A new pentagonal tiling

Just came across news of the discovery of a new pentagonal shape that can tile the plane.

Tiling the plane means that you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps. It is known that any triangle can tile the plane, as does every quadrilateral (a four-sided shape). Tiling using (convex) pentagonal shapes is an interesting problem whose history goes all the way back to 1918. Prior to the latest discovery, there were only 14 distinct tiling patterns. This newly discovered pentagonal tiling is the first discovery since 1985. The three mathematicians who discovered this new pattern are Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington at Bothell. They made the discovery after an exhaustive computer search through a large but finite set of possibilities. No one know if there are more distinct pentagonal shapes that can tile the plane. This piece from the Guardian gives a good background to this problem.

You can find a picture of the new pentagonal tiling in the above piece or this piece from npr.org or this piece from a design company.

The following is a picture of the 15 distinct pentagonal tiling patterns (the new one is on the bottom right).

PentagonTilings15.svg
Source: Wikimedia Commons

An open letter to Dr. Palmer, the killer of Cecil the Lion

Dear Dr. Walter J. Palmer:

Because of your “adventure” in Zimbabwe, you are now the most “famous” dentist in the world. You recently have admitted to killing Cecil, a 13-year-old lion that is the best known lion in Zimbabwe. Cecil was lured from a wild life sanctuary with bait, shot with an arrow by you, tracked for 40 hours, and finally shot dead with a rifle and then beheaded and skinned. No doubt, this would be another trophy to add to your huge collection (you are known to be an experienced hunter with 43 kills of exotic animals under your belt). However, Cecil was wearing a GPS collar as part of a research project run by Oxford University. This meant that sooner or later the world would find out. The news of the killing stirred up a firestorm on the Internet and sparked outrage among conservationists and other people from all corners of the globe including Bloomington, Minnesota, where your dental practice is located. It is not hard to find negative news about you in news media and social media. I do not intend to dwell on the negative. I will let other people do that. Instead, I would like to focus on something of a positive nature. It appears that the days of big-game hunting are over for you. Perhaps you can switch to another type of hunt.

\text{ }

Cecil the lion and Dr Palmer
It appears that the days of big-game hunting are over for Dr. Walter J. Palmer.
Is it time to switch to another kind of hunting?

\text{ }

You are reportedly in hiding. Now you know what it is like to be hunted. Your dental business in Bloomington, Minnesota is, for the time being, shuttered. So this episode is costing you where it hurts, your pocket book. No doubt, you are hoping that the firestorm will eventually die down and you can go back to doing braces and implants.

Of course your trouble is not just limited to the pocket book. You have a real chance of being handed over to the Zimbabwe government, thanks to an extradition treaty with Zimbabwe that was signed by President Bill Clinton (I am sure the Zimbabwe authority is very eager to talk to you). Your actions in Zimbabwe could have violated the U.S. Lacey Act, a conservation law meant to shield animals from harm. So you could also be prosecuted in the US.

It is likely that Bloomington, Minnesota is no longer a viable location for your dental practice. You may think of relocating. You are the most famous dentist in the world at the moment, but for all the wrong reasons. Your notoriety will follow you wherever you go. In the age of social media on the Internet, do you think that you will be able to work as a dentist in relative peace and obscurity from now on? It is entirely possible that dentistry may no longer be a viable career option for you. Now the positive: your economic future is not all gloom and doom (your legal situation is another matter). There is another career option for you – becoming a mathematician. I know this sounds crazy, but hear me out.

You reportedly paid over $50,000 for the hunting trip that ended with the death of Cecil the Lion. You paid big bucks for the thrill of a kill. What did that get you? Nothing but troubles. Do you know there is another hunt that pays you $1 million dollars for each successful “kill”? You don’t need to shell out big bucks. They pay you big bucks for a successful hunt. As an avid hunter with a huge ego, I am sure your ears are perked up. Let you give you more details.

