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.
For example the volume (hypervolume) gets concentrated close to the surface of the sphere when dimension grows. For example, if you have symmetric multidimensional probability distributions around the zero it becomes weird.
True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…
