They're the same. So square + circle, cube + sphere, 4D hypercube + 4D hypersphere, etc...
But cross-sections don't look the way you think they do. So if you draw a 3D intersection through the 4D case, it looks much like the 2D intersection of the 3D case, etc...
In all such "intersection" diagrams you don't see the spheres in the corners of the cubes. The conceptual diagram on that page showing the high-dimensional case shows the corner spheres and the middle sphere somehow "squished out" in a spiky way.
But cross-sections don't look the way you think they do. So if you draw a 3D intersection through the 4D case, it looks much like the 2D intersection of the 3D case, etc...
In all such "intersection" diagrams you don't see the spheres in the corners of the cubes. The conceptual diagram on that page showing the high-dimensional case shows the corner spheres and the middle sphere somehow "squished out" in a spiky way.
This is not at all what happens.