Infinite Suburbia is a Euclidean plane, P. All residents live in open unit disks which, like caravans, can travel around but are stationary most of the time. When stationary, these disks lie in a fixed subset S ⊂ P which is a countable, disjoint union of open unit disks, as dense as possible in P, subject to the following caveat:
Residents of Infinite Suburbia like to travel. Therefore, if any disk d1 in S is currently vacant, any other disk d0 should be able to travel from its current stationary position, through the "road network" P\S (where, for the purpose of travelling, S is considered not to include d0 or d1), to the position of d1.
Furthermore, the manner of travel is constrained: associated with P is a (fixed) "differentiable" vector field, such that any point p ∈ P has an associated velocity v(p) (I guess by "differentiable" I mean that both of its components have a gradient). In particular, if p is the centre of a disk in S then v(p) = 0. If a disk is moving and its centre is currently at p, then it must travel with velocity v(p). Since v is "differentiable", it has a corresponding acceleration field, whose magnitude is bounded by A.
The city planners know that, being infinite, the Suburbia's residents are liable to want to travel arbitrarily far. Therefore, the main concern when originally deciding on S and v (apart from the density of S) was that expected journey times should be "asymptotically nice", in that if E(x) is the expected journey time between two disks which are no more than x distance apart (over all such ordered pairs in S), then E(x) should be asymptotically as small as possible. For variations on the problem, consider E to be the mean square journey time, or the maximum journey time.
The question, then, is: how to choose S and v? This will presumably be a balancing act since E(x) can be made much lower if S is sufficiently sparse. But I'm imagining a compromise looking like some sort of fractal, much like the road system in real cities, but much more ordered and extending to infinity. Assuming an exact optimum can't be found, what strategies will guarantee at least a decent approximation? Am I on the wrong track with the fractal idea? Is the problem, in fact, not sufficiently well-defined?