I’ve been a little lax with the blogging lately. I’m sure you’ll forgive me. Before getting to the main topic, I have a couple of quick announcements:
- I have an column coming out Wednesday or Thursday this week. I’ll be talking about the prospects of traveling faster than light. If it sounds a little familiar, it should. My very first column was “What happens if you’re traveling at the speed of light and turn on your headlights?
This one’s a bit different. I’m looking for physically motivated FTL drives or schemes. So far, I’m planning on doing Alcubierre drives and wormholes. If you have any additional suggestions, please send me.
- I think I finally have a good title for my upcoming book on symmetry. I think I have it. Ready?
“Antimatter, Evil Twins, and Grand Unified Theories: How Symmetry Rules the Universe”
Enough of that. On to the main topic.
So, first off, forgive me if this is some well-known topic in urban anthropology, cartography or some other field, but as you may recall, I have an abiding interest in maps and map projections. I’m also very interested in traffic, and the dynamics of cities.
This gets the “technical” tag, but don’t fret. There’s only one equation.
I’ve started thinking about an interesting new way to map cities. The idea is that the distance between points in the map would minimize the travel time between them. I am particularly curious about the prospect of getting my hands on a mass transit map to see how connected various parts of a city are.
My friend, Rich Gott, who commented:
Well the trouble is that with many points the distances between all pairs cannot be shown without errors. The distance error minimizing scheme that I developed with Mugnolo and we published in that other paper I to cartographica would be optimal. It puts little springs between the points and sets the potential energy between the pairs equal to the logarithmic distance error. Substitute travel times for distances and you would have your map. The forces will bring the points to the lowest energy state if one damps the motion at each timestep.
Anyway, one can’t have lots of distances exactly correct.
Following the Gutt-Mugnolo formula, the potential should be described by:
where, is the travel time between two intersections (i and j), and is the distance in the current map.
Basically, we just keep adjusting the distances (the details aren’t important) until the total value of is minimized.
This is just in the preliminary stage (mostly because it’s not clear how I’ll generate a database of all of the transit times in, say, Philadelphia), but if you want a sneak peak about what I’m thinking about, here’s my toy model town:
8 E-W streets, and 8 N-S, all equally spaced. It takes 5 minutes to walk down a city block. There is a subway running N-S up 3rd avenue (3rd from the West). For the purposes of our toy city, the average wait for a subway is 5 minutes, and it takes 2 minutes between stops.
The “optimal” grid of intersections looks like:
This is just a toy problem, but if any of you can help me with:
- Pointing me to similar maps that people have constructed for real cities (or if this sort of map has a real name).
- A searchable, script-accessible interface of subway and bus schedules in Philadelphia (or any other city).
I’d be most appreciative.
I’m especially curious to see what parts of the city are more isolated or connected than they might otherwise appear.