Especially in urban areas, two locations may be quite close geographically but difficult to travel between. I wondered whether one could create a map where, instead of physical distances, points are arranged according to some sort of travel-time between them. This would be useful for many purposes.
Unfortunately, such a mapping is mathematically impossible in general (for topological reasons). But so is a true map of the Earth, hence the need for Mercator or other projections. The first step in constructing a useful visualization is to define an appropriate Travel-Time metric function. Navigation systems frequently compute point-to-point values, but they are not bound by the need to maintain a consistent set of Travel Times between all points. That is our challenge – to construct a Travel-Time metric.
A Place of Sand 