Clay Mathematics Institute proposed seven hard math problems at the start of the new millennium. Each of these seven problems comes with $1 million US dollars in prize money. These are the seven math problems. They are for real. They actually had put $7 million dollars in an account just in case someone comes up with valid solutions. Here’s the rule for the millennium prizes.

The dynamics of the math hunt are the exact opposite of the hunt for lion. With a successful hunt of the solution of one of these seven problems, there will be no outrage. Instead you will be an Internet and social media sensation. You will be a sought after speaker in international math conferences. You will be interviewed by all the notable media personalities and will appear in all the top news shows (not just in the United States but the world over). More importantly, there will be no need to hide. One million crisp green dollar bills are all yours if you solve just one math problem.

In short, solving one of these seven math problems is akin to winning the Nobel Prize (there is no Nobel Prize in the field of mathematics; so solving one of the Millennium problems is a good substitute). So a successful hunt means glory and adulation. Compare with the grief from the hunt of Cecil the Lion, I cannot see how you could turn this down. Of course, you probably cannot turn down $1 million USD. This is a win-win proposition.

Being a smart man that you are, you probably will figure out that these 7 math problems are no ordinary math problems. But you are no ordinary hunter. I cannot see why you will not be successful if you chase these math problems with the same zeal and the same relentlessness (and ruthlessness) in the chasing down of Cecil the Lion. Anybody who can graduate from dental school has the mental and intellectual capacity to become a mathematician (if he or she so chooses). You have the drive for the hunt for business and professional success and for the hunt for exotic animals in the wild. It is now time to apply the same drive in another kind of hunt.

If you are game, I will be happy to provide you with suggestions of where to begin – books and journal articles for you to start studying. Of course, you can also start with doing some Googling yourself. I know you are a self starter.

Some of these seven problems have been around for centuries. They are not a weekend project. Countless of mathematicians had tried in vain to hunt for their solutions. One of the problems is called the Riemann Hypothesis, which was proposed in 1859. Some of the best mathematical minds in the last 150 years had been actively hunting for a solution. You can join them in this hunt. There will be no more “canned hunt”. You will have to blaze your own trail where others had faltered. No one, even the best mathematical minds in the last centuries, can beat these problems into submission. Forget about the trophies that are mounted on the wall. Any one of the 7 millennium problems is the ultimate trophy.

One big difference between the math hunt and the big-game hunt that you are famous (or rather infamous) for. Some of your big-game hunts (including the most recent one) were based on trickery and illegality. You lured Cecil out of the protected sanctuary and killed him in a zone where hunting is not allowed. Math is an honest pursuit. Either you have the proof of a theorem or you don’t. Either you have the solution to the Riemann Hypothesis or you don’t. No amount of dirty tricks can alter the reality. The pursuit of math is a good antidote to your “win at all cost” mentality and can restore some sanity to your soul.

Without the burden that comes from running an active dental practice, I am sure you have a lot of time on hand. Why not put that to good use? If you are ever prosecuted and ended up with a lengthy prison sentence, it just means more time for your math hunt. You can turn your jail cell into a powerful math research center.

Remember: instead of slaying lions and tigers, you slay math problems. Instead of paying for the hunt, you get paid for the hunt. I know that you live for the thrill of the kill. Your new hunting career can begin today.

A fascinating read about Terry Tao

I just came across an article in New York Times about the famous mathematician Terry Tao, who, a one-time math prodigy turned Fields medalist, is a prolific researcher working in diverse areas of mathematics. The title of the article is “The Singular Mind of Terry Tao”. Here is the article.

With Terry Tao being a one of the greatest minds in 21st century mathematics, I know this would be an interesting read. Indeed, it gives a vivid picture of the world of Terry Tao – a prolific mathematician producing important work, a former child prodigy, a husband, father and so on.

