erratic Why can't you crease the surface? Is it an arbitrary complication for chronic mathturbators or will it break the universe?
Cyberblah Yes. I trust that there's a mathematical reason why you can't, but my first instinct was that it's just a bullshit rule to make things hard.
GravidWithHate I'd guess that the equation describing the surface has to be continuous and differentiable.
If I'm remembering my math right, that means if you measure the slope of the surface at any given point, and then create a graph of the slope, the graph must be a smooth line or curve with no gaps. A crease or tear creates a point where the slope will suddenly jump from one value to another, or where the slope is undefined.
I don't know why this is necessary, as I never got this far, but I'd further guess that discontinuous curves do nasty things to whatever transformation functions they're using.
gmol The "no creases" (or corners) constraint is there because the result isn't counter intuitive without it.
The constraint has to do with "immersions", that is a maps between spaces that are bi continuous and whose derivatives are injective (don't map two points to the same corner/crease).
Chalkdust It makes sense to me but I'm not sure how well I can explain it... that little pinchy loop would just be getting infinitely smaller and smaller but never actually "snapping" around to the other side.
Redlof Because a system that uses creases isn't useful for modeling the curvature of space time. So I guess that means maybe the Universe would break?
It is simple to create function that introduces creases into a volume; that's the problem, it is simple. If we don't care abou the constraint, the solution to "everting" the sphere would just be to pull it through itself and no one would care.
When we do introduce the "no creases" constraint, most of us end up being very surprised that it is possible (as this video shows it is); and that's why this is interesting.
Why does anyone care? It is interesting in the same way other neat little mathematical facts are interesting e.g. there are more irrational numbers than there are natural numbers.
Billy the Poet It's in the part of the Torah that nobody reads.
pastorofmuppets In my limited research experience, the answer for "why is that constraint there" has always been "so I'd have a problem to solve."
Triggerbaby HOW ABOUT A HYPERSPHERE HUH? HOW ABOUT THAT SMART GUY?
TheSupafly I can't believe this video managed to get me to understand this.
Comeuppance I can't believe that I sat through a 20-minute video, with rapt attention, explaining how to do something - within the constraints of arbitrary rules - to a solely hypothetical object.
