The real brain burner is, neither. A sphere isn't spikey at all, actually. It's more than we feel like it should be. However, take that n-dimensional bounding cube and start rotating it, and it'll still contact the n-sphere in the same place, with exactly the same characteristics, but now on different points, even ones that were previously made of nothing but small numbers.

The sphere is "spiky" but you can't actually "rotate" the spikes because they aren't actually spikes. They're an artifact of us poor 3D-ers trying to understand higher dimensions.

"Unfortunately", 3 isn't very many dimensions, and in many cases we're really limited to 2 (our field of vision is more-or-less two, our depth perception is certainly not a direct apprehension of a third dimension but a bit of extra hinting on a fundamentally 2D view of the world), and that ends up not being a good view on to higher dimensions. I think even a 10D being would have less trouble imagining an 11th; they may still not be able to "visualize" it but with a larger sampling of what happens as you increase the dimension size they would be less fooled by artifacts of 0, 1, and 2 the way we are. We do not get very good generalization at 3, things are still dominated by special cases and small exponents.

(Of course, those "special cases" are part of why we are here in 3-space. Also, higher dimensions cause the general problem that the "body" of the life form has an exponentially increasing amount of space in their immediate vicinity where things may affect them, and that inevitably must be growing more slowly than the computing power of the life form contained within itself even though that may seem to be growing quickly to us. Even down here in 3-space we can be blindsided by things; a 10D being would never be able to keep track of their surroundings like we take for granted.)

You're on the right track. My comment alludes to sections, but there's a lot of essentially analogous math that explains the phenomena via shadows/"projections."