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Shadow of a Rotating Tesseract
I think we're talking about two kinds of higher dimensions here. The spatial dimensions and the 10-dimension theory that Rob shows in the first video. I find both to be perfectly reasonable and understandable but I like to think about spartial dimensions more.
Its funny imagining higher spartial dimensions, I've read quite a lot of articles about it and even tried drawing (Yeah drawing shadows of tesseracts is really fun :D) but to summarize the point, try thinking about us in the 3d world being 4d objects and flat papers being the 3D world (Like Sagan does) and continue from there on. If you think about it long enough, your brain will litterally hurt. You might come to such conclusions that there might exist 4 dimensional creaturs that we just can never come in contact with, or other 3d worlds that crosses in 4-dimensional space right next to us (Like how two 2d-worlds, papers, lying on top of each other never can contact one another) that we can never contact because we're stuck in 3D. And then we havnt even begun on 5D and 6D and so on.
Amazing stuff