So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? Could this be done with any number of bulges, including 2?
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No, 8 isn't special beyond it being the choice they made for that specific video.
The software the group wrote to make that video allowed you to choose that parameter arbitrarily. I bet if you spent some time digging you could find that software somewhere on the internet, and create your own eversion videos with a different choice of the number of corrugations.
You can find a (modified) version of the source code here: http://profs.etsmtl.ca/mmcguffin/eversion/
as well as commentary from Silvio Levy on the choice of number of strips.