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Stacking and crushing

You see figures being crushed while preserving their area. Hyperbolas are involved. The two initial squares are in the golden ratio for this to work. One is a unit square while the other has a side of length phi=(1+sqrt(5))/2. See Edmund Harriss's video below for the context.

Video by Edmund Harriss and Pierre Arnoux

GIF version of the animation

GIF version of the animation

What it does to the pythagorean tiling.

Animated GIF

Animated GIF