One overlooked fact that may help with visualizing a tesseract is that each dimension is at 90º of each other; if you take a line and move it in a direction at 90º of their original one (up/down from left/right), you create a plane with 2 dimensions.
If you take this square and move it at 90º from the plane, you create a cube in the third dimension….
…Now take this cube and move it in a direction 90º from the third and you’ve arrived to the fourth dimension (and so on and so forth)
You can “preview” higher dimensions in a lower one if you make the move at 45º in the other ones, for example you can move a plane 45º in X/Y and now you have a “shadow” of a cube in 2 dimensions, if you move a cube 45º in all three X/Y/Z dimensions you get a 3 dimension shadow of a teseract, which is the popular image of a cube inside another cube, If we were able to see the forth dimension, all sides, interior and exterior would be of the same size and at 90º
Trippy, right?
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