by Jonathan Kujawa
It is little surprise that geometry goes back thousands of years. Right up there with being able to communicate with your fellow tribe members and count how many fish you have caught, you need to be able to measure off farm fields and build proper foundations for your home. It is an invaluable skill to be able to accurately work with lengths, angles and the like. When Euclid came on the scene 2200+ years ago geometry was already a well developed, sophisticated, and central part of the sciences.
Euclid wrote the book on geometry. Euclid's Elements was the textbook in geometry for over 2000 years. The Elements only covered Euclidean geometry. That is, the geometry of good ol' flat space in two and three dimensions. The sort of space where straight lines never meet. And that was plenty good for a millennia or two of surveying land, building bridges, mapping the London Underground, and whatnot.
As we saw here at 3QD, it took until the 19th century for people to finally open their mind to the fact that you can and should do geometry in non-flat space. If you're going to circumnavigate the Earth, then it matters quite a bit that it is a sphere. You can calculate distances, angles, and areas on a sphere, but Euclid isn't going to give you the right answer. If you want your calculations to be accurate you'd better use spherical geometry.
Nowadays we live in the age of Global Positioning Systems and interplanetary spacecraft. If you want to your phone's GPS to be accurate to within a few meters or land a spacecraft on an asteroid which is 2.5 miles across and whizzing through space at 34,000 miles per hour, then Einstein tells us we better take into account the bends and curves of spacetime. That is, we can and should do our calculations using Riemannian geometry.
It doesn't take much, then, to convince the most hard-nosed skeptic that even “exotic” geometries are pretty darn useful.
Topology, though, is another story. Topologists ask a question which at first sounds ridiculous: “What can you say about the shape of an object if you have no concern for lengths, angles, areas, or volumes?” They imagine a world where everything is made of silly putty. You can bend, stretch, and distort objects as much as you like. What is forbidden is cutting and gluing. Otherwise pretty much anything goes.
To a topologist a square is a circle since you can distort one into another. Or, as the old joke has it, a topologist is someone who can't tell the difference between a coffee cup and a doughnut:
Even if nobody expects the Spanish Inquisition, everybody should expect even the most trifling mathematics to eventually lead to something useful. I would never take the bet that interesting math will forever be useless. As the physicist Eugene Wigner wrote in his essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences“:
…the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious….The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Indeed, topology was invented to answer a real-world problem. In 1736 Euler asked if it was possible to take a walk through the city of Königsberg, Prussia and cross each of its seven bridges once and only once. At first glance this seems like a devilishly hard problem. There are countless routes you could take as you choose turn after turn through the city streets. Euler was a proto-topologist. He realized that to answer this problem he could ignore distances, angles, turns, and all other matters of geometry. All he really cared about was the connections made by the bridges.
Topology is the study of properties of a shape which survive even the most horrific deformations. But since you're not allowed to cut or glue, no matter how you stretch and deform a shape it doesn't change the fact that things which are connected stay connected, things which are disconnected stay disconnected.
Here on the left is a map of Königsberg and on the right each landmass is reduced to a single dot with edges representing the bridges connecting them [2].
In the picture on the right it isn't nearly so hard to answer Euler's question. It still requires some cleverness, but now that we've boiled the problem down to its essence it has become solvable. I'll leave it to you to puzzle out whether or not such a walking route exists.
Now you might rightly say that nobody gives a hoot whether or not you can walk all the bridges of Königsberg while only crossing each one once. And certainly it's doubtful that anyone at the time thought Euler's geometry-forgetting trick was of much practical use. Nowadays we find ourselves surrounded by networks where we care only who is connected to whom. Topology has spent the last couple of hundred years developing the tools we need to study computer networks, social networks, Kevin Bacon's costars, phylogenetic trees, etc.
In this era of Big Data we can build massive networks faster than we know what to do with them. Every retailer can make networks which connect people by their common purchases. The problem is to extract useful information from all this data. Here's a network made using ten million receipts from a home improvement store where the 40,000 items are linked if they appear on a receipt together. There is clearly patterns, but how do we find the useful trends in this sea of data?
