I apologize beforehand if this question is too philosophical or ill-defined. Hopefully, someone can provide some insight as to whether this concept exists in mathematics or I'm exploring a dead-end.
I am interested in whether a quantifiable measure of "maneuverability" in an arbitrary space exists. The normal Euclidean spaces provide a good way to illustrate my question.
An object in a zero dimensional space has zero maneuverability. An object (say a point) can't move outside of its universe (the identical point). This would seem to be the lower limit of maneuverability i.e. zero.
A one dimensional space seems to have more maneuverability. For example, imagine a infinte line universe with objects like points and/or line segments placed randomly on the line. The points and line segments can now move along the line until they bump into one of their neighbors. Interestingly, it is impossible for the point or line segment to "jump" over their neighbor. Hence they are confined to a predetermined subset of the one dimensional space.
A two dimension space provides even more maneuverability. Say a square and a triangle exist in this space. Now the square and triangle can move in two degrees of freedom. Importantly, now the square for example can potentially completely encircle its neighbor the triangle. Arbitrarily, the square should be able to explore any part of the two dimensional space.
I struggle to formulate whether three dimensional space inherently has more maneuverability than two dimensional space. Naively, it seems to have more maneuverability than 2D but still the best that can be accomplished is that an object can fully explore the 3D space (same as the 2D space). So a key question is does the 3D space have quantifiably more maneuverability than the 2D space?
I realize that to some extent the maneuverability is impacted by how "full" you make the space with objects; however, I am interested in the inherent maneuverability of the space concept rather than an actual configuration specific definition. The key assumption is that solid objects exist i.e. they can't be passed through by another object which in the 1D case prevents an object from arbitrarily exploring its full space.
Does my need to define maneuverability have any mathematical meaning? If so, can it be quantified? If so, how to apply to arbitrary mathematical spaces (say starting with non-Euclidean spaces?)