I didn’t know anything about topology before I entered graduate school, but I continue to see it as one of the more provocative specialties in mathematics and an important transition of thought. Most definitions of the subject describe it as the study of the properties of objects that are preserved through deformations like stretching and twisting. Cutting or tearing or gluing is not allowed. A circle is topologically equivalent to an ellipse because the circle can just be stretched into the ellipse. Removing one point, however, from either the circle or the ellipse, produces something else. We now have a line segment. A sphere is topologically equivalent to an ellipsoid, again because one can be squeezed or stretched into the other. But a doughnut, because of the hole it has, is not topologically equivalent to either, Holes, in fact, become key to creating equivalence classes of things. A well known and lighthearted equivalence, is one where a coffee cup is topologically equivalent to a doughnut. Even without any training, one can see that these equivalences depend on another level of abstraction, and one that challenges intuitive notions. While topology often considers shapes and spaces, it is not concerned with distance or size.
My affinity for this branch of mathematics may have been helped along by the fact that my favorite teacher in graduate school was a topologist. Sylvain Cappell, now still at The Courant Institute of Mathematical Sciences at NYU, introduced me to topological ideas. I’ve saved a Discover Magazine article from 1993 that landed in my mailbox not long after I left Courant. In it Cappell discusses the motivation and effectiveness of a topological approach to problems. The late Fields Medalist, William Thurston, also contributed to that article. Thurston suggested that our difficulty with perceiving the higher dimensions that are a fundamental consequence of topological ideas is primarily psychological. He believed that the mind’s eye is divided between linear, analytic thinking and geometric visualization.
Algebraic equations, for example, are like sentences. The formula that gives you the area for a cube, x times x times x, can easily be communicated in words. But the shape of the cube is another matter. You have to see it.
When we talk about higher-dimensional spaces, Thurston says, we’re learning to think in and plug into this other spatial processing system. The going back and forth is difficult because it involves two really foreign parts of the brain.
Emphasizing the value of “see it” Cappell makes the argument that even with a 2-dimensional graph, relating something like interest to consumer spending, where neither has anything to do with geometry, the shape of the line that represents their relationship gives you a better grasp of the situation.
The same holds true in five or even ten-dimensional models. Logically, it may seem like the geometry is lost, that it’s just numbers, says Cappell. But the geometry can tell you things that the numbers alone can’t: how a curve reaches a maximum, how you get from there to here. You can see hills and valleys, sharp turns and smooth transitions; holes in a doughnut-shaped nine-dimensional model might indicate realms where no solutions lie.
A few days ago, an article in Forbes tells us that topology can help us see something about how to reduce slum conditions in cities. Researchers, it explains, opt for a “shape-based” understanding of cities.
According to the team’s research, when two or more city sections have the same number of blocks, they’re topologically equivalent and can be deformed into each other.” Using that approach, sections of Mumbai can be deformed into Las Vegas suburbs or even areas of Manhattan.
How are slums and planned cities topologically different? The difference emerges essentially from better or worse access to the infrastructure and researchers claim that once cities are understood as topological spaces, the access issue can be resolved mathematically.
Their approach uses an algorithm that can be applied to any city block, they note in their paper. It applies tools from topology and graph theory — the branch of mathematics concerned with networks of points connected by lines — to neighborhood maps to diagnose and “solve critical problems of development,” they wrote.
This is an interesting peek at what can happen in or with mathematics. Topology itself requires a willingness to look differently. Using topological ideas to analyze or to address the development of slum conditions in sprawling cities is unexpected. It’s a geometry being applied to a space, but not directly. Not because it resembles the space. It says more about how abstractions can give us greater access to the real world. Or how the minds eye can see.
Both the Discover and Forbes articles are worth a look.
Great topology and slum conditions…