Too Little, Too Late?

I had a birthday about a month ago, and with that birthday I entered a new decade.  I won’t reveal which decade, but let’s just say that my age is starting to really hit me.  I am not old, but I am not young either.  And this led to me reflecting on what I’ve done in my life.  There are so many things that I have accomplished, but there are so many things I still want to accomplish.  But am I now too far along to still achieve certain dreams?  If I look closely at the world of mathematics, the message seems to scream “Yes!”  After all, the top prize of mathematics, the Fields Medal (like a Nobel Prize for mathematics), is only awarded to people under the age of 40 years.  So, what does that mean?  Are the people who established the prize saying that mathematicians cannot possibly be noteworthy after reaching that particular age?  It sure feels that way.

But, just as I was experiencing a decline in hope, I thought about what some have done well past that point.  Take Yitang Zhang, for example.  A colleague shared the story of this quiet mathematician, who had a Ph.D. but struggled to get a university position.  His struggles were so stark that he spent time building sandwiches at Subway and sleeping in his car.  But, he never let his circumstances distract him from his work.  Tirelessly and silently, he plugged along in his research.  Then, he astounded the world of professional mathematicians with his discovery regarding the Twin Prime Conjecture.  (The Twin Prime Conjecture says that there are an infinite number of pairs of primes that differ by 2, such as 3 and 5 or 11 and 13.)

Zhang didn’t prove the conjecture outright, but he made huge progress in narrowing down how often prime pairs occur.  He didn’t show that there are infinite pairs with a difference of 2, but he did say that there are infinite pairs with a difference less than 70 million.  I know this doesn’t sound impressive at first, but he showed that we can start narrowing down pairs into infinity.  And this was the foundation needed by other mathematicians to build upon and narrow down the gap even further.  Suddenly, Zhang was a sensation!  It was not too late!  He prefers to avoid the limelight and continue working in the quiet, but he now has a prestigious university position and a place in mathematics history.  And this happened in his 50’s!  It is too late for a Fields Medal, which is a shame.  But it is not too late to achieve.  And that gives me hope.

Why do I need this hope?  Well, I am a teacher of many non-traditional students.  Adults come into my classrooms, hoping to reinvent themselves in later years.  Often, they are older than me.  And I am in awe of the sacrifices these people make to improve the lives of themselves and their families.  They are reaching for their dreams and achieving, not allowing age to stand in the way.  I watch my own students and draw inspiration from them.  But, I still wonder if I can do the same.  Is it possible in my chosen field?  And figures such as Yitang Zhang tell me that it is wholly possible and never too late.

*Recently, Quanta magazine highlighted the places that some top researchers visit to do their deepest thinking.  I particularly like Zhang’s location and dream of one day being there, too.

Yitang Zhang at the Beach in Santa Barbara

Paper Folding Fun

On this Sunday morning, I am yet again the owner of a losing lottery ticket.  I know.  I know.  The probability of winning is so slim, why waste my money?  I figure that $1 a week isn’t too much to spend on a little bit of hope, and some of the proceeds fund the Bright Futures Scholarships that help some of my own students.

But, I don’t want to get into a lottery discussion here.  No, I’d rather talk a little bit about the paper on which the disappointing numbers were printed.  I have this absentminded habit of folding paper in my hands, especially if it is destined for the trash.  I did this with the lottery ticket, while my son was sitting nearby.  And I felt the need to launch into an exploration on how many times I could fold my ticket before I couldn’t go further.

I started out with 0 folds, a paper that was 1 layer in thickness.  I folded it once, in half.  This resulted in 2 layers of thickness.  I folded it again, now having 4 layers of thickness.  And again, resulting in 8 layers of thickness.  Do you see a pattern so far?  The number of folds and the number of layers are related by powers of 2.  Raise 2 to the number of folds, and you have the number of paper layers.

As I commented on this, my son groaned something along the lines of “Do you have to turn this into a math discussion?”  To which I replied, “Why, yes, yes, I do.”  And I kept folding.

I managed to get it to 6 folds or 2^6 = 64 layers.  After that, the paper was too thick to force another fold.  And then I got curious about how far people have gone in this quest.  So, I relied on trusty Google and found an article with a couple of good embedded videos.

Turns out, the myth is that the paper can only be folded 7 times.  The first video on the site below is a Mythbusters episode that sought out to test this myth.  I’m not going to share their results.  Just watch the video.  If you like big things and heavy power equipment, then you may enjoy their experiment.

As I read further, I learned about a young lady named Britney Gallivan who actually derived a paper folding theorem and then used toilet paper to prove it could be folded 12 times.  Her findings are fascinating.  Near the bottom of the page, there’s a video of a group using toilet paper at MIT to take this experiment to another level.  So fun!  I encourage you to read and view at the link below.  And, if you are inclined to do your own paper folding, please don’t hesitate to share your results in the comments.  Happy Sunday Funday!

How Many Times Can You Fold a Piece of Paper?


What if I told you that one of the top mathematicians of today makes discoveries in his field through valuable playtime?  It’s true!  And he is not the only one.  Just think about why children play.  For a child, playtime is work.  It is their way of figuring out how the world works as they grow up.  But why does it have to stop at adulthood?  It doesn’t, and it shouldn’t.  For someone such as number theorist Manjul Bhargava, playtime is his way of figuring out how the world works.  And he is very good at it!  In fact, he won a Fields Medal in 2014.  (A Fields Medal is only one of the highest honors a mathematician can receive, akin to a Nobel Prize.)

