Bar Bets and Thermonuclear Bombs

Stanislaw Ulam, along with Edward Teller, fathered the thermonuclear bomb or Hydrogen Bomb as it is more commonly known. Ulam is one of the many great mathematical minds of Polish descent. Known as a rather cantankerous fellow, he made major contributions in the fields of Set Theory, Topology, Ergodic Theory, Biology and Physics. A true polymath.

While residing in the United States during World War II, Ulam was invited to join a secret project in New Mexico by his close friend John von Neumann, one of the greatest mathematicians in history and an epic contributor to the the fields of mathematics, economics, set theory, game theory, logic, quantum mechanics, computer science and nuclear physics. Neumann created the field of Cellular Automata (see Life Goes On: Further Ruminations on Conway's Game of Life for examples) with a pencil and graph paper as tools to build the first self-replicating automata.

With his typical scientific curiosity, Ulam checked out a book on New Mexico from the University of Wisconsin–Madison library, the campus where he was a faculty member at the time. In it, he found the names of all those that had previously disappeared from the campus on the check-out card. He subsequently joined the  Manhattan Project at Los Alamos, where the development of the first atomic bomb was progressing.

As the atomic bomb project was nearing fruition, Teller had turned his attention to the creation of a vastly more powerful tool of destruction, the Hydrogen Bomb, or "Super" as it came to be known. Ulam, made aware of the "super", showed that Teller's design was faulty and devised a much more suitable design. While the design took on the moniker of "the Teller-Ulam design", most involved in the project credit Ulam with being the true father of modern thermonuclear devices, as contrasted to Teller's rumored contribution as the character model for Dr. Strangelove in the eponymous film by Stanley Kubrick.

A particularly interesting contribution of Ulam's to mathematics and topology specifically is the Borsuk-Ulam theorem, first conjectured by Ulam and later proved by Karol Borsuk in 1933. The theorem is stated as follows:

Call a continuous map f:Sm→  Sn antipode preserving if f(−x)=−f(x) for all x∈  Sm

Theorem: There exists no continuous map f:Sn→ Sn−1 which is antipode preserving for n > 0

In English, this states that any continuous function from an n-sphere to Euclidean n-space maps some pair of antipodal points to the same point.

In even more plain English, I'll try to explain why this results in things that can sound incredulous to most non-mathematicians, and why it provides for the opportunity of a "bar bet" that at the very least might get you a free beer.

Take the circle above with the two labeled points A and B. This will be the only drawing here, to start us off. I'm no artist, and my meager attempts at drawings for the steps would likely confuse rather than help. Get out your pencil and paper, put on your thinking caps, and follow along!

Imagine this is a line drawn around a sphere, such as the idealized Earth. The equator is an example of such a line. In this case, we only require that it is maximal in size, that is, it is drawn around the Earth in a way that it is as large as it can be, like the equator or any of the lines of longitude that run north to south on the globe.

You've perhaps heard the term "Great Circle" used for navigation. A Great Circle is a circle around the earth that is maximal. Any segment of such a circle that connects two points is the shortest distance between those points on the globe. So taking the great circle path is always the shortest route between points on the surface of the Earth.

A shape such as the circle we are using is known mathematically (at least by topologists) as a 1-sphere or Disc: it is a one dimensional surface embedded in two dimensional Euclidean space. What non-mathematicians call a sphere is known as a 2-sphere or ball: A two dimensional surface embedded in three dimensional Euclidean space.

We want to show that we can always find two points, opposite (antipodal) to each other on the planet, that have exactly the same temperature. If we find that points A and B in fact measure the same temperature for the circle we've drawn, we have made our point. But suppose the temperatures are different.

Let's call A the colder measurement point, and B the hotter measurement point. Now imagine moving the measuring points, keeping them diametrically opposed, around the circle until they have exchanged places. Let us assume that the measurements never share the same temperature. Because of this assumption, measurement point A must remain colder than measurement point B for the whole trip, and measurement point B must remain hotter than measurement point A.

