The Mathematics of History

“Those who cannot remember the past are doomed to repeat it.” – George Santayana

Most of us have heard the above quote or some form of it, but I don’t know if it really motivated any of us when having to memorize dates or fill out maps.  But, perhaps something else will motivate us.  It turns out that there are underlying patterns in history that can be modeled with mathematics.  Here’s a TED talk by Jean-Baptiste Michel, where he highlights a couple of such models and proposes an idea about the future relationship of history and mathematics.

The Mathematics of History


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

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!

Arthur Benjamin’s Mathemagic

Arthur Benjamin calls himself a mathemagician.  His mathematical tricks are truly fascinating, and you can experience them for yourself through various videos.  He has done a couple of TED talks, including this one.

Arthur Benjamin: A Performance of “Mathemagic”

More recently, he stopped by Huffington Post to give a peek into some of the tricks in his new book The Magic of Math.

How To Square Any Number in Less than a Second, Without a Calculator

As I watched his “explanation” of how he squares the numbers, I was not completely satisfied.  Sure, it’s cool that a magician revealed his secret, but I wanted to know the why and not just the how.  So, I used a little algebra.

Arthur started his explanation with the number 12.  He went down to an easy number, 10.  Since he had to go down 2 to get from 12 to 10, he then went up 2 to go from 12 to 14.  He multiplied 10 and 14.  Then, he added the square of the amount he had to go up and down.  So, that’s 4.  The result was the answer.  He next did the process with 97.  Sure enough, it worked!  But, how?  And does it work with any real number?

I did a little algebra and now see the why, and it does work with any real number.

Let x = the number you are squaring.  Let n = amount you have to go up and down from x.  You multiply (x + n) and (x – n).  Many of you likely recognize the difference of squares.  x^2 – n^2.  Now, he added the square of that difference or n^2.  Well, this gives us x^2 – n^2 + n^2 = x^2.  This is exactly what we are trying to find!

Often, these mathematical parlor tricks can be seen with some variables and a little bit of algebra.  I hope that I didn’t disappoint anyone by revealing the secret, but I am not a magician and figure it is okay.

Ladies and Gentlemen, the next number on the runway is truly perfect!

In the previous post, we looked at numbers that are amicable.  Along a similar line of thinking, let’s look at perfect numbers.  A small example of a perfect number is 6.  What makes it perfect?  Once again, I am going to ask you to take a minute and find the proper positive divisors of 6.  Go on!  I’ll give you time.  (Don’t peek until you are done!)

6:  1, 2, 3

Now, like we did previously, find the sum of these proper divisors.

1 + 2 + 3 = 6

Ta da!

The proper divisors of 6 add up to 6.  Cool!

Any positive integer that has proper divisors adding up to itself is a perfect number.  Can you think of others?  The first 4 were found over 2,000 years ago.  Current findings use a similar line of thinking to that for finding Mersenne primes.  Turns out that perfect numbers and Mersenne primes are closely related.  I will share more on that in the next post.  Until then, happy hunting!

If numbers can get along, why can’t we? We can work it out!

Several years ago, I read the delightful book The Housekeeper and the Professor by Yoko Ogawa.  The story wasn’t exciting, but it was relaxing and enjoyable.  I was impressed by how well the author worked real mathematics into the plot.  I didn’t pick up the book again after completion, as I am not one for reading books or watching movies multiple times.  There’s too much out there, and life’s too short.  Anyhow, I was reminded yesterday that there was a movie made in Japan that stuck closely to the story.  It is called The Professor and His Beloved Equation, and you can find it on YouTube.  You may need subtitles since it is in Japanese.  I watched it in the afternoon, and I was once again struck by how well the story presents some mathematics.  The mathematical concepts are very clearly explained, and they are at a level that most casual viewers can follow.  And I don’t feel like the math was forced in for the sake of including math.  The film flowed naturally.  I do hope that you will take the time to watch, but I want to go ahead and present some math spoilers this week.  A few of the concepts are too good to not write about them.

Let’s start with something called amicable numbers.  Take the numbers 220 and 284 and find the proper divisors (all divisors excluding the number itself).  Go ahead, I will give you a minute.  (Don’t peek until you are done!)

