Any representation of an object in fewer dimensions is actually a representation, and not a real thing. That's how you can draw a cube in 2D, but you lose some aspect of it - see how you can't see the opposite side of the drawn cube. The drawing is an orthogonal projection of the image - just like a shadow, really.
4D can be represented into 2D, but instead of losing only one dimension, it actually loses 2, making it even less accurate a representation. It is a valid representation, nonetheless.
So, yes, that is the drawing of a tesseract.
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Funny as physics can be, you can actually substitute one of the dimensions in a projection for a dimension of time. That's what allows us to represent a perfect cube in 2D - you make it an animation (making use of the time dimension), and rotate it around. Unfortunately, we still haven't invented hologram monitors, but once we do, it'd be much easier to represent tesseracts.
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Regarding the original question - what's the use of a fourth physical dimension. Well, You know you can actually work with each dimension separately, including movement even. I don't see any other practical use for it right now, but maybe it can help in explanations for fundamental physics, the origin of the universe, etc. Finally, it could be used in art
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