>then perhaps sqrt(x^2+y^2+z^2+w^2) gives the distance in 4D.

That is definitely the case. In fact, in spaces of higher dimensions (beyond 4) the distance is measured in the same manner.

>I started searching the internet, and after a while I found
>your Java aplets on
>http://www.cut-the-knot.org/ctk/Tesseract.html. I find this
>absolutely fascinating.

Thank you. Truth be told, I, too, spent a lot of time with those applets.

>I was wondering: If a tessaret can be rotated in four
>dimensions, perhaps it is also possible to let objects
>interact with eachother in 4D. perhaps it is possible to
>assign values to an object: depth, height, length,
>hyperDepth.

Of course.

>and then use the same programs used for making
>for example three dimensional gaming programs, and adapt
>them to the fourth dimension, to simulate a real time four
>dimensional environment.

Yes, probably. Depends on how this programs are written.

>Also, perhaps it is possible to then develop a virtual
>camera within the program, that 'films' this virtual 4D
>environment, and projects it onto a 2D image, for your
>screen, the same way we use it to film virtual 3D
>environments in for example computer games. Is this
>possible, to film it in 2D? or would it have to be 3D,
>because 2 dimensions difference is just too much? we can
>show 3D on 2D images, but not 3D on 1D, so perhaps 4D on 3D
>can be done, but 4D on 2D cannot.

Wikipedia has a good article on 4d and reeferences

http://en.wikipedia.org/wiki/Fourth_dimension

>Perhaps my ideas are crazy,

No, not at all. Modern physics, based on the string theory, implies that the world we are living in is actually 20D or so.

>but I have been thinking about
>this for very long, without being able to get my mind off of
>it, and I just wanted to share it.

Thank you