But after rotating the measurement points 180°, we know that measurement point B must be measuring a colder temperature than measurement point A and vice versa. This contradicts our assumption, and by the mathematical technique of proof by contradiction we now know that there must have been some point along the path where measurement points A and B showed the same temperature.

The same proof technique can be extended to any 2-sphere/ball like the Earth. When moving to the higher dimensions, we are allowed more degrees of freedom in our chosen path for the antipodal points. We can choose any arbitrary meandering path, so long as the two measuring points remain antipodal.

Imagine we choose some path, again resulting in the exchanging of the positions of the two measurement points. By the same methods, we can see that again, there must be two antipodal points somewhere on that path that share the same measurement. This is critical to our emerging bar bet. This means that every path taken by measurement point A on it's way to measurement point B, with measurement point B following along by remaining diametrically opposed (antipodal) on the opposite site of the Earth must have a point where the two points shared the same temperature.

Let's consider the set of all possible pairs of antipodal points on the Earth with the same temperature. These points need not have the same temperatures, they must only have the same temperature as their matching antipodal point on the opposite side if the Earth to be a member of this special set.

You may be thinking that the result might be some myriad of dots scattered across the surface of the ball where the matching antipodal dot shares the same temperature. But this would be wrong!

If this was the case, we could always find some path from the original measurement point A to measurement point B without needing to touch any of the dots in the set. But as we have already shown, this set must contain a pair of the dots opposite each other on the Earth that share the same temperature along whatever path we take from A to B.

From this, we can make the conclusion that the set of dots must contain some set of dots that result in a continuous path dividing our ball (the Earth) into two sections, one containing the original measurement point A and the other containing measurement point B.

Now imagine applying the same line of thinking to this path, except we'll use some other continuous function such as barometric pressure. We pick two antipodal measurement points, call them C and D, and take our measurement. If the result is the same, we are finished showing the same fact that we proved for temperature holds true for barometric pressure. If they differ, we again let measurement point C follow the path to measurement point D, with point D following along on the opposite side of the globe. As with temperature, we will find some point along the path that has matching measurements for measurement points C and D.

But as we know, this point is on the path that contains all of the points where the temperature measurements are equal for the matching antipodal points of the path.

This means that that point, and its matching antipodal point, simultaneously have the same temperature and barometric pressure! Such points can always be shown to exist on a 2-sphere/ball, in our case the Earth, for any two continuous functions on its surface. We could have just as well used temperature and wind speed, wind speed and barometric pressure, etc.

The Borsuk-Ulman theorem proves that this is true for any pair of continuous functions on any 2-sphere! As can be seen in the theorem, it can be extended to any number of dimensions, with a corresponding increase in the number of continuous functions that will share a pair of antipodal points with equivalent values for each function.

This means for example that when that big plump helium-filled party balloon diffuses to death and is lying on the floor in a crumpled mass, there is a pair of points that were antipodal when the balloon was inflated that are now lying precisely on top of each other. Always.

So the next time you find yourself at the local watering hole with your bright companions, bet them a beer that there are two places on the opposite sides of the Earth, right now, that have exactly the same temperature and barometric pressure.

The bet is almost surely to be taken, and you'll earn yourself a free beer. Or maybe have one tossed at you!

While you might curse Ulam for his contributions to nuclear weaponry, remember to raise a toast to him before drinking your free beer! 

For more uses on this most interesting theorem, the book Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry by Jiri Matousek and published by Springer provides a rewarding overview in an approachable text at the elementary level.

Excellent introductory topology books can be had in Topology by K. Jänich (also from Springer) and the classic Topology by James R. Munkres, published by Prentice Hall. 

Topology and Its Applications by William F. Basener published by Wiley provides more examples of useful results from general topology.

The Topology entry at Wolfram Mathworld provides a one page description of the field with copious references. You may find yourself wandering the site following one interesting link after another.