From Wikipedia: https://en.m.wikipedia.org/wiki/Hypercube "4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract)."
How do we draw an orthogonal line to the three orthogonal linas that we have?
Draw a square on paper. The lines will be orthogonal. Draw a cube. The third dimension won’t be orthogonal. It can’t be. But, that doesn’t mean a third dimension doesn’t exist or can’t exist.
The same thing happens with a hypercube. The fourth dimension won’t be orthogonal on paper. It won’t be orthogonal in three dimensions (you can build a 3D projection of a hypercube). This doesn’t mean it isn’t real or can’t be real.
Whether you think of 4D space as something real we don’t have access to or something imaginary isn’t really too important. It’s just very helpful to realize it won’t have the concreteness of 3D objects in 3D space for you because you don’t have direct access to it.
Sagan's video proves that it's always helpful to investigate abstract concepts with an experiment.
But the video also raises an important question: Can we derive true conclusions from wrong assumptions?
Here the assumption that the "flatlanders" are really flat is wrong.
Sagan notices that too and he says that his objects are not really flat, but they have thickness.
Now these little cutouts have some little height
but let’s ignore that, let’s imagine that these
are absolutely flat.
So the thickness of his objects are never zero. His flatlanders has a notion of third dimension because they have the third dimension. Their height is not zero.
This is the typical rhetorical sophistry widely used in physics. We may also call it equivocation on the word zero. Here, Sagan defined the word zero both as nothing and something and he wants us to go along with this equivocation.
You cannot have non-zero height and zero height at the same time. If you have zero height you do not exist. A door is either open or closed.
Instead of flatlanders, Sagan may have used shadowlanders. Shadow is closer to two dimensional objects but even shadow has thickness.
So, Sagan's assumption that his objects have no height is definitely wrong. Can we arrive at a correct conclusion from this wrong assumption? I guess not.
Also it is clear that when we try to transform a 3-D object into a 4-D object, the object's morphology changes. A cube becomes something else. A cube does not cross the dimensions as a cube.
This also raises questions about physicists' claim that we are living in a 4 dimensional continuum called spacetime. If so, our current morphology in the 3-D world and 4-D world of spacetime cannot be the same.
Then the question is, is our present anatomy 3-D anatomy or 4-D anatomy?
Are you sure? In a 3-D coordinate system all axes are orthogonal because they make 90 degrees angles with each other.
Edit: I see that you mean "cannot be drawn as orthogonal lines on paper." But, in reality they are orthogonal e.g., when I construct a 3-D model of a cube.
They are orthogonal.
> I cannot conceive a geometrical image of higher dimensions.
This is normal, and essential to the point of the article. If you could visualize 10-dimensional space, it wouldn’t be so counterintuitive.
Try looking up images and videos of 4D objects projected into 3D and 2D. That might help. Hypercubes are maybe the easiest.