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Saturday, July 16th 2016





The time here in Mirkwood* is slowly coming to an end. It's been a good stay. The weather has been extremely cooperative. 

(*I assume you realize that I'm not using real place names.)

The restaurant here attached to the "Prancing Pony" in Bree has gone bonkers. It used to be a basic country-style place where you could get a good meal for a decent price. Now they're charging $10 dollars for just a cheese burger. That's like Hilton Hotel prices, and not exactly in line with out budget. So we take food from the local grocery store to our room.  But that's not what I wanted to tell you about.

The first thing I've done every morning has been to go to the stable for my day's ride, while June slept in. I've mentioned the names of some of the horses I've ridden in this park: Whiskey, Bailey, Dan, Levy, Chip, and Bonnie are ones that come to mind. I rode Bonnie once a year ago, and she was my first horse on Monday of this week. As you can imagine if you know me, I always try to strike up a friendship with whatever horse I'm on, talk to him or her, and treat them to some of my cowboy songs, hoping I'm not annoying whoever is riding right before or behind me. So, today I was sitting on a bench by the horses' enclosure, just a board nailed to the fence, with my back to the equines, watching the world as I was waiting for a few more participants to show up for the ride. All of a sudden, I felt a horse nestling the back of my neck. I turned around, and there was Bonnie, saying hello to me. We chatted for a few moments, and then she ambled over for her breakfast hay. I mentioned that little serendipity to Sierra, one of the trail guides, and I added that I did not fantasize that Bonnie really remembered me. But Sierra said she very well might have. Either way, it was nice to have a horse be nice to me.

And now back to the miracle of phi and the golden ratio.

As I said in the last entry, we need to come back to the appearance of phi in the natural realm. Earlier on I skipped most of it, except to give a mere mention of a few examples and a couple of websites. However, I didn't go any further into depth with it, eager to do more math and philosophy than to catalog the appearance of the Fibonacci numbers in nature. That's been done and overdone, I mused. However, on second thought, to do justice to some of those occurrences, we can steep ourselves a little bit more in conceptual matters, and so I'm returning to it. This material may eventually wind up towards the middle of the completed website, right along the other relevant material. One of the first things I've done is to get rid of my earlier rather awkward animation of a golden spiral and to substitute one from Wolfram Alpha, which I then subsequently animated.

I'll reproduce it here so that you don't have to shuttle back and forth:

golden spiral growth

There are two important groups of spirals, mathematically speaking, and most spirals that we encounter fall into one or the other group: Archimedean spirals or logarithmic spirals. The former are named after Archimedes of Syracuse (287-212 BC), and one member of this set bears his name, "Archimedes' Spiral." Archimedes was frequently engaged in applied math (e.g. war catapults, discovering the principle of density displacement), but also made some important contributions to the more theoretical side of math. He came up with a break-through method of calculating the value of pi. As I'm thinking of him and his work on spirals, I cannot help but think of the occasion of his death when he was killed by a Roman soldier. According to a popular, albeit unreliable story, when the soldier entered his room, he was engaged in studying some geometric figures, and he was supposed to have said, "Do not disturb my circles!" Was he maybe making further advances in his study of spirals?

Logarithmic spirals are associated with Jakob Bernoulli (1655-1705), who makes another appearance in this series in connection with the "Basel Problem." Bernoulli was so impressed by this kind of spiral that he called it the "miracle spiracle" spira mirabilis, and asked for it to be a decoration on his tombstone. The tombstone artisan clearly had not studied up on the nature of the spiral in question or perhaps did not understand it. He made a "plain old" Archimedian spiral instead. I don't think that Bernoulli cared any longer at that point, but the difference was very important to him during his life time because it illustrated for him the way in which a thing can be changed and yet remain the same. Specifically, he saw in the logarithmic spiral the renewal of the person entering eternity in heaven. And actually, if it hadn't been for that mistake, we probably would not be talking about it as much. 

