How is a tesseract 4D if it can be modelled in 3 dimensions? Attached is an image of a tesseract. It consists of two cubes placed diagonally adjacent to each other, with each corresponding vertex connected with a line.
My question is, how is this a four-dimensional object if it can be modelled in 3D? I don't have a 3D printer or anything to test this, but it seems pretty easy to be able to make two cubes and just join their respective corners.
 A: The tesseract is a 4D object and can be though of as the surface generated by a cube when it is translated. In reality however, if you were to print a tesseract using a 3D printer, this will still be 3D since embedding a 4D surface in 3D is not feasible. Making 2 cubes and joining their respective corners will still result in a 3D surface, unfortunately. These are the limitations in living in a 3D world!
A: You could 3D-print two cubes with their corresponding corners joined, but that wouldn't be a tesseract because in the tesseract ALL the angles between the edges are right angles.
A: What you're seeing in three dimensions is not a four-dimensional hypercube but a projection of a four-dimensional hypercube into three-dimensional space.
Suppose you have a glass three-dimensional cube that's perfectly transparent except the edges have been marked so you could see a shadow if you were to shine a light through the cube. Now shine the light from a concentrated source directly along a fourfold axis towards a white table passing behind the cube and perpendicular to this axis (thus parallel to the faces punctured by the chosen axis).
The shadow formed by the edges then constitutes not the cube but a two-dimensional projection of it, and it appears as two concentric squares connected by the remaining four edges like a picture frame. The projection commonly shown for the four-dimensional figure is the next higher dimensional analogies of this "picture frame".
Exercise for the reader: Suppose you have not a cube but a regular tetrahedron and you shine the light along a threefold axis. Draw the resulting projection and then suggest what a projection of the pentachoron (the four-dimensional simplex) might look like.
