Hello Alex Bogomolny,

My name is Chris, and I am a high school student, fascinated by the fourth dimension.
At first I thought it was impossible, even mathematically, but then I thought: If sqrt(x^2+y^2) gives the distance between a point and the Origin in 2D, and sqrt(x^2+y^2+z^2) gives the distance between a point and the Origin in 3D, then perhaps sqrt(x^2+y^2+z^2+w^2) gives the distance in 4D.

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.

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. 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.

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.

Perhaps my ideas are crazy, 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.

Chris

>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