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If there were an infinite number of dimensions, would that mean there was also a path, if we could find it, from any point in 3d space to any other point in 3d
by ineedasername 6y ago
If there were an infinite number of dimensions, would that mean there was also a path, if we could find it, from any point in 3d space to any other point in 3d space that was of arbitrary length?
- pas 6y agoIn general no unless there's one that's warped just the right way. Which is unlikely. (Also it comes down to cardinality of the involved infinities. If we have countably many infinite dimensions but uncountably many points in each one of them, then you have a "lot more" point-point pairs than dimensions.)
- caf 6y agoAdditional dimensions don't give you additional ways to be closer to something, they give you additional ways to be further away. If we're displaced from each other by X, Y and Z units along axes in the usual 3 spatial dimensions, the closest we can be is if we're at a coincident position in each of the additional dimensions. Any displacement along those additional axes just moves us further away from each other.