Author Archives: jgparrott

Musings: The Poincaré Conjecture

Mathematics is no stranger to unsolved problems. Time and time again, equations, conjectures, and theorems have stumped mathematicians for generations. Perhaps the most famous of these problems was Fermat’s Last Theorem, which stated there is no solution for the equation xp+yp=zp, where x, y, and z are positive integers and p is an integer greater than 2. Pierre de Fermat proposed this theorem in 1637, and for over three hundred fifty years, it baffled mathematicians around the globe. It was not until 1994 that Andrew Wiles finally solved the centuries-old theorem.

Though the most famous, Fermat’s Last Theorem was by no means the only unsolved problem in mathematics. Many problems remain unsolved to this day, driving many institutions throughout the world to offer up prizes for the first person to present a working solution for any of the problems. Some few are general questions, such as “Are there infinitely many real quadratic number fields with unique factorization?” However, most of the problems are specific equations proposed by a single or multiple mathematicians and are generally named after their proposer(s), such as the Jacobian Conjecture or Hilbert’s Sixteenth Problem.

One such problem, proposed by Henri Poincaré in 1904 and thus named the Poincaré Conjecture, remained unsolved until 2002.  In order to encourage work on the conjecture, the Clay Mathematics Institute made it a part of the Millennium Problems, which included several of the most difficult mathematics problems without proofs. A proof to any of the problems, including the Poincaré Conjecture, came with a reward of one million US dollars. To this day, the Poincaré Conjecture remains the only problem solved.

The Poincaré Conjecture is a problem in geometry but concerns a concept that, for many, is difficult to comprehend and all but impossible to visualize. The best means to approach it is to imagine a sphere, perfectly smooth and perfectly proportioned. Now, imagine an infinitesimally-thin, perfectly flat sheet of cardboard cuts into the sphere. If you were to take a pen and draw on the cardboard where the sphere and the cardboard intersect, you would produce a circle. If you were to take the sheet of cardboard and move it up through the sphere, the circle where it and the sphere intersect would gradually shrink. Eventually, just as the cardboard is at the edge of the sphere, the circle will have shrunk to a single point.

Plane-sphere intersection. Image: Zephyris and Pbroks13, via Wikimedia Commons.

Note that in the field of topology, this visualization applies to any shape that is homeomorphic to a three dimensional sphere (referred to as a 2-sphere in topology since its surface locally looks like a two dimensional plane, much as how the Earth appears flat while standing on its surface). Homeomorphic refers to a concept in the field of topology concerning, what is essentially, the distortion of a shape. For instance, one of the simplest examples in three dimensions is that a cube is homeomorphic to a sphere, since if you were to compress and mold the cube (much as you would your childhood PlayDoh), you could eventually shape it into a sphere. However, in topology, you are not allowed to create or close holes in a shape. This is why shapes such as a donut or a cinder-block are not homeomorphic to a sphere, due to the holes that go through them.

Poincaré proposed a concept concerning homeomorphism and the previously described visualization, and it is here where imagining the problem no longer becomes possible. We live in a three-dimensional world, where any position in space can be plotted based on relativity to three axes, all perpendicular to each other. To imagine a fourth spatial dimension perpendicular to those three is mentally impossible, as is any shape with higher dimensions, and yet many problems in geometry and physics relate to a fourth and even higher dimensions. The Poincaré Conjecture relates to these higher dimensional shapes, specifically closed 3-manifolds (shapes with a locally three dimensional surface). It states that, if a loop can be drawn on a closed 3-manifold and then be constricted to a single point, much like the intersection of the cardboard plane and the sphere in the aforementioned example, then the closed 3-manifold is homeomorphic to a 3-sphere, the set of points equidistant from a central point in four dimensions (Morgan).

If the concept of the Poincaré Conjecture is difficult to conceive, its solution by Russian mathematician Grigori Perelman in 2002 is almost incomprehensible. Due to the number of variables involved, one could not simply set up a system of equations between a three-dimensional space and a 3-sphere. Instead, Perelman used a differential geometry concept called Ricci Flow, developed by American mathematician Richard Hamilton. In short, it is a system which automatically contracts to a point on any surface, and it proved to be the precise tool needed to prove the Poincaré Conjecture. (THIS video does a good job of explaining it in layman’s terms) (Numberphile)

An example of Ricci flow. Image: CBM, via Wikimedia Commons.

Interestingly, despite the immense difficulty of solving such an abstract problem as the Poincaré Conjecture, Perelman refused the prize awarded to him for his accomplishment. His solution to the problem was an exercise in his own enjoyment, and as he later stated upon being offered the Fields Medal (the mathematician equivalent of the Nobel) and the immense monetary prize,  “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” Later, he also argued that his contribution to the solution of the Poincaré Conjecture was “no greater than that of… Richard Hamilton,” and that he felt the organized mathematical community was “unjust.” (BBC News, Ritter)

To this day, the Poincaré Conjecture remains the only Millennium Problem solved. Its proof wound up leading to the solution of various other related geometrical problems and closed a century-old mystery. As the field of mathematics continues to grow and progress, it is only a matter of time until other unsolved problems come to resolution.

Works Cited

Morgan, John W. “RECENT PROGRESS ON THE POINCARÉ CONJECTURE AND THE CLASSIFICATION OF 3-MANIFOLDS.” The American Mathematical Society 42.1 (2004): 57-78. The American Mathematical Society. The American Mathematical Society, 29 Oct. 2004. Web. 9 Oct. 2014.

Jaffe, Arthur M. “The Millennium Prize Problems.” The Clay Mathematics Institute. The Clay Mathematics Institute, 4 May 2000. Web. 09 Oct. 2014.

