A geometric fallacy
A geometric fallacy is basically a geometric construction that leads to a false result. I once found an obscure russian treatise on this matter (published by мир), that kept me entertained for weeks: Ya.S.Dubnov’s “Errores en las demostraciones geométricas.” The edition I have is a decent translation into Spanish, made by K.Medkov in 1993, although unfortunately some of the translated statements had to be treated with a grain of salt. Nonetheless, I did enjoy greatly browsing through the different geometric problems, and learned quite a bit about mathematical reasoning.
Most of the geometric fallacies rely on the use of correct reasoning on the incorrect figures. But I would like to present here one of the most amusing ones, not only for the fact that it didn’t use a wrong figure, but also because it takes us to deliver a contradiction to the Pythagorean Theorem! It goes like this:
In all right triangles, the sum of the lengths of the legs equals the length of the hypothenuse. That is, the sides
,
and
satisfy
, where the largest side is conventionally denoted
(see the page in e.g. wikipedia for more information about triangles)
The “proof” of this absurd result is done by taking to the limit a clever construction. Let us start with a given right triangle, and consider its two legs alone:
 |
 |
Starting in the vertex 
 |
 |
Repeat the same staircase construction starting in the vertex
 |
 |
By taking this construction to the limit, new staircases of length
QED
This is obviously not true. What is wrong with this proof?
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Francisco Blanco-Silva 
Two curves being “as close to each other as desired” doesn’t mean they have the same length… With a straight line, and an epsilon of wiggle room, you can fit a line that’s twice as long (even a continuous one if you want).
Spot on! So, why will this limiting process work in certain cases, and not in others? Consider the example of the computation of the length of a circle by approximation with regular polygons (see my reply to the comment of Daniel Maxin)
The fallacy is due to the fact that there is no definition of what it means for the staircase to be as close as possible to the hypotenuse. The division process does not change at all the setting of the problem. At step n, you simply have 2^n triangles with sides a/2^n, b/2^n and c/2^n. As n goes to infinity, these sides converge to zero so you do have lim(a/2^n+b/2^n)=lim(c/2^n)=0 but of course this does not imply a+b=c. In other words this limiting process does not offer a bound on a+b-c
The definition of “being as close to the hypotenuse” is technically a uniform approximation. As Aaron points out in his post, think of “an epsilon of wiggle room” around the curve. This is a standard definition and is used, for example, to compute the length of the circle:
As you can see, the limiting process does indeed offer the proper answer. Why does this procedure works in this case, and not in the case of the hypotenuse of a right triangle?
Along the same lines: you can “prove” that Pi=4 by repeatedly removing corners from a circumscribed square to the unit circle.
Yup, same idea: isn’t it grand? So the key question is: why does this fail?
A variation on this is one that my dad taught me. Begin with a circle of diameter 2. The circumference is therefore 2Pi, and the circumference of the semicircle is therefore Pi. Now draw 2 “half-size” semicircles spanning the original diameter. The circumference of each smaller semicircle is now one-half Pi, and they sum up to Pi. Repeat this process with 4 semicircles, then 8. As each sum is still Pi, one can “take this to the limit” and declare that ultimately that the line approaches the original diameter…equating Pi with 2.
Very nice!