Dissection Fallacy

A dissection fallacy is an apparent paradox arising when two plane figures with different areas seem to be composed by the same finite set of parts. In order to produce this illusion, the pieces have to be cut and reassembled so skillfully, that the missing or exceeding area is hidden by tiny, negligible imperfections of shape.
A strikingly simple and enlightening example can be constructed by dissecting an  checkerboard in four pieces as depicted (left figure). The middle and right figures then seem to demonstrate that the same pieces can give rise to two different polygons having area and , respectively. This would imply that .
However, a closer look at the slanted sides of the trapezoidal and triangular pieces shows that they cannot be aligned as implied in the above fallacious illustrations. In fact, they are the diagonals of two dissimilar rectangles of sizes  and , respectively, and hence have distinct slopes. But the difference of the ratios ( versus ) is too small to be perceived by the eye.
Note that the dissection cuts the sides of the  squares according to the proportion 5:3. The illusion becomes even more effective if the numbers 3, 5, 8 are replaced by a triple of higher consecutive Fibonacci numbers.