Mapping and conservation law What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus. 
The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.
$ F : ((\Bbb R \to \Bbb R^2) \to \Bbb R^3) \hookrightarrow \Bbb T^4 $.
Thanks.

If I understand correctly you have first a mapping $\mathbb R \stackrel{f}{\longrightarrow} \mathscr P(\mathbb R^2),$ where $\mathscr P$ denotes power set, and then a chain of mappings 
$\mathbb R^2 \stackrel{g}{\longrightarrow} \mathbb R^3 \stackrel{\iota} {\hookrightarrow} \mathbb T^4$ acting on each element in $f(t)$ for $t \in \mathbb R.$
