> Roger, Ken, Rob, Horse, Fintan and all interested Pirsigers:
>
> The structure of Intellectual patterns isn't fundamentally different
> than any other level of reality. Does the brain structure the mind,
> or
> does the mind structure the brain? The only answer in YES!
>
> My thoughts are based on the riddle of the golden mean, as it
> was explained to me by a mystic art professor back in college, and on
> a
> movie called pi (The movie is out on video and I urge you to see it.
> The title on the box is actually the Greek
> symbol, which I can't make with this keyboard, but you know what it
> looks like.) I believe this is all relevant to the MOQ.
>
> I forget who introduced "the golden mean" in a recent post. My
> understanding of "the golden mean" is totally unrelated to the idea of
> Aristotelain moderation described in said posting. I only mention it
> to
> prevent confusion.
>
> The Academy founded by Plato asked just one question as an entrance
> exam. To be admitted into Plato's Academy one had to solve the riddle
> of
> the golden mean. The riddle was permanently presented to everyone,
> aspiring student or not, at the stone gates of the first university.
> There, carved into stone, were two basic geometic shapes; a circle and
> a
> square. These two ideal forms were positioned in a particular way with
> respect to each other, and solving the riddle requires an
> understanding
> of that positional relationship and it's meaning. BTW, this is
> supposedly real history.
>
> It'll be very difficult to describe this geometic riddle in words. I
> know this sounds wierd, but it will help if you get yourself a pencil
> and a sheet of paper to help work it out and visualize it better.
>
> Draw, or imagine, a geometrically perfect circle. Size doesn't matter.
> :-) Make a line across the circle, from side to side, that disects it
> into perfectly equal parts. On top of the line, but just inside the
> cirlce, draw a geometrically perfect square. The square also has to be
> resting and centered on the disecting line. Because the verticle lines
> of the square have to be the same length as the base, by definition,
> there is only one square that will fit in any given circle. The
> verticle
> lines of the square move up from the disecting line base to reach the
> edge of the circle. Naturally, the top of the square is parallel to
> the
> base and the disecting line it rests upon. So, we have a disected
> circle
> with a sqaure on top of the line, but within the circle.(There could
> be
> a square UNDER the line too, but it would otherwise be exactly the
> same
> and so it would be pointless repetition.) That's what was carved into
> stone at the entrance of Plato's academy. If and when you could
> explain
> the meaning of this geometric diagram, you'd be admitted as a
> student.
> (sorry if that was tedious)
>
> I'll tell you what it means, just in case you ever go back in time.
>
> Size really doesn't matter, the solution to the riddle is in the
> relationship of these geometric forms. Since there is always just one
> square that fits in any given circle their relationship is
> mathematical.
> The circle and the tightly fitted square represent a ratio. You could
> express it in terms of the total area of each form or the length of
> their boundries. But the solution to the riddle of the golden mean was
> actually found in the disecting line. Their idea of a trick question?
> Still have your paper and pencil?
>
> The disecting line intersects the left and right edges of the circle.
> Two more points are created as the "correct" square's verticle
> segments
> touch the disecting line. That gives us four points. (That have been
> derived by the relationship between three geometric shapes, a line, a
> circle and a square.) Going from left to right on the disecting line,
> point A is at the left edge of the circle. Points B and C are where
> the
> square's two verticle segments meet the disectling line and point D is
> at the right edge of the circle. Just to be crystal clear, all four
> points are on the disecting line and read ABCD from left to right.
>
> The line has been divided into three segments, AB, BC and CD. Segment
> AB
> is the distance between the left edge of the circle and the left edge
> of
> the square. Segment BC is simply the length of the square's base. (And
> the length of all its sides, by definition.) Segment CD is exactly the
> same length as segment AB and is the distance between the right edge
> of
> the square and the right edge of the circle. The total segment, from A
> to D is simply the diameter of the circle.
>
> We don't even need segment CD to solve the riddle, it would just be a
> pointless repetition like another square under the line instead of on
> top of it. Just throw segment CD out the window and forget about it.
>
> Sorrythe length of this explaination. Its so hard with words. Wish I
> could draw a picture. It would be so much easier. But please indulge
> me,
> we're very close to the riddle's solution now.
>
> The length of our two remaining segments, AB and BC, is the key that
> unlocks the whole riddle. More precisely, the ratio of the two lengths
> is the key. The ancient Greeks didn't have a zero in their numbering
> system, but today we can express the ratio as 1 to 1.6180. If segment
> AB were 1 meter long, segment BC would be 1.6180 meters long. (You've
> probably noticed that this is alot like pi, and this info will help in
> appreciating the movie by that name.)
>
> Plato described the ratio with words instead of numbers, saying
> enigmatically, "the lesser is to the greater as the greater is to the
> whole". He somehow saw that without zeros or decimals. The lesser is
> segment AB, which is 1 meter long in our example. The greater segment
> is
> BC and is 1.6180 meters long. The whole is segment AC and simply the
> sum
> of AB and BC. Segment AC is 2.6180 meters long. You can check this
> with
> a calculator. The ratio between the length of segments AB and BC is
> the
> same as the ratio between segments BC and AC.
>
> "The lesser is to the greater, as the greater is to the whole."
>
> The riddle's solution is in the recognition that this relationship,
> this
> ratio derived from ideal geometiric forms that don't really exist
> anywhere except the human intellect, is found througout nature! The
> ratio is exhibited in the shape of galaxies, the structure of human
> bodies, fingerprints, DNA, sun flowers and most conspicuously in the
> structure of the chambered nautilus. It is a pattern that seems to
> underly all other patterns.
>
**************************************************
I already posted everything up to this point, but ran out of
time and had to stop here. I'm replying to my own post so the whole
thing is in one place.
The exact ways in which this mother of all patterns shows itself
in nature really can't be show properly in this medium. One can get a
ruler, a calculator and lots of natural objects to test. Or one can just
see that movie called pi.
The movie is ou on the edge and not for the faint of heart. It
tells the story of an obsessed mathematician. He is looking for a
formula to predict the stock market. Along the way, he makes some
amazing discoveries. His work is sought after by moneyed interests,
whose ruthless attempts add a little action to an otherwise cerebral
trip. Also on his trail are a group of Hebrew mystics who believe the
mathematician may have learned the true name of God. His mathematical
discoveries are extemely attractive to wall streeters and mystics alike!
The golden mean, the magic ratio of Plato's academy is all over the
movie and there are plenty of examples of its' appearance in nature. In
the movie they never reveal the actual numbers involved in the ratio and
they call it the golden spiral instead of the golden mean, but its very
recognizable nevertheless. Oddly, the movie has almost nothing to do
with finding out the area of circles.
Geometry and mathematics are purely intellectual constructs. It
strikes me as more than strange that nature would exhibit such patterns.
It sends shivers down my spine when I think that all the universe is
built on patterns we supposedly invented just a few thousand years ago.
Seems we invented nothing at all, we just discovered it.
I don't think I can really do this justice, but maybe the movie
can amaze you.
Thanks for taking the time,
David
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