comprehending beyond the 3rd Dimension

[Ultimater]

I couldn't help but wikipedia a 4th or 5th dimension.
http://en.wikipedia.org/wiki/Fifth_dimension
I had a good laugh at:
http://upload.wikimedia.org/wikipedia/en/5/5e/Dice_analogy-_1_to_5_dimensions.svg
in case your browser cannot render SVG images, here's a PNG:
http://i42.photobucket.com/albums/e317/Ultimater/public_stuff/5d.png
To me, expressing a 3D cube on a paper is already illegal since the lines overlap. If you've ever tried creating an image with a 3D graphics program, you'll know how 3D images are composed of a 3D pixel in different arrangements among the x,y, and z axises. Since the computer screen can only project a 2D segment of the 3D graphic, it is essentially just displaying a side of a cube and to view the whole cube would require rotation. I've been thinking about the possibility of creating a 4D graphics program which uses a 4D pixel as a unit which can be arranged among 4 dimensions. Then you'd be able to move the object among any of the 4 dimensions to get a new 2D view of the object. I figure there are four points on a Tetrahedron:
http://i42.photobucket.com/albums/e317/Ultimater/public_stuff/Tetrahedron.gif
so each point can represent a dimension of a 4D realm.
Then I came across the Tesseract:
http://i42.photobucket.com/albums/e317/Ultimater/public_stuff/Tesseract.gif
The way it is folded is even weirder:
http://i42.photobucket.com/albums/e317/Ultimater/public_stuff/Tesseract_fold.gif

[NogDog]

If you want to blow your mind some more, look into Calabi-Yau manifolds (http://en.wikipedia.org/wiki/Calabi-Yau_manifold), one use of which is to describe the 5-6 (+/-?) additional, "compactified" spatial dimensions posited by various string theory and gauge theory versions to exist in addition to the four dimensions we experience at normal scales (3 spatial + 1 time dimension).

[Ultimater]

The tesseract is already weird enough when viewed from a random angle
http://i42.photobucket.com/albums/e317/Ultimater/public_stuff/Tesseract_angle.png
and that is supposed to be a four-dimensional hypercube!? Its just weird... it looks abnormal.
Maybe if you don't gaze at the see-through inside so much you can make more sense of the object by the 4 of the 8 surfaces/sides visible to the plane.

The thing about matter is that it is contained within 3 spatial dimensions and time isn't a dimension what provides space for matter. So technically a tesseract should look weird since matter is contained within the 3 spatial dimensions.