A tesseract can be represented in normal 3D space, but it’s labelled a 4D object – why is this?
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As an analogy; a cube is a 3D shape, but you can draw a picture of one on a 2D sheet of paper. The 3D representation of the tesseract is the equivalent of the drawing of a cube.
A tesseract is the 4D version of a cube, which in turn is the 3D version of a square, which is the 2D version of a line, which is the 1D version of a dot.
The thing in common with all these shapes, is that each corner connects lines in a L shape, that’s a right angle. The number of lines it connects depends on its dimension. A square connects 2 lines, a cube 3 and a tesseract 4.
You can draw a cube (3D) onto a square of paper (2D) if you’re good at drawing, but you’d only see the cube from one point of view. This is called projection because it’s like you’ve drawn the shadow of the cube on the paper. Imagine the cube was floating between a bright light and the paper.
You don’t really appreciate this when you’re drawing because your brain is amazing, it takes shortcuts and just draws what seems natural in our 3D world.
A tesseract is definitely a 4D object, but you can project it onto a 3D space like drawing the cube. It’s a bit more involved and requires using mathematical rules rather than visualisation but if you think about what information you need to draw a cube (3D) on a square of paper (2D) and how that changes with dimension, then you’re on the right track.
A tesseract could be best represented in a 3D space when there’s a time component involved, making it 4D. Pretty much the only way we’ll ever see it, however, is in a 2D space (computer screen or paper) with time, making it 3D, and our minds just extrapolate the third spacial dimension.
Think of it like architecture drawings where a 3D object is translated to a 2D sheet of paper with no time dimension. There’s one less dimension but contractors and architects (and most people in the general population, really) could imagine what it would mean in 3D space.
People kindof hit a limit at 4 dimensions though, including time, since that’s all our minds have ever had to comprehend.
Imagine holding up a wireframe of a cube to the light so it casts a shadow. That shadow might look like [this](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcR22utzNY-yTYMfzwEhGbi8W02kSGVy1o-8YojjDo9a2kYXv79Xb_BFrJJyvRQf8KrlQe0&usqp=CAU). Rotate it another way and you might get [this](https://www.researchgate.net/profile/Nicole-Panorkou/publication/307617685/figure/fig12/AS:402998749614091@1473093797815/A-cube-or-a-hexagon.png) depending on the angle you hold it. The 3D cube casts a 2D shadow. In math they call this a projection.
The tesseracts you see aren’t a 4D shape because we can’t see in 4 spatial dimensions at once. The closest we can get to representing one visually is a projection. So when you are looking at [this projection of rotating tesseract](https://www.youtube.com/watch?v=g8ypKnaC3xA), you are looking at the shadow a 4D tesseract would cast onto 3D space.
Analogous question: How is a sphere a 3D shape when it can be represented in 2D space?
The key is the word *represented*. [Here’s a sphere](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/1200px-Sphere_wireframe_10deg_6r.svg.png), which can be represented just fine in 2D space. But that’s the thing: it’s a *2D* ***representation***, not an actual sphere! You can’t pick it up and hold it, roll it, or kick it down the road. A 2D representation can look like a sphere, but having the properties of actually *being* a sphere requires 3D.
It’s the same with your question. Yes a tesseract can be *represented* in normal 3D space, but that representation you have seen *is not an actual tesseract*, the same way that a flat image of a sphere is not actually a sphere. Being an actual tesseract requires a 4D object.
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