It turns out that it is also a gentle introduction of various math concepts. For example, the article has a great short introduction of prime numbers that gives readers a sense that prime numbers are a simple construct that arises out of a concept of numbers and the four arithmetic operations (addition, subtraction, multiplication, and division). Knowing these basic number concepts is all you need to spot the prime numbers. Thus prime numbers are elemental objects in mathematics. It goes on to say that any alien species in other parts of the universe is probably very different from us but “we can be almost certain that their mathematicians have discovered the primes and puzzled over them”. It also points out that “scientists have uncovered deep connections between primes and quantum mechanics that remain unexplained”.

The short intro to prime numbers is to lead the readers to the discussion of the Green-Tao Theorem, which led to the award of the Fields medal for Terry Tao.

Another interesting thing about the article is that it gives the readers a sense of what it means to do mathematics. It is not a static pursuit of solving algebra problems from stale old math books. In fact, “The ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians”. Mathematical research is a fundamentally creative act. It is a very difficult pursuit. It is akin to a struggle with the devil (as the article playfully suggests). Mathematics research is a long game. Doing math research requires courage; it may take weeks, months and years if success comes at all.

Another interesting tidbit of information is the letter of recommendation written by Paul Erdos, the revered Hungarian mathematician, in supporting Terry Tao’s application to Princeton.

    “I am sure he will develop into a first-rate mathematician and perhaps into a really great one,’’ read Erdos’ brief, typewritten note. ‘‘I recommend him in the highest possible terms.”

Based on what we know of Tao, Erdos’ prediction is spot on. The full article is here.

Better math skills may help you keep your home

Math is in the news again. It’s not about the solution of a century old math problem. It is not about the math behind an earth shaking new technology. This time the focus is on the humble basic math skills such as how to calculate the price of a sofa in a clearance sale or how to calculate the earned interest in a savings account. The math news that is in the spotlight points to a tantalizing possibility that if enough people could practice this kind of simple basic math, the deep global recession that occurred a few years ago might have been averted, or its impact might have been greatly reduced.

According to a new study published in the National Academy of Sciences, there is a strong negative correlation between basic math and quantitative skills and the likelihood of defaulting on a subprime mortgage. In other words, borrowers with lower math skills are much more likely to default on their loans and borrowers with higher math skills are much less likely to default on their loans. It is well known that a massive increase in the volume of subprime mortgage defaults in years 2006 and 2007 helped push the economy into a deep global recession in 2008.

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The Study

The study is based on data from subprime mortgages originated in 2006 and 2007 in greater New England area as well as data from a phone survey involving 339 holders of these subprime mortgages. The phone interviews assessed the numerical ability, financial literacy, verbal skills and cognitive ability of the 339 participants. Numerical ability in the study refers to the proficiency of a borrower for solving basic mathematical calculations.

One of the results of the study is that the numerical ability is a strong predictor of which borrowers will be more likely to be delinquent in loan payments or to be foreclosed upon (the lower the numerical ability, the higher the likelihood of default).

“Our analysis raises the possibility that limitations in numerical ability may have significantly contributed to the massive amount of defaults on subprime mortgages in the recent financial crisis,” concluded the researchers of the study.

Could it be that the loan delinquency of the participants of the phone survey was due to other factors such as income, education and credit scores? For example, could the results of the study be due to the low credit scores of the participants? If the participants of the study had defaulted prior to the study, they would be more likely to default again.

After the researchers of the study adjusted for these factors, the connection between poor math skills and higher default rates remain the same. As a result, the researchers concluded that it is likely the effect of the math skills that is driving the results and not the credit scores and other factors.

The researchers of the study also did not find any significant connection between the poor math skills and the types of mortgage loans among the participants of the study. The correlation between poor math skills and higher loan delinquency is present across all the loan types. So it is not that the borrowers in default happened to choose risky loan types such as adjustable mortgages or negative amortization mortgages. So the study suggests that restricting these risky types of mortgages may not prevent future mortgage default epidemics.

