so all of our information comes from observations made through telescopes either here on Earth, or ones we’ve sent out in orbit (like the Hubble Space Telescope). What we’ve observed that we’re interested in is things like Sunspots. Sunspots are large eruptions of solar material, magnetic fluxes in the Sun’s field.

In other words, they’re complex developments in a large scale phenomena. The Sun is a massive nuclear reactor, it does one thing and does it on a gigantic scale. However, once that process is determined the size of the Sun is no longer interesting as the entire volume is going through the exact same process. It’s the complex results that it causes that become interesting. Our Sun, and most stars are much, much larger than our planet, yet it is much easier to understand them than our planet. Why, because stars are fundamentally very simple things. Complexities, like sunspots, are almost impossible to predict long term, but try to predict long term behavior and interaction of a crab population off the coast of Maine and the former problem of sunspots seems easy in comparison.

Let’s travel down one more track and hopefully things will come into focus.

Information, for this example, specifically mathematical/computer information is now the avenue of thought for this essay. Let’s first look at picture. Well since Hatrack can’t host images let’s pretend we’re looking at a picture. It’s 200x200 pixels and it’s all black (RGB -> 0,0,0). Can you picture it? It’s just a big, black rectangle on your screen.

Each pixel (for those who are interested, pixel means picture element) the computer stores either 24 bits (a bit is a ‘1’ or a ‘0’, those good on/off things you’ve been hearing about) or 32 bits. Let’s assume 32 just for simplicity and because on probably 95% of your computers that’s the word size of the computer (don’t worry about what that means if you don’t know). The computer certainly stores all that information for the picture. There are 40,000 pixels (200*200) and that comes out to 1,280,000 bits of information. All right, now let’s take another picture, say one of a flower garden. It’s also 200x200 pixels in size, only now instead of being completely black it has a myriad of colors in it. However, the computer doesn’t change how much storage it needs, it still takes that 1,280,000 bits to store that image on the computer, and certainly if you’re displaying this image in your browser or an image editor all that memory will be used to store the picture, completely black or of a beautiful flower garden.

But what happens when you’re done working with the image want to save it on your computer? Well you can certainly save all those bits on your disk (it would come out to 160,000 bytes or 156.25 KB or 0.15259 MB) but chances are that you’ll want to compress it some how, so you can put lots more of those pictures on your disk.

Well I’m not going to try and explain JPEG or WinZip compression (especially seeing as I don’t understand them myself ) so let’s go with something much, much, much simpler, let’s go with repeated number compression.

The theory behind this is very easy to understand. Let’s say you have a series of numbers: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, and you want to read this series off to a friend (you’re really bored, Hatrack just went down or something). You can read off all those 1s (equivalent in theory, to storing all those 1,280,000 bits on your computer) or you could say “There are twenty number ones in a row.” Much simpler, much faster, much easier, and you just used repeated number compression.

It’s exactly the same with those pictures, exact same formula, so let’s look at the results. Take the flowers first. This picture is quite complex, many colors, almost no repeated numbers. As our theoretical compressor goes to work it will find almost nothing to compress. It starts off and finds some variety of red in the flower, then reads the next color off to discover that it is also red, but a little brighter, so it does not compress it. Perhaps there are a few repeated colors, but they are few and far between. This picture remains about 1,280,000 bits when you store it.

Now let’s look at the all black picture. The compressor goes through and finds that every single color is the same as the previous one. So what does it store, the number 40,000 (the number of individual colors in the picture) and the number for the color ‘black’. Now instead of taking up 40,000 numbers (156.2KB) the computer stores 2 numbers (0.00195KB).

Now what do we learn from this? We learn how much information is stored in each picture. In one picture we find all most no information, in fact the size of the picture stores exactly as much information (two numbers) as what’s in the picture. Whereas in our other picture of flowers we find that it has a great deal of information for us to look at. The amount of information in each picture is directly proportional to the amount of complexity in it.

Of course the size of both of these images will change depending on what type of compression you use, but I can grantee that the all black picture will always take up far less space than the picture of the flowerbed. For a more detailed look at what randomness or complexity actually is I suggest reading Wolfram’s A New Kind of Science, but basically you can pretty much follow your intuition on the amount of complexity something has, be it in a picture or anything else in life.

So what’s the point here? Well the first point made is that the amount of information something stores (a computer file, a physical object, an idea) is related to it’s complexity. This is because the less complex it is the more it can be compressed, and written as a sum of rules. Looking back I don’t think I’ve explained this fully yet so maybe a little more time on it and I’ll move on.

A sin wave, one that goes up and down and up and down like a series of semi-circles is a pattern that repeats infinitely. The sin (x) where x is any real number between infinity and negative infinity results in another number. Sin (0) will result in the number 0. Sin (pi/2) (there’s our complex friend again) will result in the number 1.

It just took me about a paragraph to describe the function sin, and I’m not really sure I did a stellar job of it but that’s not really the point. I could describe it to you, and explain what it looks like, and that’s one thing. Or, I could write out “sin (x)” and the same amount of information would technically be conveyed. Just like I could show the entire black picture or I could say “a 200x200 picture, all black”, and the same information is conveyed. So the point here is that the amount of information conveyed in any medium of conversation is equal to the least amount of information that can be used to describe it accurately.