Numberphile. “Ricci Flow – Numberphile.” YouTube. YouTube, 23 Apr. 2014. Web. 09 Oct. 2014.

“Russian Maths Genius Perelman Urged to Take $1m Prize.” BBC News. BBC, 24 Mar. 2010. Web. 09 Oct. 2014.

Ritter, Malcom. “Russian Mathematician Rejects $1 Million Prize.” Russian Mathematician Rejects $1 Million Prize. The Associated Press, 1 July 2010. Web. 09 Oct. 2014.

Musings: Ancient Thinking

If there’s one realization that’s guaranteed to send a mind reeling, it’s getting a glimpse at just how different human beings from one culture to another are at thinking. Even in modern day, the contrasts between two societies can be striking and staggering. Envision an American visiting Japan for the first time with no clue of just how different Japanese culture is from American culture. Coming from a culture that prizes individual accomplishment and direct communication, the tendency for Japanese culture to assign worth based on relationships and to consider addressing a subject bluntly as ‘clumsy’ would be shocking, if not dazzling (Western Washington University).

Archeologists deal with the difference of thought on a regular basis. In studying now-extinct civilizations, much of their deciphering of the artifacts they discover cannot be done by relying on modern ideas and understandings. Even in the subject of math, perhaps the most definite rules of the universe understood by mankind, ideas and understanding of mathematical concepts and uses have drastically evolved.

When I began this class, I was excited for the topics that I expected would be covered. I’ve still enjoyed the classes thus far, but I can say with absolute certainty that I was not prepared for the culture shock that would come with dealing with mathematical concepts that, in our modern society, are so basic and fundamental.

Mesopotamia. Image: Giusi Barbiani, via Flickr.

Mesopotamia. Image: Giusi Barbiani, via Flickr.

Zero. Zero is perhaps the most important digit in our system of numerals. It’s a place holder; it’s a starting point; it’s the middle of a number line that goes on for infinity in each direction. But go back to the inception of mathematics beyond 2+2=4, and you will find zero is as nonexistent as that which it represents. And boy, what a difference NOT having a zero makes. Perhaps the first function of zero that one misses when working without it is its job as a placeholder. How does one write 10, or 100, or 1000 without a zero? The Mesopotamians used a base 60 system, which meant instead of 10, 100, etc. their digits went up 60, 3600, 21600, and beyond. But still, the problem becomes: How does one write those without a zero?

Like this: 1. That’s it. 1. When the Mesopotamians landed on a power of 60, they wrote it as 1, because just as if you took the zeroes out from behind 10, 100, and 1000, all that’s left is a 1. This both creates a problem, but at the same time it provides a fascinating workaround. Since any power of 60 can be written as 1, the numbers prior to them can treated as fractions. For instance–and it’d be useful to think of a clock for this–if you wanted to write ‘half’ using Mesopotamian numbers, you would not write 1/2, but rather 30. Think minutes; thirty minutes is half of an hour, which is 60 minutes. 1/4 would be 15. 1/8 is a little more complex, as it comes out as 7;30 (that’s 7 sixties and 30 ones), but it’s still exactly like 7 minutes and thirty seconds is one eighth of an hour.

I could be a millionaire! If this was Mesopotamian. Image: David Guo, via Flickr.

I could be a millionaire! If this was Mesopotamian. Image: David Guo, via Flickr.

Which brings the subject to reciprocals; reciprocals are fractions that, when multipled to a number, produce a 1. Again, because the Mesopotamians didn’t have a zero, their representations of 60, 3600, 216000, etc. all appear as 1. Because of this, reciprocals in the Mesopotamian numerals are sets of numbers that multiply not just to one, but any power of 60. Some examples would be 4 and 15, which multiply to 60, or 16 and 225, which produce 3600. Because these powers of 60 appear as 1, these sets count as reciprocals. It’s truly staggering what not having a zero does to math.

But when you consider the applications these ancient civilizations, such as the Mesopotamians, used math for, it does not make much sense to have a zero. For their purposes, a representation for zero would be irrelevant. Using a base 60 system, they could count to far higher with their digits before needing place holders. And when you’re counting cattle or grains or simple transactions at the market, zero is the last number you want to see on your accounting clay. This was a society that dealt entirely in positive numbers and practical, tangible concepts. We can look back at the Mesopotamian number system now and think, “Look at how hard it was for them to do even basic operations like completing the square,” but in a time when each man could only farm as much land as they and their family could do themselves and the technology for giant architectures was not common, completing the square was about as advanced in mathematics as any one person ever needed to attempt.

Like writing, mathematics is a largely intangible concept, and thus got off to slow start purely for practical purposes. Archaeological evidence indicates it would have simply started in counting animals and crops for the purposes of trade, or perhaps for counting people in some sort of rudimentary census. It wasn’t until humanity’s capability for the written language had advanced enough to express and record complex ideas that math began to see use for architecture and infrastructure. For the Mesopotamians, perhaps one of the most important uses of mathematics was in irrigation. Mathematical standards enabled uniform construction of materials, which was essential for carrying water the long distances necessary to hydrate the numerous farms of Mesopotamia. Advanced accounting and inventory ensured that construction had all of the materials a project would require without being oversupplied, as well as pay and supply the necessary manpower to work the construction project (Melville, Robson).

The mathematical system of the Mesopotamians can be quite a culture shock for American students. I myself was lost on the concept for the first week. It took a lot of practice for me to understand and comprehend the ‘reciprocals’ required due to the lack of a Mesopotaian zero. But it’s truly fascinating, regardless of its difficulty. It’s amazing to think this is one of the first advanced number systems to exist in human history. Its differences are shocking; for the unprepared mind, they can leave one feeling numb and lost. But once one manages to cross the bridge from the present to the past, the concepts ready to be rediscovered  are truly staggering.