In a book on differential forms, I read,
"After all, there were times when people took geometric intuition as proof, and later found that their intuition was wrong".
I would like to see an example where intuitive geometric proof fails. (In short, I am trying to clarify the above quote by an example.)
There are several historically important examples around the idea of continuity, although I am not sure how seriously to take them. At some point in history people did not have a rigorous enough idea of what they meant by continuity for them to be wrong, strictly speaking. And then at some further point they had the modern definition of continuity, but arguably the modern definition is absurdly general and doesn't really successfully pin down the intuitive notion. Anyway, some examples:
It is geometrically intuitive (e.g. from drawing examples of continuous curves), and many mathematicians believed in the old days, that a continuous function must be differentiable almost everywhere, say away from an isolated set of points (think e.g. of a piecewise linear curve). This belief was proven very wrong when Weierstrass constructed the Weierstrass function, which is continuous but differentiable nowhere.
It is also geometrically intuitive that, say, the interval $[0, 1]$ is "one-dimensional" while the square $[0, 1]^2$ is "two-dimensional" and so surely $[0, 1]$ is "too small" to fill up $[0, 1]^2$. But first Cantor showed that the two had the same cardinality, and then Peano showed that one could even find a continuous surjection $[0, 1] \to [0, 1]^2$. (On the other hand, we have invariance of domain and also Sard's theorem.)
There is also a somewhat infamous set of examples involving the Italian algebraic geometers, but I'm not qualified to give details. See this MO question and this MO question for some details.
Before pointing out a flaw in intuition, let me show an example of where intuition is right: Suppose we want to, formally, find a length of a curve, say an elliptic arc. An (intuitively) good idea is to approximate the curve with series of line segments with endpoints on the curve. We know (by intuitive extension of triangle inequality) that the length of the curve will be greater than sum of lengths of these segments, but at the same time we should be able to get this approximation as close to the true value as we like. So we will expect this length to be the least upper bound of all lengths of these line-segment-approximations. Turns out, this works for great majority of curves we work with in practice.
Now, suppose we want to do the same trick with approximating surface area. Instead of taking line segments, we could divide the surface into triangles and look at sum of their areas, and take supremum again. Should work just as well, right? The only problem is - it doesn't really work. The problem arises with quite simple example, namely with a circular cylinder. It can be shown that these triangular approximations can lead to arbitrarily large finite area, so with that we would have that simple cylinder would have infinite area, which it (obviously?) doesn't! This is known as Schwarz's paradox, and in my opinion is an excellent example of how our intuition can be incorrect. For your information, there are ways of formally defining surface area, but it's nowhere as simple as defining length of a curve.
Euclid Axiom
For a long time geometric intuition told people this must be a theorem.
The classic "String around the Earth" (or "String Girdling Earth" as Wikipedia calls it). It (roughly) asks this:
(assuming an "Earth" that is perfectly spherical and solid for example... -- incorrect assumptions of course, but this is just a problem for fun)
Most people's intuition suggests that the extra one meter does almost nothing of difference to a string around the Earth, but it actually does! Not to spoil anymore, I will left to each one the job of collecting the data and making the calculations!
