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?