Public Nakedness

One of the greatest mathematicians of our world is best known by a possibly fictional story about the time he ran through town stark naked.  The Greek mathematician Archimedes of Syracuse has many accomplishments to his name, including an excellent approximation of pi and some outstanding inventions.  The anecdote referenced earlier, though, has to do with how he reacted when figuring out the solution to a problem concerning the king’s crown.  The king had recently ordered a crown made of pure gold, but the king was not sure if the maker of the crown had been honest and used pure gold.  Archimedes needed to find out without putting so much as a scratch on the object.  It is said that Archimedes came to the solution while bathing.  He sat down in a tub and saw how his body displaced the water.  From this, he knew that he could measure the displacement of water with a submerged crown and compare to what an equal amount of gold would displace.  Archimedes was so thrilled with his discovery that he jumped out of the tub and ran through town shouting “Eureka!  I have found it!”  It turned out that the maker of the crown was indeed dishonest and had not used pure gold.

Is this a true story?  Historians are not sure, but Archimedes did write about principles of buoyancy in his work On Floating Bodies.  So, even if that particular event did not occur, Archimedes did know the math and physics behind it.

Doughnut = Coffee Cup

Did you know that a doughnut and a coffee cup are one and the same?  You didn’t?  Well, to a topologist, they are.  Topology is an area of math that is concerned with properties of space.  These mathematicians like to take shapes and see what can be made of them by deforming them without tearing.  For example, supposed a doughnut is made of a soft, pliable rubber.  The topologist will start to turn that rubber into other shapes without changing the number of holes already present.  In this way, a doughnut with a hole in the middle can eventually form into a coffee cup with a hole in the handle.  Items with the same number of holes are of the same genus.

What the heck sort of use is this crazy math?  Well, in topology, you have people who analyze knots all the time.  They like to look at messy tangles of string and try to see how many knots are in the tangle and how to possible straighten out the string without cutting it.  This sounds like child’s play but has an important application in fighting infectious diseases and cancers.  How can we untangle the messy DNA involved with these illnesses without cutting the strings?

Queen of the Sciences

Often, mathematicians are well-rounded folks, being able to work with numbers and words simultaneously.  One famous mathematician and brilliant writer was Carl Friedrich Gauss.  He penned many lovely letters, including some romantic gems during and after the lives of his wives.  It is from him that mathematics gets the fitting moniker “The Queen of the Sciences”.

Fermat’s Last Theorem: What is it?

What exactly did Fermat’s Last Theorem say?  I think many of you are familiar with the Pythagorean Theorem.  That one basically says that you can find a missing side on a right triangle so long as you know the other two sides by using the formula

In working with right triangles, there are several cases where the solutions for a, b, and c are all positive integers (not decimals).  For instance, you can have a = 3, b = 4, and c = 5.  Another example is a = 5, b = 12, and c = 13.  Well, what if the exponent were not 2 but something higher?  Fermat’s theorem says that there are no solutions consisting of entirely positive integers.  It took mathematicians many years to prove this, but Andrew Wiles showed it to be true!

Long Proof

What was the longest paper that you had to write in school?  I bet it comes nowhere close to the paper written by Andrew Wiles.  This mathematician wrote up a proof for Fermat’s Last Theorem that took 109 pages!  He presented it in 1995, made some corrections, and then found approval from his peers.  The funny thing about this proof is that it pertains to a problem that Fermat had encountered in a book and then written in the margin that he could not quite fit his proof there.  Unfortunately, Fermat died before giving his proof, but one does have to wonder what he was going to write.  It sounds much shorter than 109 pages!