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Some Comments

So all eyes now point to math and quantitative skills. What kind of numerical questions were asked in the phone survey? They are not calculus problems and they are certainly not advanced math in financial modeling. They are:

  1. In a sale, a shop is selling all items at half price. Before the sale, a sofa costs $300. How much will it cost in the sale?
  2. \text{ }

  3. If the chance of getting a disease is 10 per cent, how many people out of 1,000 would be expected to get the disease?
  4. \text{ }

  5. A second hand car dealer is selling a car for $6,000. This is two-thirds of what it cost new. How much did the car cost new?
  6. \text{ }

  7. If 5 people all have the winning numbers in the lottery and the prize is $2 million, how much will each of them get?
  8. \text{ }

  9. Let’s say you have $200 in a savings account. The account earns ten per cent interest per year. How much will you have in the account at the end of two years?

We know math is useful. In fact, this study provides empirical evidence that math pays. Having proficient level of math and quantitative skills can lead to favorable and optimal financial outcome in a person’s life. Having good number skills can help any person makes better financial decisions and better manage his or her financial affairs. At least, having good quantitative skills can help avoid costly financial dislocation that comes from defaulting on mortgages, as the study suggests.

The authors of the research study also indicated that their “results indicate possibly large benefits from increased financial education of homeowners.” Certainly, greater emphasis in financial education (and quantitative skills) in high school can have a positive effect on financial outcomes later in life.

Our view is that we do not need to wait for school reform to take place. We can make changes at the individual level. The above five questions are not math problems that can only be taught in school. These are problems that we encounter in our daily life. So in order to improve on quantitative skills, we do not necessarily need to go back to school to take a course in finance. We can improve on quantitative skills one purchase at a time and one bank transaction at a time. Come up with a right question. Then take out a calculator and a notepad and try to resolve the issue that is at hand.

Some students cannot wait for schooling to be over so that they don’t have to deal with math any more. This jaundiced view of math is unfortunate. There is school math and there is math in everyday life. This study published in the National Academy of Sciences shows that it pays to have skills in the latter.

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Further Information

Here’s a few news outlets that reported on the same study.

This is the study published in the National Academy of Sciences.

Cheering for Yitang Zhang

If you Google the name of “Yitang Zhang” recently, you will find many entries with keywords such as “twin primes”, “twin prime conjecture”, “bounded gaps between primes”, along with “University of New Hampshire”. Here’s three of the entries: UNH professor solves ancient mathematics riddle; UNH lecturer stuns the math world; The Beauty of Bounded Gaps. Here’s a piece from New York Times: Solving a Riddle of Primes. The reason for all this attention from the mainstream media that normally don’t pay attention to math? Mr. Zhang made a huge breakthrough toward solving an ancient problem about prime numbers. This is a brief account of his achievement.

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A prime number is an integer greater than 1 that has no divisor other than 1 and the number itself. For example, 3 is a prime number because 1 and 3 are the only positive integers that evenly divide 3. However 6 is not a prime number since 3 evenly divides 6 in addition to 1 and 6. The first several prime numbers are: 2, 3, 5, 7, 11, 13, …

A pair of prime numbers are twin primes if they differ by 2, for examples, 3 and 5, 5 and 7, 11 and 13, 17 and 19 and so on.

The Greek mathematician Euclid gave the oldest known proof that there are infinitely many prime numbers around 300 BC. Euclid also conjectured that there are infinitely many pairs of twin primes.

The statement that there are infinitely many pairs of twin primes is called the Twin Prime Conjecture. Note that the Twin Prime Conjecture is saying that there are infinitely many pairs of primes such that the gap within each pair is exactly 2. Many great mathematical minds since ancient time had been trying to prove this conjecture but to no avail. But there were incremental progress throughout the twentieth century and in the first decade of the new century.

Mr. Yitang Zhang did not prove the Twin Prime Conjecture. He proved that there are infinitely many pairs of prime numbers such that the gap within each pair is bounded is at most 70,000,000 (70 millions). In other words, Mr. Yitang Zhang proved the Bounded Gaps Conjecture.

The gap of 70 million seems like a large number and may not seem all that significant to non-mathematicians. Now that there is now a proof that the gaps between pairs of primes do not have to increase without bound (as some mathematicians had suspected), mathematicians can work on narrowing the gaps (may be even reducing the gaps down to 2).