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

284:  1, 2, 4, 71, 142

Now, find the sum of the proper divisors for each number.  (Again, don’t peek until you are done!)

The sum for 220 is 284.

The sum for 284 is 220.

Wow!  The numbers 220 and 284 are an amicable pair.  The sum of the proper divisors of one number is the other number!

This particular pair was discovered by Nicolo Paganini in 1866.  He was only 16 years old at the time!  He was not the first to discover amicable pairs, but he was the first to find this small pair.  Long before Paganini, Arab mathematicians had discovered amicable pairs, some of which were discovered again later in the 17th and 18th centuries by Fermat, Descartes, and Euler.  I encourage you to do some searching.  Other amicable pairs are larger numbers, which are harder to discover but can still be fun to verify.

The New MRI: Faster Than A ?

In the summer of 2012, my young son had a brain MRI to check for a chiari malformation.  (He has a long medical history, due to inheriting a craniofacial syndrome from me.)  After having been through several CT scans in his short life, I was accustomed to worries about sedation so that he would be perfectly still for the imaging.  I was very surprised when we arrived at the hospital, for they did not need to sedate, and the scan was not going to take very long.  He needed the short version.  (The long version would have still required sedation.)  We went through the scan and waited about an hour for the results.  Thankfully, there was no chiari malformation.  He did have to stay in the hospital, as we were also there for ICP (intracranial pressure) monitoring and a skull surgery.  He “just” needed a bone graft and not a full cranial-vault remodeling, and we now knew this from the MRI results.

In the months and year that followed, I didn’t think too much about the speed of the MRI, until a colleague came into my office and shared about Terry Tao and compressed sensing.  Terry Tao is a person who appears frequently in mathematical news, as he is the top mathematician of our age.  Typically, a mathematician picks one small area in which to be an expert and focuses on that for a career.  Terry Tao is rare in that he can pick up a topic of interest, quickly become an expert, and then make large contributions to that area of mathematics.  This just gives you a hint of why my colleague was sharing about Terry Tao.  Anyhow, I have digressed and must get back to compressed sensing.

What is compressed sensing, and how was it possibly related to my son’s MRI?  Compressed sensing involves an algorithm that utilizes l1 minimization.  It takes too much time and storage space to collect every pixel of data for an image.  So, a camera or other device collects a fraction of these pixels.  The l1 algorithm starts arbitrarily picking effective ways of filling in the missing pixels.  Then, the algorithm starts putting in layers of colored shapes over the randomly chosen pixels while seeking sparsity.  It wants to use the simplest kinds of shapes to closely match the existing pixels.  Each new layer will have smaller and smaller shapes.  Eventually, with enough layers, the resulting image will be extremely close to the original.  The simplest or sparsest image is the closest to the original.

So, collecting the data during the MRI scan does not take as long, as fewer pixels are needed.  Processing takes a little while, because the l1 algorithm needs time to work.  I don’t know if this discovery was in place for my son’s MRI, but this story did give me something to relate to.  His scan took only a few minutes, and we had to wait about an hour to see the results.  All but one picture was clear, and we were relieved that he does not have a chiari malformation.

I am eager to see what all developments will come out of this discovery.  Emmanuel Candes, Justin Romberg, and Terry Tao have laid groundwork and proven mathematically that the resulting image after running the l1 algorithm will be extremely close to the original.  Now, people are looking at all kinds of applications.  Besides constructing medical images quickly, people may restore old files, construct images of space, eliminate the need for compression software, and accomplish much more with speed and accuracy.

If you would like to read more, here are 3 articles that helped me to gain better understanding.

Fill in the Blanks: Using Math to Turn Lo-Res Datasets into High-Res Samples

Better Math Could Make Medical Diagnostics 6X Faster

Compressed Sensing Makes Every Pixel Count

Jim Simons: Renaissance Mathematician

Jim Simons is a man who has lived a full life as a mathematician.  While doing mathematical research and laying the foundation for string theory, he also cracked codes for the NSA.  He then went into cracking Wall Street, applying mathematics to investments and becoming wealthy.  He is now active with his Simons Foundation, which supports math and science education and autism research.  Watch this TED talk to find out more.

A Rare Interview with the Mathematician who Cracked Wall Street