Squaring the Circle; Pi; Infinite Series THE NUMBER PI Source: www.squaringthecircle.com, contact iterate@aol.com
"This idea of a construction in steps that gets closer and closer to pi is a delightful idea, --- Martin Gardner In
the mansions of mathematics today there is no room for one who still
thinks about the most famous problem of all time - squaring the circle.
For one hundred and ten years mathematicians have been convinced that
the nearly four thousand year old problem of squaring the circle is
sufficiently understood and that it is insoluble. With a prickliness
perhaps betraying some lingering anxiety, anyone who wastes further
time on the puzzle is regarded as mathematically incompetent. Mathematicians
have had no success searching for a pattern throughout more than 2
billion decimal digits of pi. The straight edge and compass no longer
play any role at all in this modern quest. The kind of method discussed
here offers a different vantage point for observing pi. It reintroduces
the classical straight edge and compass, redefining the problem of
squaring the circle to avoid Lindemann's dead end. Perhaps it will
yield something interesting, maybe even some tool needed for solution
of a scientific problem. A mathematical explanation of why 4 relations
of the 1:2 right triangle can be arranged in an alternating series to
approach square root (pi/2) - 1 would be intrinsically valuable. But,
to offer him another prodigy equally astounding, let him look into the
tiniest things he knows. Let a mite show him in its minute body
incomparably more minute parts, legs with joints, veins in its legs,
blood in the veins, humours in the blood, drops in the humours, vapours
in the drops; let him divide these things still further until he has
exhausted his powers of imagination, and let the last thing he comes
down to now be the subject of our discourse. He will perhaps think that
this is the ultimate of minuteness in nature. I want to show him a new
abyss. I want to depict to him not only the visible universe, but all
the conceivable immensity of nature enclosed in this miniature atom.
Let him see there an infinity of universes, each with its firmament,
its planets, its earth, in the same proportions as in the visible
world, and on that earth animals, and finally mites, in which he will
find again the same results as in the first; and finding the same thing
yet again in the others, he will be lost in such wonders, as astounding
in their minuteness as the others in their amplitude. For who will not
marvel that our body, a moment ago imperceptible in a universe, itself
imperceptible in the bosom of the whole, should now be a colossus, a
world, or rather a whole, compared to the nothingness beyond our reach?
Anyone who considers himself in this way will be terrified at himself,
and, seeing his mass, as given him by nature, supporting him between
these two abysses of infinity and nothingness, will tremble at these
marvels. I believe that with his curiosity changing into wonder he will
be more disposed to contemplate them in silence than investigate them
with presumption.
See also: How to Unroll a Circle
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