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?)

  • $\begingroup$ It seems like what you're looking for is something like the minimum dimension of an object that you can remove and disconnect the space. For example, in 2 dimensions, removing a point always leaves the space connected, but if you remove various one-dimensional objects you can break it into two pieces. In other words, the one-dimensional object provides a "barrier" which you can't move across. In general this will just be one less than the dimension of the space, so it doesn't give any more information than that. $\endgroup$ – mdp Apr 12 '12 at 16:21
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    $\begingroup$ A couple of thoughts: (1) You can tie a knot in three dimensions that can't be unknotted, whereas all knots in 4 or more dimensions are deformable into the trivial knot (the 'unknot') i.e. they are topological circles. Topology may provide a starting point. (2) You can move two $n-2$ dimensional objects past each other in $n$ dimensions, whereas you cannot in $n-1$ dimensions (if they have infinite extent) so in that sense $n$ dimensional spaces are more 'manoeuvrable' than $n-1$ dimensional spaces. $\endgroup$ – Chris Taylor Apr 12 '12 at 16:24
  • $\begingroup$ @MattPressland Sorry, I obviously need to study topology. Wouldn't a finite length 1D object (a line segment) still leave the space connected? (I guess the line segment is topologically similar to a point?) It seems you would need a infinite line to disconnect the space. But my model doesn't allow for infinite objects within the space. $\endgroup$ – Jason Waldrop Apr 12 '12 at 16:46
  • $\begingroup$ Not every 1-D object will work, it's just possible to disconnect the space with 1-D objects, but impossible with 0-D objects. Even if you insist upon finite length, a circle will still disconnect a 2-d space. $\endgroup$ – mdp Apr 12 '12 at 17:11
  • $\begingroup$ @MattPressland OK, this helps alot. I see what you're saying that a finite circle disconnects a 2D space but that assumes the circle is fixed in space. My "maneuverability" model would allow the two objects to "work together" such that the circle could "move out of the way" allowing say a square to attain the points in space previously occupied by the circle. Occupying arbitrary space is only part of what I'm interested in. In some sense I can't articulate very well, the objects need to be able to "jump" or "pass" over each other. In this respect, 2D and 3D space seem identical. $\endgroup$ – Jason Waldrop Apr 12 '12 at 17:28

You are asking about topology of configuration spaces, especially the fundamental group ($\pi_1$), connected components ($\pi_0)$, and dimension. The following assumes knowledge of what a group is, but not any knowledge of topology.

There are topological quantities that measure different ways and extents to which a particular space can be connected, or ways that one space is more connected than another. These quantities are not necessarily numbers, but they are defined purely in terms of the topology of the space without reference to maneuvering of rigid objects. For example, the plane and sphere are "simply connected" because loops can be shrunk to a point, but the circle has only the weaker property of being "connected", where any point can be continuously moved along a path to any other. This connectivity difference is reflected in quantities such as the fundamental group of the space (among others) being different in the plane/sphere case and that of the circle. Introductory books on topology, cover this material. There are also higher connectivity measures such as homology or Betti numbers that account for "holes" in the space.

Motion of objects makes sense in spaces that are placed inside n-dimensional Euclidean space, or inside some other homogeneous space with a family of motions (such as a sphere, a flat torus, a hyperbolic plane, etc). Then, for a given list of objects, there is the configuration space of their possible placements, and the maneuverability is asking for the topological connectivity measures of the configuration space:

  • the dimension of the configuration space is the number of degrees of freedom to make a small (local) motion of a typical placement of the objects

  • if the configuration space has only one component, it means any configuration can be continuously maneuvered into any other.

  • if not, there are types of configuration that cannot be reached from each other, such as the two placements of the square and triangle both of height 1, in a long tube of diameter 1.001.

  • If the fundamental group of any component of the configuration space is non-trivial, it means that there are qualitatively different ways for the square and triangle to dance around.

  • In the plane, there are also qualitative differences in the dances with different numbers of times that the objects rotate around their own centers, and this reflects the possibility of topologically nontrivial paths in the configuration space of a square or triangle with one point pinned in place. Where there are geometrically relevant subgroups of the group of motions, the topology of the pinned (or other restricted) configuration space can play a role, or tracking of motions in subgroups or quotient groups, such as center of mass motion.

  • a geometry-free, purely topological notion of a measure of freedom to maneuver, is to take, for each $n$, the configuration space of $n$ ordered points in the space, and consider the topology of these configuration spaces. This is the same as the limit of infinitesimally small objects, but it requires no embedding in a Euclidean or geometric space.

Robotics and motion planning are subjects where this type of theory is used.

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