I'll have a limited space like $1 km^3$ that have (through handwaving) more then three dimensions. But anyone entering it would perceive it as a three dimensional space. A little bit like this video, but more or less a flat version.

This results in two people walking around the same tree in clockwise or anti clockwise direction would end up at different positions.

What I would like to know is how big is my surface area on which I can walk at maximum inside the multidimensional space.

My first guess was, if I have a $1 km^n$ hypercube and ignore height I would have a surface area of $1^{n-1} $. With an 4 dimensional example I would have a surface area of $1km^3$. Unfortunately this is a unit of volume. So my formula is missing something to get $km^2$. I would also assume that the result would be something greater than $1km^2$


Well, what you seem to be after is the number of 2D faces in an nD-cube? A 3D cube has six 2D faces, and a 4D cube has 24 2D faces. You can find a table and the mathematics here.

| improve this answer | |
  • $\begingroup$ Some how I feel dump now. It seems to be so obvious. What I actually searching is the number of faces someone can walk on. For a 3-cube that would be one, for a tesseract it would be 6. It seems that the number of faces someone can walk on for an $n$-cube is the number of faces of the $n-1$-cube. $\endgroup$ – lokimidgard Jun 26 '16 at 12:56
  • $\begingroup$ Anyone can miss something. Don't worry about it. But you need to think a bit more carefully about gravity for the number of faces that can be walked on; I'm not sure your answer is right. $\endgroup$ – John Dallman Jun 26 '16 at 13:34
  • $\begingroup$ hm... I need to think about this. In my head I'm unwrapping the multidimensional space to a simple bigger 3d space and applying normal physics to it. But your right that this is most likely wrong. $\endgroup$ – lokimidgard Jun 26 '16 at 14:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.