Projecting maps by travel time

Maps where something else is used instead of distance or area (e.g. population, GDP, travel time) are called cartograms. Apparently maps where distances correspond to travel times are called linear cartograms, but this seems like kind of a bad name. I'll call them travel-time cartograms instead.

You could think about different parts of the 2D surface of the Earth as having different speeds when you cross them by some mode of travel (trails are easy to walk on, mountains are hard to walk up but easy to walk down, and water can't be walked on at all), and try to make a map that way. The mountain case tells us that when travel time depends on direction, a travel-time cartogram can't be made, i.e. the distance between the top and bottom of a mountain must be two things at once. I guess you could take an average, but I don't like that much -- it seems to destroy or hide a lot of information. Instead, I'll consider different modes of travel, and assume that direction doesn't matter.

I think three modes of travel (plane, car, walking) might be the sweet spot in terms of accuracy vs. difficulty, but two modes of travel is complicated enough to bring out the conceptual issues, and airplanes change our travel time around the world much more than cars do. Airplanes are pretty weird -- do you consider the points along the plane's trajectory to be quickly reachable, or do you imagine the plane disappearing from one airport and then reappearing at the other end? I'll use the time-delayed teleportation model, since I can't really get places by parachuting out of a plane. So, my first-approximation travel-time cartogram should have this property:

The distance between any two points is the minimum travel time by a combination of air travel and "land travel" (a continuous form of travel at 30 miles per hour, a rough average of car and foot travel)

What happens when I try to build this map?

• Major cities and their neighborhoods should be near each other.
• Far from airports, the map should approximate a globe (since only land travel is relevant there).
• Any shortest line between two points (along the surface) should correspond to the shortest trip between them (i.e. it should point-for-point cover the same route as the shortest trip).
• There are points on the surface between the major cities that don't correspond to points on the globe, so that travel between airports isn't immediate, but you also can't parachute out of the planes.

Instead of working with a globe, let's start with a circle (a slice of the sphere, where travellers can move only along the circle between points) and add two airports that can be moved between quickly. What we need to do, it seems to me, is bend the circle through the third dimension until the airports are near one another in 3D space, then add a line between them of length equal to the travel time. It's important to note that distance on this cartogram isn't measured in raw Euclidean terms, but instead by distance along the surface itself (just like on a globe, you measure distance along the surface instead of tunneling through the earth). If you add more airports, then it's like folding the circle up so that many points on its circumference nearly meet.

In fact, if we assume that airplanes all travel at the same speed along this circle, then you can make a separate surface corresponding only to air-travel, with points corresponding to airports and edges corresponding to flights, and this surface will be curved overall like a circle. So, we have a large "land-travel circle" and a smaller, nodes-and-edges "air-travel sphere", and we can make the final map by folding the larger circle through 3D space so that its airports meet the smaller circle's airports.

To get back to the real world, "all you have to do" is make a land-travel globe (which looks like a normal globe), an air-travel "globe" (a smaller web of connections between airports that is overall curved like a sphere), then fold the land-travel globe through the fourth dimension so that its airports meet up with the air-travel globe's points. That's pretty awkward, because now we have a 4D map that is going to be really hard for humans to read and get intuitions about!

To flatten a globe into a map in a way that lets humans understand distances, we sometimes put a grid on the sphere as guidelines for humans. Can we do a similar thing here -- put a grid on our 4D map, cut and flatten it, and then print it out? Not sure, but that seems like what we want to do!