this post was submitted on 20 Oct 2024
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A lot of things seem obvious until someone questions your assumptions. Are these closed forms on the Euclidean plane? Are we using Cartesian coordinates? Can I use the 3rd dimension? Can I use 27 dimensions? Can I (ab)use infinities? Is the embedded space well defined, and can I poke a hole in the embedded space?
What if the parts don't self-intersect, but they're so close that when printed as physical parts the materials fuse so that for practical purposes they do intersect because this isn't just an abstract problem but one with real-world tolerances and consequences?
Yes, the paradox of Gabriel's Horn presumes that a volume of paint translates to an area of paint (and that paint when used is infinitely flat). Often mathematics and physics make strange bedfellows.