Go Take a Hike!

The people of Konigsberg, Prussia were prone to taking walks after dinner.  Their town had the Pregel River running through, dividing the town into 4 land masses: one on each side of the river and two islands in the river.  The land masses were connected by 7 different foot bridges.  The people wondered, could they walk around and cross each bridge exactly one time each?  

When Leonhard Euler came into the picture, they no longer had to ask.  They knew!  You see, Euler turned it into a graph theory problem.  The land masses were represented with points called vertices, and the bridges were represented with segments or curves called edges.  It turns out that every land mass had an odd number of bridges coming out of it.  So, the graph had more than two odd vertices.  This tells us that it is impossible to take the walk that the Prussians wanted to take.  Always, a bridge would have to be used more than once in order to cross every bridge.  

We can thank Euler for not only solving this problem but developing the field of graph theory, which has many uses today.  Some instances where graph theory comes in handy?  Graphs can be used to map out family trees, train tracks, highways, data networks, and much more.  Local to me, an application has been the planning of walking paths at Bok Tower Gardens in Lake Wales, Florida.  The garden planners created a graph of the various points of interest and the paths that may be used.  They are striving to make this picturesque place more accessible to those with handicaps, and graph theory is a vital tool in that process.

Mathematical Spelunking

spelunk:  to explore caves, especially as a hobby

Spelunkers are like ocean explorers or space travelers.  They are venturing into places in our universe where other people have never been, and those places can be mighty dangerous.  The thrill of the journey and the beauty of the environment are intense and make the risks worthwhile for those who boldly go forth.

Much planning goes into cave exploration, and some of these explorers pull double duty by making maps of their travels.  The mapping is actually very mathematical.  Much like the surveyors we see out by the sides of roads, a cave surveyor must use trigonometry to accurately mark where they have been and where they are going.  Yup, that course whose title literally means “triangle measure” is the heart of the work.  You don’t believe me?   Ask these guys.

Eddy Cartaya Glacier Cave Discoveries and Trig TED Talk

One Edge, One Side, One Strip

A Mobius strip or Mobius band is a surface with only one edge and one side.

What sort of practical application might this have?  I am glad you asked.  Perhaps it can be really useful as an automobile belt.  With only 1 side, the wear and tear should be pretty even.  Or, if you want to keep your kids busy, have them make one and then start cutting it down the middle.  All kinds of cool stuff can happen!

http://www.youtube.com/watch?v=BVsIAa2XNKc

If you glue the edge of one Mobius strip to the edge of another, you get an object that only exists in 4-D space called a Klein Bottle.  Check out some 3-D representations at www.kleinbottle.com

Nearly Twin Primes

Twin primes are primes that occur is succession with one composite between them.  For example, 3 and 5 are twin primes.  They are primes in that they only have factors 1 and themselves, and then they are twins because they are separated only by one composite which is 4.  The Twin Prime Conjecture claims that we can find this sort of pattern to infinity.  No matter how far we go out on the number line, there will be another pair of twin primes to find.  To view a really cheesy but fun video about this, check out the NOVA Science Now link.

http://video.pbs.org/video/1511294183/

Recently, the math world has exploded with discovery concerning this conjecture.  Mathematicians had not been able to prove the Twin Prime Conjecture exactly as is, but they were able to prove that primes go on in a skippy pattern with more than one composite between them.  Efforts are being made to narrow the gap, and hopefully eventually prove the gap of only one.

In May of 2013, it was revealed that Tom Zhang of the University of New Hampshire proved infinite pairs of primes that are 70 million apart.  This seems like a big gap, but it was remarkable as it was the closest anyone had come.  (It is definitely a smaller gap than the previous infinity.)  To make it even more astounding, Professor Zhang was relatively unknown until this.  He has been quietly working on his mathematics for years as he went about life and work in jobs such as sandwich maker for Subway.

The big reveal of Dr. Zhang’s discovery started a flurry of work.  Dr. Terry Tao started a Polymath Project to enlist many volunteers to continue work on this project.  The team narrowed down the gap from 70 million to 4,680 by July.

As of September, James Maynard had reduced the gap to 600.

In November, Thomas Engelsma says he reduced the gap to 576.

And the work goes on!