And how does he play?  For one, he is an artist.  He studies Sanskrit poetry and is an accomplished musician.  Both of these pursuits are rich in mathematics.  Did you know that Sanskrit poetry has the Fibonacci numbers?  I talked about Fibonacci numbers in the previous posts Flowers and Fibonacci and Fibonacci Fun.  Except, in India, the numbers in the famous sequence are called the Hemachandra numbers.

How else does he play?  His office at Princeton University is littered with mathematical toys such as Rubik’s Cubes, Zometools, and puzzles.  And what studies about Fibonacci/Hemachandra numbers would be complete without a collection of pine cones?  These toys are not fun and games.  But, maybe they really are?  Either way, Rubik’s cubes helped Bhargava to solve a 200-year-old number theory problem while he was still a graduate student at Princeton.  Talk about the power of play!

If you want to read more about this accomplished mathematician, check out the following article in Quanta magazine:   2014 Fields Medal and Nevanlinna Prize Winners Announced

The Conception Algorithm

How do you feel about someone pinpointing when you had sex?  Too personal?  Not an appropriate question for this blog?  Actually, it’s a very appropriate question when doctors are trying to pinpoint the fetal age of a baby.  Judgment of when a woman became pregnant can be inaccurate by as much as several weeks, and this can negatively affect care decisions in cases of premature births.

But, mathematics has come to the rescue!  Just last year, scientists Abbas Ourmazd and Russell Fung published their algorithm in the journal Nature, and the algorithm is now being used at the University of Wisconsin-Milwaukee to estimate time of conception.  Thanks to funding from the Bill and Melinda Gates Foundation, these academics are well on the way to pinpointing exactly when a pregnant woman had sex.

If you would like to read a little more, check out the story in Science Daily at

Physicists to use their unique tool to improve neonatal health

Pathway to a Cure for Diabetes

“Scientific breakthroughs are often facilitated by mathematics.” – Richard Bertram, FSU Professor of Mathematics

And mathematics is leading science on the path to finding a cure for diabetes.  At Florida State University, Richard Bertram created the models for reactivating the oscillations in insulin-producing pancreatic beta cells.  Then, he collaborated with Michael Roper, FSU Associate Professor in Chemistry and Biochemistry.  Roper built a device that can deliver a liquid glucose solution to dormant pancreatic beta cells.  With Bertram’s models, the device was programmed to deliver precisely the right amount of glucose in rhythmic pulses of exactly the right size and frequency.  This programming mimics how the body delivers the glucose, and it triggered the dormant cells to start working again in a healthy manner.

Such a breakthrough is amazing news for those who struggle with Type 2 diabetes.  And, the models and device are a stepping stone in finding a cure.

If you would like to read more about it, check out the article at ScienceDaily – Crunching the numbers: Researchers use math in search for diabetes cure.

The Man Who Knew Infinity

I watched a movie yesterday and highly recommend it.  The Man Who Knew Infinity is based on the true story of the mathematician Srinivasa Ramanujan.  I don’t want to say too much for fear of giving it away, but the movie takes us through the development of Ramanujan’s mathematical career.  He was a brilliant, self-taught man whose work is still being pored over.  Although the applications of some of his discoveries were not apparent at his time, his insights are being applied today to the study of black holes.  The movie does have an abrupt, sad ending, which I should have seen coming.  Once I shook off enough of the depressed feelings from the end, I could think for a while on one of the mathematical truths in the movie.

Really, this mathematical truth is one for which G.H. Hardy and Ramanujan are quotably famous.  G.H. Hardy had gone to visit Ramanujan in the hospital, and he remarked to Ramanujan that his cab number of 1729 was a dull one and hopefully not an omen.  Ramanujan replied, “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”  I had heard this quote many times before the movie, but I hadn’t taken the number 1729 and played with it.  So, I did so for a little while yesterday evening.  I made a list of the cubes from 1 to 1,728.  I stopped there, because a larger cube would take me beyond 1,729.  I then examined my list to see which ones would add up to 1,729.  I quickly found the sums 729+1000 and 1+1728.

The process was not difficult, but it did leave me to ponder how Ramanujan had come to his conclusion.  Was it really as simple as that?  The movie indicated that he was brilliantly fast when it came to computations in his head.  He said that he would pray to his family god and then have the results appear in his head.  But, what was really happening?  Did he just recently work with this number 1,729 so that it was fresh in his mind, or did he do an analysis that quickly in his interaction with Hardy?  Either way, it’s impressive.

I went on to stare a while longer at the numbers.  The prime factorization of 1,729 is 7 x 13 x 19.  I thought it interesting that 13-7=6 and 19-13=6.  And, if you factor the sum of cubes, you end up with either 19×91 or 13×133.  The digits of the first number are found in the first and last digits of the second number.  Likely a coincidence but still fun to ponder.  After this, I grew tired and quit thinking on 1,729.  I believe that I will revisit it sometime soon and will post when I do.

DNA and Fractional Dimension

Japanese scientists recently made a breakthrough in the study of DNA movement through living cells.  They applied mathematical analysis and derived a formula that not only describes the DNA movement through living cells but may lead to other significant discoveries such as the revelation of the 3D architecture of the human genome.  You can read more about the discovery at the following link:  Mathematical Analysis Reveals Architecture of the Human Genome

As I read the article, I saw the words fractional dimension and immediately perked up.  I first heard about fractional dimension when I started to learn about fractals, and I love fractals!  If you would like to learn about fractional dimension, check out this video.

Fractional Dimension

As the scientists study how densely DNA is packed in a cell, they take into consideration the DNA’s fractal dimension.  They believe that the fractal dimension will lead to an understanding of how cells use certain genes.  Stay tuned for their next discovery!

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