#topological space

[v5learning]

Exterior Point in Topological Space 📝 Learn BS Mathematics

[brewinggeographer]

Geography and Topological Space

I'm not a mathematician. As much as I'd love to be able to understand and communicate the language and logic of concepts like "graph theory", it usually doesn't take more than a page or two of such abstract thinking—as elegant as it may be—before my aching brain yearns for the historical geographer's chaotic world of empirical facts and subjective stories.

I nonetheless get excited by examples of abstract mathematical order made visible in my seemingly messy "real world". Such is the case with a recent blog post by The Atlantic Cities' Emily Badger, in which Badger summarizes the theoretical epidemiological geography of Northwestern University's Dirk Brockmann. Applying graph-theory principles, Brockmann argues that a hypothetical global pandemic jumping around the planet in seemingly random, chaotic fashion via globe-trotting air travelers is, in fact, following a very conventional contagious pattern, like the radiating ripples on a pond made by the splash of single stone. We just need to look at this pattern of diffusion from the right perspective.

That perspective is topological space, which is about as abstract a concept as I introduce to my students in introductory human geography. The idea of topology typically is traced back to a classic 18th-century math problem involving the Seven Bridges of Königsberg. Part of this story's appeal to me is that it sounds like a problem a bunch of old Prussians would have debated over tankards of Baltic Porter some evening: Was it possible to cross all seven bridges in the city without passing over any of the bridges a second time? The answer is no, but it took the mathematician Leonhard Euler to prove it. Euler's key observation (illustrated below) is that the problem could be reduced to what we now call topology; the length and location of the bridges, in a conventional geographic sense, were irrelevant, and all that mattered for solving the problem was how each of the bridges connected two of this spatial network's four nodes (i.e., the left and right banks of the Pregel River, plus the two islands on which Königsberg sat.)

When I introduce topological space to my students, and its application to more than just drunken debates at the bar, I discuss the cartography of mass-transit systems. Probably the most famous example is the map of the London Underground, which set the standard for efficiently displaying metropolitan rail lines topologically, distorting conventional geographical space in the process. Applying the concept to my own college in Santa Monica, and its location relative to downtown Los Angeles, one might observe that the college is a bit more than 13 miles away from City Hall, as the crow flies, from a west-southwest direction. That's geographical space. But in terms of topological space, particularly that defined by the networks of mass-transit systems on which many of my students commute, SMC is functionally further from downtown than its crow-flying distance would suggest. It would take at least two bus rides, and/or a fair bit of walking, to make such a journey. And if one wanted to ride the train; forget about it. Despite the recent opening of the Expo Line to Culver City, Santa Monica remains miles (and years) away from being rail-connected to downtown. Indeed, one could reasonably argue that, topologically speaking, Long Beach is actually closer than SMC to the historic core of L.A., since the 20-mile distance between the two cities' downtowns can be covered by a simple, single ride along the Metro Blue Line.

Any sort of spatial interaction that is mediated by networks is perhaps better understood in terms of topological space than conventional geographic space. This is exactly the idea exploited by Professor Brockmann modeling the theoretical spread of pandemic disease. If one were to map, as Brockmann does, the global diffusion (via air travel) of a theoretical disease starting on Cyprus in the eastern Mediterranean, the resulting animated map appears chaotic; the disease literally pops up all over the place, with no apparent geographic pattern.

But if one alternatively envisions the world connected to Cyprus topologically, then the disease's theoretical spread follows a very tidy contagious pattern.

Sometimes math can be quite beautiful. (Although in this case, it does require momentarily forgetting that all those animated red dots represent lots of people getting sick with the flu.)