Note that the OP is about cuts that are nice, cutting polygons into smaller polygons, that could be done using a scissor.
The article you linked is about using magical cuts that be non measurable, that look like magical interlocking 2D sponges. They are somewhat like the figures in the gif, but they actually more weird. It's somewhat similar to the Banach–Tarski Paradox. If you like that, I recommend to watch the video by Vsause https://www.youtube.com/watch?v=s86-Z-CbaHA that explains the Banach–Tarski Paradox with some technical details, but not too many.