That Eureka Moment

One of my long-term projects has been developing a mathematical model for scheduling in Jarrett Walker-type transit systems. Today, I made a breakthrough--

I had previously realized that a model of such a system would yield two intersecting planes in my defined 3-space, and that desired overall frequencies could be model by stacking instances of these planes atop each other. But I'd been at a loss as to how to model the routes within the planes (other than the fact that they were vectors).

What I realized today, however, is that the planes are actually a set of null space vectors; this means that a prescribed eigenvalue in the null space will seed an eigenvector that corresponds to that value. What that means is that the entire system can be modeled with nothing more than the equations of the two intersecting planes, and the set of eigenvalues that yield the eigenvectors corresponding to the known bus routes.

Fortunately, we remain in vector space so far, but it appears the apparatus I'm constructing will wind up generalizing into a differential equation, to accommodate "gridlike" systems which attempt to install a mass transit grid even over non-grid street networks.