Ok, imagine having a finite line segment from point (a) to point (b) in $R^2$ . I'm not familiar with mathematical terminology of this kind, but let me state that the line we began with is the geometric interpretation of $A^1$. The geometric interpretation of $A^2$ is a square with sides $A^1$. We could go on by saying that $ A^3$ is a cube with edge length $ A^{1}$ again. 


I wonder what the geometric interpration of $ \sqrt[2]{A}=A^{1/2}$ would look like. Is it a (straight) line? Is it constructible? Of course, we could extend this question by asking ourselves what $A^{k} $ would look like, in which $k \in \mathbb{R}$ or even $\mathbb{C}$ . When $k>2$, $A^k$ probably isn't constructible anymore on a sheet of paper, but one can still think about how these constructions of $A$ would 'look like'.


Thanks in advance, 


Max Muller


P.S. I realize I ask more than one question now, which is an indictation I don't know a lot about this (yet) and I'd like to know more about this subject. Should this be a community wiki? Feel free to modify the Tags, I don't know how to classify this question exactly.



The Root of a Line
Here's my proposal for the square root of a line segment.


It's the Cantor set obtaining by repeatedly 
splitting the intervals into four, and removing the two middle pieces.
When you take the cartesian square of that space, you obtain something whose projection
is exactly an interval:


[alt text http://www.staff.science.uu.nl/~henri105/drawing.pdf](http://www.staff.science.uu.nl/~henri105/drawing.pdf)


