originally posted by: theabsolutetruth
...Squaring the circle...
Squaring the circle usually refers to an old geometrical problem that planted grey hairs in the scalps of the ancient mathematicians and
geometricians, until finally in the 19th century it was utterly debunked as bloody impossible-- there is apparently no way you can construct a square
with area π (or sides equal to √π in other words) in a finite number of steps using a square and compass. Drawing a circle with area=π is a walk
in the park though; you just draw a circle with radius=1 and the circle's area is π, or draw a circle where the diameter is 1 and the circumference
equals π. But with squares-- not that simple.
Back when Euclid and Pythagoras and the others searched for the ultimate fraction for π this problem of squaring the circle wasn't such an absurdity
as it may seem today. In ancient Egypt 255/81 was a fraction used for π, and Plato argued how 22/7 was not an accurate enough fraction for π. But
they still believed it was possible, that π could be fractioned absolutely and elegantly, they simply couldn't live with the idea of irrational
numbers, or as in the case of π, transcendental numbers. Everything in their world could be broken down into simple fractions of natural numbers, it
was like a religion, they saw certain fractions everywhere, in music, astronomy, arithmetics and certainly in geometry. For instance, Archimedes
proved that π lay somewhere between 3 + 1/7 and 3 + 10/71 in his third proposition in his work on Measurement of a Circle found on p.93 in 'The Works
of Archimedes' edited by T. L. Heath, ISBN 978-0-486-42084-4.
Squaring the circle and many of the other mathematical enigmas of Antiquity belongs within the borders of pseudoscience, for π just isn't a rational
number and you can't really construct a straight line with length π using just a square and compass, although, one turn of a cylinder where the
diameter is 1 will give you a straight line of length π, but to square it sensibly using fractions? Nope. The cylinder example was probably how they
measured the Great Pyramid, since nearly all measures are relations between the royal cubit and π. They measured up the thing using a drum with
diameter equalling the royal cubit, thus 100 turns will give you a multiple of 100 x π, or 314.1592654... royal cubits.
Or so they say anyway. For there aren't any lack of attempts at rationalising π and constructing a square with sides equalling √π. It was said
that all the secrets of geometry and the universe would unravel as you'd squared the circle. And what do you know, up through the ages, mathematicians
have devised just about every utility available to solve this problem. So in a way the old saying is probably right. For even if you knew everything
there is to know about math, you'd still be unable to
Square the Circle.
Leonardo also had his take on this, you may have seen The Vitruvian Man before:
edit on 25-2-2015 by Utnapisjtim because: edited drawing, added Vitruvian Man, changed some words and corrected a few typos + π cylinder
pyramid