While I think 3D representations of 4D objects (like the tesseracts shown on that wikipedia page) are interesting, I feel there is something fundamentally distorted about every 3D depiction of a tesseract. You can "represent" higher dimensions conceptually in 3D, but you can't see them in 3D, so it is a misconception for someone to look at a tesseract depiction on a 2D plane, like a computer screen, and say, "ah this is what a 4D cube looks like!"
First off, all the connecting lines need to be the same "length" for it to be a true hypercube. And of course, all the angles would have to be 90 degrees perpendicular to one another. This, alone, simply cannot be shown in 3D. And the same problem for the "rotations" they attempt to show mathematically. I have no solution for this problem, aside from stepping away from trying to understand the 4th dimension tangibly, and instead try to only see it from a conceptual/intuitive standpoint.
Another issue I have with 4D representations: It is often assumed that a 4D object entering our plane would start off smaller, and once it had intersected our plane at its greatest diameter it would then be at full size, and again, as it passed through it would shrink until it disappeared. This is how it is commonly visualized for a lower dimension, such as a 2D plane, if something like say, a 3D sphere, were to intersect that plane. The 2D beings would, theoretically, see a point appear that gradually grew into a line of whatever the diameter of the sphere was.
I have an intuitive feeling, however, that this is wrong, and that the actual appearance in 3D would instead be a gradual fading in of the object, rather than a growing/shrinking type phenomenon.
Interesting that we see this very appearance with things like ghostly apparitions, and UFO's...
In reality, I suppose it could appear in a variety of ways as it intersected our plane, depending on the shape and type of material of the 4D object.
First off, all the connecting lines need to be the same "length" for it to be a true hypercube. And of course, all the angles would have to be 90 degrees perpendicular to one another. This, alone, simply cannot be shown in 3D. And the same problem for the "rotations" they attempt to show mathematically. I have no solution for this problem, aside from stepping away from trying to understand the 4th dimension tangibly, and instead try to only see it from a conceptual/intuitive standpoint.
Another issue I have with 4D representations: It is often assumed that a 4D object entering our plane would start off smaller, and once it had intersected our plane at its greatest diameter it would then be at full size, and again, as it passed through it would shrink until it disappeared. This is how it is commonly visualized for a lower dimension, such as a 2D plane, if something like say, a 3D sphere, were to intersect that plane. The 2D beings would, theoretically, see a point appear that gradually grew into a line of whatever the diameter of the sphere was.
I have an intuitive feeling, however, that this is wrong, and that the actual appearance in 3D would instead be a gradual fading in of the object, rather than a growing/shrinking type phenomenon.
Interesting that we see this very appearance with things like ghostly apparitions, and UFO's...
In reality, I suppose it could appear in a variety of ways as it intersected our plane, depending on the shape and type of material of the 4D object.