[lmm]

Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.

[thaumasiotes]

Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.