3d >2d is the easiest one to grasp, since we deal with it every day. A photo, for instance, could be seen as a projection of a 3d object onto a 2d plane (film, screen, paper, etc). Given just that image you might be able to guess that it a cube, but it in a weird perspective where it might not actually be a cube. Also, there parts of the cube hidden from view. But we already dealt with real http://goal365news.com/2017/12/17/while-investigating-sakais-disappearance-with-a-bprd-team/, live, cubes all the time, we have plenty of intuition with them now. But the tesseract? Ugh. 4d hurts. So, we do the same thing that we did with a cube. We project the tesseract into a lower dimension (3d space) to get one perspective on what it looks like. We can look at it (indeed, walk around it, since it 3d), but it not a lot of information about what it really like. So, again like the cube, we rotate it around some point and take a bunch of projections. Imagine that it really is just a bunch of quadrilaterals undulating in 2d. Holy shit, what the hell is this thing? They growing and shrinking and appearing and disappearing out of god damn nowhere! What the hell is the logic of this mess? Oh wait, it stuff just coming and going out of view or moving to a different angle from the camera. I can easily construct a cube in 100 spatial dimensions. That just the collection of all points (x1, ., x100) with xi equals l or l (l being the half length of a side). Making a 2D projection out of this is equally trivial but it will be rather boring to look at.
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