# Size of a multi dimensional Labyrinth

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 $1km^2$

• 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. Commented Jun 26, 2016 at 12:56