This is why you pay $2000 for 8 pounds of computer, and pay about 20 cents for 8 pounds of concrete, the complexity matters, not the size. What type of complexity does each one have? The laptop has, what is to you, useful complexity, whereas the concrete has very little interesting complexity (for the average person, us civil engineers feel differently about that ).

Well I’m not going to try and define what types of complexity are interesting and what aren’t, that’s too much of a philosophical discussion and I’m far enough off track already. So instead I’ll acknowledge that some complexity is more important than other types. Like I said, I wont try and explain why the complexity you’ll find in Portland cement is somehow less interesting than the silicon complexity that makes up all those megabytes of RAM in your computer. Instead I’ll provide a more general rule for looking at complexity.

We’ve seen that the amount of useful information something contains will be no more than how complex it is, that complexity can determine the interest in something. But these are side points really. What’s interesting is to see the way people look at the world through complexity.

Take you’re watch and look at it (if you’re not wearing a watch, pretend!) Even if it’s digital there are some very closely define markings on it (letter composition actually has to be very specific to be readable, slight errors can garble the language easily). The important of these marking is absolutely essential to the watch’s usefulness, should they be off, the watch’s function is rendered useless. So take this watch, that only works based on the complex markings, put it down, and walk about 30 feet away from it (or if you’re a slacker like me, imagine walking 30 feet away from it). Now look at it and see what’s important.

Is it the small, complex markings on the face that has your attention? Well unless your eyesight is unusually good (or your watch it unusually large) it’s not going to be that. Instead it will be the larger part of the watch, the shape of it. If you’re watch looks like mine what you’ll see, what will make it important is the rounded shape in the middle and the two, longish, straps that come out to the side.

Now that you’re farther away from it, it’s still the complexity of the watch that matters to you (6 inches long might be a description of use when trying to fit it into the world view, but what makes it special or interesting is not the size but how it’s different from the norm) only it’s a different magnitude of complexity. When you were holding it up to your face you noticed complexity on a much smaller scale, probably not even half an inch. When you moved away from it and were unable to clearly make out that level of complexity, now about an inch is the scale you’re noticing.

It’s my opinion that this is a general truth, that something is interesting to humans, conveys important information based completely upon that thing’s complexity. Only the amount of complexity is not an absolute, but measured at a relative distance. When you’re 1 AU (astronomical unit, the distance from the Earth to the Sun) away from the Sun, a few square feet of nuclear reaction loose all interest to you, you notice the way the sun as an entire (gigantic) body is different from that which is around it (nothing), and even then, the self similarity of the sun at the magnitude from which you view it is rather uninteresting. It’s a big ball in the sky, yellowish for the most part. The effects are very important (look at all the tall, healthy, complex corn it grew!) but the sun itself holds almost no interest as something to be studied.

However, when you’re about 10 feet away from the many 1000 degree Celsius square foot of nuclear reaction, your evaluation of it’s importance will change drastically. You’ll notice a lot more about the area, and find that the complex turbulence that develops in such a highly energized gas (or plasma) is actually quiet fascinating. Now the entire Sun around you isn’t important as one thing. It’s too big to study at this level, only the complexity that can be takes in from this distance is interesting.

All right, well I kind of balked at explaining that last part because I didn’t really think I could do it right, and I don’t really think I’ve done it any justice, but no one else was going to write it so I guess it was up to me. The last thing I want to do here is summarize my main points, since I certainly did wander around a bit.

The amount of information conveyed by anything is determined by it’s complexity, something monotonous can only deliver as much information as it takes to describe it’s repeatability.

A thing is interesting to human beings based on it’s complexity, but the way to measure that complexity isn’t wholes tic, but is once again based on that person’s perspective, where they stand in relation to it, something I feel is both a literal reality and a metaphorical position in relation to something even as abstract as an idea. When you’re really far away from an idea you wont be able to appreciate the intricacies of the idea but only the outer edges of what it means. Like an ultra-conservative person viewing the benefits of socialism, they can make out the broad plan and its planned effect, maybe even see what the other side thinks this idea will accomplish, but until they really examine and understand this idea the true benefits and complexity in building up a government will be lost.

Hobbes

[ May 19, 2004, 12:27 AM: Message edited by: Hobbes ]

Posts: 10602 | Registered: Oct 1999  |  IP: Logged | 

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 348790238579013509246502378
4092835098327497340982349872735782
3572337777777777777777777777777777
7777772937523741209837410298401236
5091275092374080723498172356102

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What did you notice there? Presumably it was the long string of 7s, right? Not very complex, those 7s. But the fact that they appear in the midst of such stagering complexity is what makes them grab your eye. So once again the complexity fo the system is the important, and the anomoly it produces becomes interesting. In fact those 7s are incredibly complex because there's no way to describe how they fit into the system. It's a problem that would permanently stump mathematicians (unless of course they knew I deliberatly planted them there and just randomly pressed keys for the rest ). The simplicity of the 7s in such a complex system is actually more complex than the numbers around it because it does not follow the same pattern they follow and yet is somehow still in the same group as them.

Hobbes

[ May 18, 2004, 02:52 PM: Message edited by: Hobbes ]

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I am mostly thinking of asthetic (sp?) tastes. There is something beautiful about a simple, elegant dress that a fancy dress cannot compare to.

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Posts: 26071 | Registered: Oct 2003  |  IP: Logged | 

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The five colors blind the eye.
The five tones deafen the ear.
The five flavors dull the taste.

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