Saturday, September 19, 2009
When does a question have no answer?
Have you ever had a feeling that one of your math homework problems couldn't be answered? You're not alone. Psychologists will tell you that this is a common human trait; when something is too hard for us to do, we'll often try to convince ourselves and others that no one can do it. Mathematicians are not immune to this feeling.
Sometime in the 1600's, Pierre de Fermat claimed to have a proof that xn+yn=zn has no natural number solutions when n>2. He never demonstrated his proof though, leading other mathematicians of his day to attempt their own proofs, to no avail. This result became known as Fermat's Last Theorem.
When n=2, this equation has infinitely many natural number solutions. Can you find one?
One of the most prolific mathematicians of all time, Leonhard Euler, was said to be so frustrated by his inability to prove Fermat's Last Theorem that he had Fermat's house torn apart, looking for any scrap of paper that may contain hints of Fermat's proof. Fortunately, Fermat was long dead by the time Euler did this.
Over the centuries, many of the world's most accomplished mathematicians tried to prove Fermat's Last Theorem. It wasn't until 1995 that Andrew Wiles, working in secret for seven years, finally provided a proof. Wiles worked in secret, because he didn't know if he'd ever find a proof and he didn't want others thinking he was wasting his time. To many mathematicians, the theorem seemed unprovable. After all, the world's best tried and failed for over 300 years.
What do we mean by unprovable? We don't mean that humans are unable to do it. We mean that, even in theory, no proof can possibly be given. Is it possible that such unprovable ideas or unanswerable questions could exist? In the early 1930's, the mathematician Kurt Göedel proved that not only is it possible that some mathematical questions can't be answered, but that there definitely are unanswerable questions. Essentially, Göedel's Incompleteness Theorem says that there are mathematical hypotheses that can never be proved or disproved. The worst part is, there seems to be no way of knowing which questions have this undesirable quality.
In the late 1850's, Bernhard Riemann proposed an idea that is now called the Riemann Hypothesis. He was never able to prove his idea. In the past 150 years, many of the world's best mathematicians have attempted to provide a proof. There have been hundreds of mathematical papers and books written on the subject. Some brilliant people have devoted their entire careers to finding a proof. Yet proof still remains to be found. In fact, if someone does prove it, they can cash in on a $1,000,000 prize.
Find the complex roots, win a prize!
Some people worry that the Riemann Hypothesis may be one of Göedel's predicted mathematical enigmas. But there is no known way to decide if it is or not. So, the frustration that you feel when you encounter a difficult homework problem, is felt the world over by mathematicians facing some of the most mind-boggling questions.
Sometime in the 1600's, Pierre de Fermat claimed to have a proof that xn+yn=zn has no natural number solutions when n>2. He never demonstrated his proof though, leading other mathematicians of his day to attempt their own proofs, to no avail. This result became known as Fermat's Last Theorem.
When n=2, this equation has infinitely many natural number solutions. Can you find one?
One of the most prolific mathematicians of all time, Leonhard Euler, was said to be so frustrated by his inability to prove Fermat's Last Theorem that he had Fermat's house torn apart, looking for any scrap of paper that may contain hints of Fermat's proof. Fortunately, Fermat was long dead by the time Euler did this.
Over the centuries, many of the world's most accomplished mathematicians tried to prove Fermat's Last Theorem. It wasn't until 1995 that Andrew Wiles, working in secret for seven years, finally provided a proof. Wiles worked in secret, because he didn't know if he'd ever find a proof and he didn't want others thinking he was wasting his time. To many mathematicians, the theorem seemed unprovable. After all, the world's best tried and failed for over 300 years.
What do we mean by unprovable? We don't mean that humans are unable to do it. We mean that, even in theory, no proof can possibly be given. Is it possible that such unprovable ideas or unanswerable questions could exist? In the early 1930's, the mathematician Kurt Göedel proved that not only is it possible that some mathematical questions can't be answered, but that there definitely are unanswerable questions. Essentially, Göedel's Incompleteness Theorem says that there are mathematical hypotheses that can never be proved or disproved. The worst part is, there seems to be no way of knowing which questions have this undesirable quality.
