A lot of things seem to happen in 5D:
1) There's only 3 regular polytopes in 5D and above
2) there infinitly many Johnson polytopes in 5D and above (cartesian product of polygons to the Johnson solids and above)
3) the demicube ceases to be regular in 5D and is the only dimension where it is semiregular (regular faceted)
4)the unit-radius hypersphere is at maximum volume in a dmension between 5 and 6 but in whole numbers 5 is the largest size.
5) uncomfirmed: the sausage conjecture states that in 5D and above the comvex hull of any number of hypersheres is smallest when they're arranged in a line (i.e. when the convex hull is a sausage)