I've got a couple, maybe (depends on your intuition i guess):
In 4d a (topological) sphere can be knotted.
Hyugens's principle: When a wave is created in a field in N-dimensional space, if N is even, it will disturb an ever-expanding region forever (think a pebble hitting a pond's surface) whereas if N is odd the wavefront will propagate forever but leave no disturbances in its wake (think of a flashbulb, or of someone shouting in an infinite space full of air but no solids to echo off of).
Measure a group of humans on N traits and take the individual average of each trait. For surprisingly small N (think 10-ish, but obviously depending on your group size), it's highly likely that no human in your group (or even in existence) falls within 10% of the average in every trait. This is roughly equivalent to the statement that less and less of the volume of an N-sphere is near the center as N increases.
Sometimes called "the flaw of averages". Of course I learned about this from another HN post recently:
Arguably you can do that in 3D, if you accept the horned sphere as a knot. Though I suppose that does raise the question of what you are willing to call a sphere.
Regardless it's an embedding of a sphere that cannot be deformed into a unit sphere so I think the analogy holds.
I've got a couple, maybe (depends on your intuition i guess):
In 4d a (topological) sphere can be knotted.
Hyugens's principle: When a wave is created in a field in N-dimensional space, if N is even, it will disturb an ever-expanding region forever (think a pebble hitting a pond's surface) whereas if N is odd the wavefront will propagate forever but leave no disturbances in its wake (think of a flashbulb, or of someone shouting in an infinite space full of air but no solids to echo off of).