Authored by: Tkilgore on Tuesday, March 19 2013 @ 06:00 PM EDT |
I keep thinking that the above-described argument that the square root of two is
two can be found in one of the Dialogues of Plato. I keep thinking that years
ago I read one of the dialogues and saw it there. Socrates, as I remember, used
this argument in order to confound the "wisdom" of a certain Sophist
-- and succeeded thereby in reducing him to red-faced, spluttering confusion.
But I have been unable to find what I "remember" to be the case.
So, does anyone else happen to know what is the earliest known invocation of
this argument?
Oh, and BTW there is a very similar proof that pi is equal to two. Start with
the upper half of the unit circle on the xy plane. Its length is pi, as we all
know. Now, take two semicircles, one centered at (-1/2, 0) and the other
centered at (1/2, 0), in which the y value remains non-negative. Proceed from
(1, 0) to (-1, 0) along the resulting path. This time, the length is
pi/2 + pi/2 = pi.
Now use 4 semicircles, then 8, and keep going. The length of the path, by direct
calculation, is pi every time. But the path is "getting closer and closer
to the line segment connecting (-1, 0) and (1, 0)" and the length of that
line segment is, as we all know, 2.
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Authored by: J.F. on Wednesday, March 20 2013 @ 06:27 PM EDT |
To people who don't recognize this, it's called "Taxi Cab Geometry".
You'll find that reference easy enough. It's sometimes used in computer games to
simplify tasks like computing the distance of entities from the player.[ Reply to This | Parent | # ]
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