I think of high-dimensional spaces in terms of projection. Projecting a 3-dimensional space onto a 2-dimensional space loses information and the results depend on perspective. Same with an 11-dimensional space being projected onto a 10-dimensional space.

I find that this metaphor works pretty well for visualizing how a vector-space search engine represents how two documents can be "similar" in N-dimensional term-space: look at them from the right angle and they appear close together.

Yeah, stopping that need to visualize everything is one of the mechanisms usually adopted for working in high-dimensional space.