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In 3-D we can formulate laws about spinning things in terms of the one direction that isn’t spinning (h), the “axis of rotation”.
But we can always formulate laws in terms of the two directions that are spinning, regardless of dimension.
But atomic orbitals, which are determined by spherical harmonics (which in turn are more complicated in higher dimensions), would generally be able to hold more electrons.
However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves.
You can see this in ripples on the surface of water (2-D waves).
It so happens that because of the nature of orbits, a stable orbit can only exist in 2 or 3 dimensions.
The “effective potential” representing the balance between the gravity and centrifugal forces of an orbiting object. In all other dimensions planets and moons will always either spiral in or fly away.