Topology gives tools we can use and, vice versa, we are led to interesting topology questions we never thought to ask. Once you start looking, you see topology everywhere. A few months ago we ran across Lord Kelvin and his Atomic Vortices. There we were interested in the “knottiness” of knots and didn't want to count them as different if they were only superficially so. We allowed ourselves to stretch, bend, and untangle them as much as we liked. In retrospect we see that we were thinking about knots like a topologist.
If you ride the London Underground you quite likely don't care about the precise route the lines run, how far it is between stops, or any other extraneous geometric details. You care about which stops are on which lines and which lines connect to each other. In contrast to the map at the top of the page, riders usually prefer a nice, clean, easy-to-read map like this one:
This is a topologist's map. Distances, angles, and the like have been sacrificed in favor of clarity.
One of my favorite uses of topology is that it gives an easy trick to escape any maze. Imagine you are like Theseus and find yourself having to navigate a labyrinth. Unfortunately, you have no Ariadne to provide you with a spool of thread and, worse, it's pitch dark.
How to escape? Put your hand on the wall and walk! If you keep your hand on the wall, eventually you'll find the exit. How can this be? If the maze is actually just a single big room, then it's easy to see that if you keep a hand on the wall and walk, eventually you'll make your way to the exit. Now imagine that the big room has a wall which goes halfway across the middle, then as you walk you'll go to the middle along one side of the wall and back along the other side. A topologist would point out that the same is true no matter how long the wall. The length of the wall doesn't matter, so why not imagine it shrinking back and eventually being absorbed into the exterior wall? This certainly doesn't change the effectiveness of keeping one hand on the wall. And if it's a very complicated maze, why not imagine all the walls slowly shrinking back to the exterior walls? To a topologist a complicated maze is no different than a large empty room and, either way, the hand on the wall trick works just as well [5]!
A remarkable new application of topology has emerged in the last few years. Gunnar Carlsson is a mathematician at Stanford who uses topology to extract meaningful information from large data sets. He and others invented a new field of mathematics called Topological data analysis. They use the tools of topology to wrangle huge data sets.
In addition to the networks mentioned above, Big Data has given us Brobdinagian sized data sets in which, for example, we would like to be able to identify clusters. We might be able to visually identify clusters if the data points depend on only one or two variables so that they can be drawn in two or three dimensions.
But imagine you're a researcher who knows that diabetes, say, responds to medicine in various unpredictable ways. Obviously you'd like to identify these complicating factors. You can gather all sorts of data on these patients. Each one provides tens or hundreds of variables which measure differences in diet, exercise, environment, genetics, treatment history, etc.
Imagine each patient is a dot in a cloud, except that cloud exists in a hundred dimensional space. You can't even picture such a thing, how are you going to spot any clustering or other patterns? To a topologist size doesn't matter. All of the dots can be slowly inflated. At first they are all distinct, but as they get larger and larger nearby ones merge into ever bigger blobs. Importantly, the nearer dots merge first. These are your clusters. By studying how these blobs merge as they are inflated, we can identify meaningful structures within the data.
Fortunately, topologists have spent the past couple of centuries working hard to develop tools we can use to tell us information about the structure of shapes. One of the main tools, homology, is a century old. But only in the past decade or so have Carlsson and others shown us that we can use homology to extra meaningful information from real world data!
[1] The London Underground maps can be found along with a wealth of other Underground maps at Edward Tufte's website.
[2] Image borrowed from this article in Science which uses graph theory/topology on the problem of genome assembly.
[3] Image borrowed from here. A bunch of other cool pictures of networks can be found here.
[4] Image borrowed from here. They also have a bunch of cool pictures here.
[5] Of course this method is could be extremely inefficient. After all, you could be standing near the exit and have the bad luck to walk in the opposite direction! You'll traverse the entire maze before coming back around to the exit. Ii should also say that if the maze has walls which are not connected to the exterior walls, then the trick has to be modified. If you have the bad luck of starting on such a wall, you'll never get to an exterior wall. Instead you'll end up back where you started. If that happens, just pick another wall you haven't yet used and try again. Eventually you'll end up using a wall connected to the exterior and you'll finally find your way to the exit.
[6] Image from Gunnar Carlsson's Topological Data Analytics company AYASDI.