In the late 1850's, Bernhard Riemann proposed an idea that is now called the Riemann Hypothesis. He was never able to prove his idea. In the past 150 years, many of the world's best mathematicians have attempted to provide a proof. There have been hundreds of mathematical papers and books written on the subject. Some brilliant people have devoted their entire careers to finding a proof. Yet proof still remains to be found. In fact, if someone does prove it, they can cash in on a $1,000,000 prize.
Find the complex roots, win a prize!
Some people worry that the Riemann Hypothesis may be one of Göedel's predicted mathematical enigmas. But there is no known way to decide if it is or not. So, the frustration that you feel when you encounter a difficult homework problem, is felt the world over by mathematicians facing some of the most mind-boggling questions.
Labels: Euler, fermat, goedel, homework, riemann, wiles
Friday, June 26, 2009
Magnets, Liquids, and the Exploding Surface of Our Sun
Dynamic sculpture by ferrofluid artist Sachiko Kodama.
The study of fluid flow probably started long before Archimedes' invention of the water screw about 2300 years ago. Though it was a long time after that before mathematicians put serious thought into describing fluids with equations. One of the early successes came in the 1700's, when Leonard Euler produced a breakthrough mathematical model of a fluid with no viscosity. His equations were later modified to include the effects of viscosity and entropy. Mathematicians, engineers and physicists have been struggling to understand the solutions to these equations ever since. One open question is worth $1,000,000 to anyone who answers it.
By comparison, the study of electromagnetism is fairly young. I've heard tales that certain ancient peoples may have understood how to construct simple chemical batteries. But as recently as the 1750's, Ben Franklin was electrocuting kites with lightening bolts, with no idea of what was happening. It wasn't until the 1870's that James Clerk Maxwell constructed the first detailed mathematical model of electromagnetism. Relativistic effects weren't included until after Einstein's work in the early 1900's.
So what happens when a fluid is influenced by electromagnetic fields? What strange world is born from this fusion of complex systems? Scientists, mathematicians, and hobbyists have discovered only a small subset of the answers, but what they do know is fascinating.
The partial differential equations of magnetohydrodynamics: the marriage of Euler and Maxwell.
Ferrofluids
By mixing certain ferromagnetic metal powders with soaps and oils, you can make a high viscosity fluid with a lot of surface tension that responds in beautiful ways to the influence of magnets. The video at the top of this article shows one of these fluids producing beautiful spiked dynamic patterns in response to a magnetic field controlled by artist Sachiko Kodama. The high viscosity and surface tension keeps the metal powder particles from being ripped out of suspension by the magnets. The spikes arise because the metal particles are being forced along magnetic field lines. The fluid part of a ferrofluid does not respond well to magnets. The behavior of a ferrofluid is due solely to the embedded metal particles.
Paramagnetic Liquids
Paramagnetic liquid oxygen being attracted to a strong magnet.
Certain liquids, liquid oxygen for example, have inherent magnetic properties. Unlike ferrofluids, which are magnetic because of particles suspended in them, paramagnetic fluids respond to magnets because every molecule in the fluid has unpaired electrons.
Magnetofluids
The violent magnetohydrodynamics of the sun's surface.
When electrons move, a magnetic field is created. So, if a fluid contains streaming free electrons and is also able to respond to magnetic fields, it becomes a self-driven magnetic dynamo. Magnetic fields cause changes in fluid flow and fluid flow causes changes in the magnetic field. This type of fluid is found in stars and fusion reactors. Much of our own sun's strange behavior, including sunspots and solar flares, is caused by this dynamo. Online science news magazine Wired.com recently had an article describing the bizarre activity on our sun's surface. Read the full article to see some beautiful photos and learn why sunspots usually occur in 11-year cycles.
Labels: Euler, ferrofluid, liquid, magnet, Maxwell, sunspots
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