Just a day or two ago I saw on an apologetics website a notice of a new book, which supports the claim that evidence for God can be found throughout nature. I don't remember author or title, and I don't want to embarass him or his publisher or me in case I'm going to be wrong on this point. All I know about this book at the moment is what the notice said and the picture on the cover, which includes an Archimedian spiral. Books of this nature usually include discussions of the Fibonacci numbers and the various golden angles, rectangles, triangles, and spirals. So, as I glanced at the picture, I remembered Han Solo's words, "I have a funny feeling about this."

So, what's the difference? I'll put it in terms that I can understand and spare us the formulas.

1. Archimedean spirals. Imagine a garden hose that lies flat on the ground, neatly arranged in ever-expanding circles. The circumference grows with each rotation, but the distance between each individual piece of the hose and its adjoining ones remains the same. The radius from the center increases, and so does length of each arc. Consequently the angle of expansion (slope, derivative) of the arc flattens. Here is a picture of the Archimedes spiral.

Archimedes Spiral

2. Logarithmic spirals. Take the same garden hose. Start to coil it up a little. Measure the relationship between the radius and the arc. Now coil the hose a little more and maintain the same ratio. Well, the segment increases, but you're maintaining the same ratio and thus the same angle of expansion (slope, derivative).Then the distance between segments must increase concomitantly. Extend the coil some more. Again, keep the same ratio, but extend the length of hose. Your distance between hose segments will be increasing some more. As you keep going in that fashion, the spiral becomes looser and looser, and that's because it keeps its ratio. If you were by chance to see just one segment, and you had no idea of how large the magnification was, you would not be able to tell the position of that segment relative to the point of origin of the spiral. Here is a picture that's not in perpetual motion.

Logarithmic Spiral

A logarithm is the flip side of an exponential number. Express the number 100 in exponential terms, using 10 as your base. You will write 102.

Now you can say that 2 is "the logarithm of 100 to the base 10."

Logarithms are helpful in many ways. They decrease the distance between numbers to a manageable size.

Expressed exponentially 1,000 is 103, and so its log to the base 10 is 3. A difference of 900 is expressed with just one integer, going from 2 to 3.

10,000 is 104, and this time it's a gap of 9,000 units that's expressed with an increase of yet a single number, namely its log 4 (base 10).

The pattern goes on in the same way. Furthermore, if you need to multiply two numbers, such as 100 x 10,000, you can take their logs, 2 and 4 respectively, and just add them. Then you can go back to the exponential version and write out the result as 106, which is a million. Obviously you don't need logarithms for making simple multiplications or divisions for the powers of 10. However, all numbers have a base 10 logarithm, and so a table of logs can help you with slightly more complicated procedures.

Some of us who have lived through more history than other readers may remember the good old slide rule, which was eventually replaced by the pocket calculator, though not for a few years after I had finished my undergrad as a science major (zoology). It was extremely useful and amazingly accurate. I won't go into its design or functionality now, except to show how the display on a number line (and it had several) was based on a logarithmic scale. You've barely begun with the number 3 by the time you get halfway, but the the distance for each number goes down on a logarithmic function, and you get all of the first ten numbers on each bar. slide rule scale

Actually, I need to tell you that more often than not, it's not quite that simple. Mathematicians and scientists usually prefer to work with a base different from 10, namely a number that goes by the name of e, called such in honor of the magnificent Leonhard Euler, who is considered the greatest mathematician ever by some people. e is also an irrational number, its value is approximately 2.71828..., and I'm not going into its properties any further. I will just say that e holds as many surprises as phi and pi. 

The ratio of increase in a logarithmic spiral can vary from spiral to spiral. As we observed already, the chambered nautilus grows according to a logarithmic pattern, but it's not phi. However, we can find logarithmic spirals in many other parts of nature, where they follow the Fibonacci numbers, and, thus the golden ratio and phi.

It's gotten way too late again. More next time

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