The Sizes of Infinity

You may have read this title and went “Huh?  Why is this about sizes of infinity?  Isn’t infinity just…infinity?  It’s the concept of something that goes on and on and never stops?  It’s not a number, so how can we talk about size?”

Well, you can thank a gentleman named Georg Cantor for this discussion.  He was the founder of set theory, and he showed that the infinite set of real numbers is actually larger than the infinite set of natural numbers.  If both go on forever and ever, how does one compare the sets?

Cantor had an ingenious explanation.  Take an infinite set such as the whole numbers {0, 1, 2, 3, 4, …} and an infinite set such as the natural numbers {1, 2, 3, 4, …}.  Each element in the whole numbers can be paired with an element of the natural numbers.

0 –> 1

1 –> 2

2 –> 3

And so on.

There is a one-to-one correspondence between the elements of the 2 sets.  Also, each set is countable.  You can assign a counting number to each element in each set.  These 2 sets are the same size.

Now, make the same comparison of the natural numbers with the real numbers.

1 –> 0.12345…

2 –> 1.45286…

3 –> 2.93789…

4 –> 3.19887…

And so forth.  Try to match each natural number with each real number.  Just when you think you’ve covered them all, take the tenths digit of the first real number, the hundredths digit of the second real number, the thousandths digit of the third real number, and so on to get 0.1578…  Now add 1 to each digit to get 0.2689…  Uh-oh!  You get a new number that isn’t on that “exhaustive” list of real numbers!  This new number is different in the tenths place from the first number, different in the hundredths place from the second number, different in the thousandths place from the third number, and so forth.  You will never find a real number in the “exhaustive” real number list that is exactly the same, because it will differ from this new number in at least one decimal place.  We found a real number that wasn’t included in the list of real numbers!  It’s a paradox, and it shows that the real numbers are uncountably infinite.  And this infinity is bigger than one that is countable.  It can be demonstrated as a continuum on a number line, while a set that is countably infinite cannot.

So, Cantor showed that some infinities are larger than others.  From there, he couldn’t decide if there was another size of infinity between the size of the integers and the size of the real numbers.  He came up with the Continuum Hypothesis, which says there is not.  But we don’t know for sure.  It’s at the top of the list of unsolved questions in mathematics to this day.

The fact that we can’t answer this one particular question doesn’t stop mathematicians from asking more related questions and seeing if those can be solved or even lead to the sought-after answer.  One such question was recently answered and made the news.  Mathematicians were asking whether 2 orders of infinity, p and t, were equal or not.  These 2 orders are the minimum sizes of collections of infinite sets of the natural numbers with particular properties, and both are larger than the set of natural numbers.  If someone showed that p < t, then this would prove the existence of an infinity between that of the natural numbers and that of the real numbers, thus disproving the Continuum Hypothesis.  What happened?  They proved that p = t.  So, the Continuum Hypothesis still stands and remains unproven.  But the work will continue.  Some mathematicians feel unsettled by this result.  They have a feeling that more infinities exist, but so far they just keep coming up with the same 2.  They won’t let it rest until the Continuum Hypothesis is proven or disproven.

If you would like to read more about this recent discovery, check out the article in Quanta Magazine at Mathematicians Measure Infinities and Find They’re Equal




Hurricane Math

My corner of the world is currently gearing up for a record-breaking storm: Hurricane Irma.  I live in south Florida, not too far from Miami.  Food, water, supplies have been flying off shelves for over a week.  Long lines for gas are the immediate norm.  And the NOAA hurricane site is one of my favorites.

While eating lunch today, I decided to turn on the Weather Channel for a bit.  Normally, I avoid TV reports.  The excitement of the weather reporters does little to assuage my anxiety, and I quickly grow tired of the same images being used to generate fear.  But something came over me.  Anyhow, I did benefit a little from my viewing time.  I learned something about the math of the categories they use to describe the storms.

I already have an innate sense of what Category 1, Category 2, Category 3, Category 4, and Category 5 mean.  I’m a native Floridian, and I was one of the “lucky” people who lived in a centralized inland location and had 3 hurricanes pass over directly in 2004 (Charlie, Frances, Jeanne).  And I know that each category up has higher wind speeds with greater potential.  I also knew that this scale, the Saffir-Simpson Scale, is approximately linear in nature when describing the sustained wind speeds.

Category 1:  74 – 95 mph

Category 2:  96 -110 mph

Category 3:  111 – 129 mph

Category 4:  130 – 156 mph

Category 5:  157 mph and higher

(Information courtesy of the NOAA at

What I didn’t know is that the destruction caused by these winds actually goes up exponentially with each category.  The good weather man on TV mentioned this, and a cool graphic popped up.  Yup, the graph was definitely an exponential curve.  Probably not the best thing to ponder at the moment, considering that the storm is a Category 5 destroying the Caribbean as I write this and will likely be a Category 4 if/when it hits my area.

But it is incentive to keep preparing.  Shutters are going up tomorrow morning.  We have water, food, flashlights, manual can opener, radio.  And lots of prayers are being said.  The storm shifted a little east.  If it does a little more, then it can still go up in the Atlantic.  And maybe, just maybe, it will curve away and dissipate.  There’s still time for that miracle.  In the meantime, while I still have power and internet, I plan to do a little more research on hurricane math.  I’ll post again if I find anything worth sharing.


Tantalizing Tessellations

Several years ago, I mentioned tessellations or tilings in Bee’s Needs .  For that post, I focused on the regular hexagons of honeycombs, but I mentioned that squares and equilateral triangles are the other regular polygons that may be used to create a tiling.  In fact, those are the only 3 regular polygons.  Why?  Well, it has to do with the measures of the interior angles.  These are the only 3 regular (all sides same length, all interior angles same measure) polygons where the interior angle measures are factors of 360.  An interior angle of a square is 90 degrees, and 90 x 4 = 360.  An interior angle of an equilateral triangle is 60 degrees, and 60 x 6 = 360.  An interior angle of a regular hexagon is 120 degrees, and 120 x 3 = 360.

Why is this necessary?  If you tessellate these figures so that they touch at a corner and fit tightly, the touching angles go all the way around with no gaps, a revolution of 360 degrees (like a circle).  Other regular polygons come up short and leave a gap.  You try to fit regular pentagons so that they touch at the corners?  Before you make a revolution, your angles are at 108 + 108 + 108 or 324.  There’s a gap of 36 degrees!

It’s not to say that you can’t tessellate other shapes.  It’s just these are the limits for regular polygons.  Now, if you allow yourself to use a combination of different regular polygons or even irregular polygons, then so many more options exist!  And mathematicians have been on a quest to identify and categorize more tessellations, especially ones where the same polygon is used over and over.

Recently, French mathematician Michael Rao built on the works of several mathematicians before him, and proved that only 15 types of convex pentagons may be used to create a tiling.  He also proved that only 3 types of convex hexagons plus all types of quadrilaterals and triangles may be used.  Again, these are the tessellations where the same shape is being used over and over, without other shapes. And he was working beyond regular polygons.  The work has been many years and many mathematicians in the making, and Rao did use a computer to assist.

Rao’s proof must withstand the tests of his colleagues, but other mathematicians seem hopeful that the proof will hold up.  If you would like to read more about it, check out the article in Quanta magazine.

Pentagon Tiling Proof Solves Century-Old Math Problem

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!