Talking Trees

Graph Theory is an area of mathematics that uses constructs called graphs to model many relations and processes found in real-world problems.  The graphs consist of vertices (points or nodes) connected by edges (segments or arcs or loops).  This area of mathematics has applications in computer science, biology, chemistry, linguistics, physics, chemistry, sociology, and more.  Can I dare say that Graph Theory is ubiquitous?  So, imagine my lack of surprise when I was watching a TED talk today and saw Graph Theory in action.  The link below takes you to the talk “How Trees Talk to Each Other” by Suzanne Simard.  The talk does not get into the mathematics, but there are some images that caught my attention.  One was that of a fractal, starting at 8:36 in the video.  (This grabbed me, as my Master’s thesis talked about using fractals to monitor the health of forests.  Sadly, the image had nothing to do with her mathematics.)  The next was a graph, which appears at 10:40 in the video.  She does talk about how that graph is used to model connections in a forest, giving me a bit of mathematical excitement.  Despite the lack of a whole lot of mathematics, this video is well worth a watch.  Check it out, and watch out for the bit of Graph Theory.

How Trees Talk to Each Other

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Prime Time

sieve:  a utensil with meshes or holes to separate finer particles from coarser ones or solids from liquids. – Merriam Webster Dictionary

Every student who has completed grade school learned about prime numbers.  We learned what they are:  natural numbers greater than 1 with factors of only 1 and themselves.  We learned methods to find them.  We learned to use them in factoring.  An elementary exposure was part of the ticket out of elementary mathematics.

Not everyone, though, was exposed to the Sieve of Eratosthenes.  The Sieve is an ancient method (circa 240 B.C.) that gives us a systematic way to find primes, starting with 2.  You write the natural numbers from 1 up to however high you want to go.  You cross 1 out, as it is neither prime nor composite.  Start with 2, your first prime.  (I usually have students circle it.)  Then, cross out all multiples of 2, as they have additional factors other than 1 and themselves.  Go back, and the next available number, 3, is prime.  Circle it.  Cross out all multiples of 3.  Go back, and the next available number, 5, is prime.  Circle it.  Cross out all multiples of 5.  Go back.  Repeat over and over until you reach the end of your sieve.

The method is pretty simple in its execution, but it’s time-consuming when trying to sort out primes for large numbers.  Even with today’s super computers, the method takes up too much memory and time.  But, there’s hope!  Peruvian mathematician Harald Helfgott has found a way to modify the sieve so that computers don’t need to use as much memory space.  Scientific American published an article on the discovery, and it is worth a read.

New Take on an Ancient Method Improves Way to Find Prime Numbers

Swimmer’s Algorithm?

How many of you watched the Olympics in Rio?  :raises hand:  I wasn’t obsessed with it, but I definitely watched the highlights like men’s swimming and women’s gymnastics.  I caught a little bit of the doubles kayaking (not sure what it’s officially called), thanks to the TV on the stationary bike at the gym.  Oh, and I also showed some of the archery to my son, who is finally taking some lessons after years of interest.

The most interesting to me was the men’s swimming.  And before you go to thinking that it had to do with the swimsuits, just stop right there.  😀  I actually prefer human specimens to be covered.  A full-body wet suit looks way better than a Speedo bikini.  But, anyhoo, I digress.  I watched the swimming in fascination of the athletes and what they might accomplish this year.  As usual, Michael Phelps and Ryan Lochte certainly impressed.  And, in the women’s swimming, who could ignore Katie Ledecky?!  Her races became something to follow, and I will likely find myself a fan of women’s swimming in the future.  You go, girl!

But, as I watched these incredible performances, I couldn’t help to wonder if the odds were stacked in favor of some swimmers and against other swimmers.  From a casual glance, it appears that they all are on a level playing field.  They have fancy swimsuits, caps, and goggles.  They have the same starting blocks and signals.  They are all in the same pool.  But, what about the lanes?  Does the position in the pool make a difference?  In these races, I quickly got a gut feeling that it did.  And, I got to wondering what determined a swimmer’s position in the pool.  It seemed like Phelps and Lochte were often in the middle lanes?  But then, not always?

Imagine my pleasure and surprise, then, when I surfed Facebook today and came across this posting by the MAA.

Charts Clearly Show Some Swimmers Had Unfair Advantage

Seems my gut feeling was onto something.  I was nowhere near working out the mathematics of it, but this is quite interesting.  And, as a friend commented, it might make a good undergraduate research project for someone.  There’s still some math to be found!  I am eager to see their further findings down the road.  I have taken up swimming as part of my exercise routine, and I definitely notice a difference when I am in the pool alone versus with others.  I also notice a difference in what part of the pool I am using.  With too many people in the pool, there’s an erratic current to fight.  Surely 8 lanes of swimmers will have some effect on the water patterns?  It’s time for those versed in fluid dynamics to figure it out!