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PMCIIII

The second dimension extends a line at 90 degrees from the first, creating a plane.

Adding yet another line 90 degrees from the first two lends a cube. You may spare no great expense visualising the corner of a 'box' to see the corner has three line elements; all at 90 degrees from each other, that lend it the property of the cube.

Now here comes the tricky part: Add another line to the corner of our cube; 90 degrees in relation to the other three lines, and we obtain the hypercube ("Tesseract"), which is the fourth-dimension analogue of our regular, ol' cube.

Now I'm asking if the human mind is capable of visualising something like this. I have a hypothesis that answers this in the affermative, but I still need help. But first, some background.

- A 3-D sphere entering then leaving the second dimention would appear as a circle that starts from a point, grows to the max diameter of the sphere, then shrinks back into nothingness
- Likewise, a hypersphere entering our 3-D space would appear to us as a regular ol' sphere beginning at a point, growing to the maximum diameter of the hypersphere, then decreasing in size to nothingness.
- A 2-D organism would observe the sphere entering his universe as an object with depth and height, but no width. We may interpret this as an infinitely-thin sliver; an infinitely thin slice of a 3-D sphere.
- As 3-D organisms, we percieve the 4-D hypersphere entering our universe as a regular ol' sphere having height, depth and width but not omega (omega is the fourth-dimensional line element. Thus, we have width, length, height and omega). The fourth-dimensional organism would interpret the sphere we see as an infinitely thin slice of his hypersphere.

Interesting properties:

When we, as 3-D organisms, look at any surface in our universe, what we're actually looking at is second-dimentional *planes.*Just look at a box if you don't believe me: The box you're looking at is consisting of three

*planes*joined in 3-space to form a cube.

Likewise, the organism in 2-space, when looking at his square, is actually looking at lines. Two of them, provided the corner of the square is seen.

It stands to reason, therefore, that any fourth-dimentional organism looking at the surface of his hypercube would actually be looking at

*cubes.*That's right: The surface of the hypercube is the

*cube,*just like the surface of our cube is the

*plane.*

Now my question is this: Since we are 3-D organisms with 3-D imaginations and 3-D powers of visualisation, is it possible for us to actually visualise a hypercube? Or, does our 3-D brain prevent us from fathoming such an object? Remember now, regardless of size, our cube (think box or something) to the 4-D organism is an infinitely thin sliver of his 4-D hypercube. Think of the possibilities: I can stuff a box of cereal into a box that measures 14" by 10" by 3". But in the fourth dimension, I can actually stuff an infinite number of cereal boxes in the hypercube, even though the dimensions of the hypercube are finite (14" by 10" by 3" by X" omega)!

I have a lot more to say on the subject. But do you guys think that visualising such a space is possible? I mean, Dr. Sagan once was quoted as saying he can visualise 4-space. Then Dr. Hawking turns around and says, "I have a hard time visualising the third dimension -- let alone the fourth!"