Indeed, since the announcement of Mr. Yitang Zhang’s result in May, some mathematicians had narrowed the gaps down to mere millions (in one case down to hundreds of thousands)!

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His achievement is even more impressive considering the background of Mr. Yitang Zhang. Mr. Zhang did earn a PhD in mathematics in 1991 from Purdue University. In the tough academic job market after graduation, he could not find any academic position. He worked as an accountant for a period. He even made sandwiches at a Subway shop at one point. He is currently a part-time math lecturer at the University of New Hampshire, a nice school for sure, but far from the elite club of mathematicians working at places like Harvard, Princeton and Stanford.

Some people have this notion that monumental math problems such as the Twin Prime Conjecture are the domain of young math whiz. Mr. Zhang is over the age of 50.

Another handicap for Mr. Zhang is that he is not in the math specialty of number theory, which would be the specialty of the mathematicians who work on problems like Twin Prime Conjecture.

The last academic publication of Mr. Yitang Zhang is from the year 2001. So he is considered an inactive researcher by many in the math community.

Given his background, no one in the math community expects great results from Mr. Yitang Zhang. Apparently he was oblivious of his supposedly “handicaps”. He certainly did not make any excuses for himself. He just kept plugging away, building upon the advances made by other mathematicians.

The story of Yitang Zhang is remarkable in two fronts. One is his mathematical work. The other is the human dimension of the story, which makes his achievement all the more remarkable. Mr. Yitang Zhang is an inspiration to us all.

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More information about Yitang Zhang can be found on the Internet. Here’s a few more entries from an Internet search.

Who Wants to be a Millionaire?

It is rare to find reportings of math in the mainstream media that explain an actual algebra equation. It is rarer still to find such reportings that involve a math problem attached with serious prize money. The other day I found just such a piece in time.com – a math problem with an award of $1 million (US dollars). I would like to discuss this math problem and its significance.

It turns out this is a number theory problem that is intimately connected to the famous math problem called Fermat’s Last Theorem. So that is where we start.

    Fermat’s Last Theorem
    There are no positive integers A, B, C and x that satisfy the following equation

      \displaystyle A^x+B^x=C^x

    where x is greater than 2.

A bit of perspective. The requirement that x>2 is necessary. If x = 2, then the equation \displaystyle A^2+B^2=C^2 would have many solutions. For examples, \displaystyle 3^2+4^2=5^2 and \displaystyle 5^2+12^2=13^2 (note that this is the familiar Pythagorean Theorem).

Fermat’s Last Theorem was stated as a true mathematical fact by Pierre de Fermat in 1637 (at least he implied that he had a proof). He scribbled in the margin of his own copy of the Greek text Arithmetica by Diophantus commenting that a proof of this statement was too large to fit in the margin.

No one knows whether Fermat really had a proof of his conjecture. But we do know that no one could prove Fermat's last Theorem until 1995. For several hundred years (until 1995), Fermat's Last Theorem was regarded as one of the most difficult mathematical problems.

In 1993 Andrew Wiles, a British mathematician, had formulated a proof of Fermat's Last Theorem after working for 6 years in secret. But a flaw was discovered in one part of the proof. In 1995, after working with a former student for over a year, Wiles finally had patched the hole that was in the original proof. Thus Fermat’s Last Theorem had ceased to be a conjecture and became a true mathematical statement.

Apparently the successful resolution of Fermat’s Last Theorem was not the end of the story. Soon after, people began to hunt for more research opportunities by generalizing Fermat’s Last Theorem. Someone proposed a new problem, which is now called Beal’s Conjecture (after Andrew Beal, the person who proposed it). Consider the following statement.

    Beal’s Conjecture
    There are no positive integers A, B, C, x, y and z that satisfy the following equation

      \displaystyle A^x+B^y=C^z

    where

    x, y and z are greater than 2, and

    A, B and C share no common prime factor.

Note the similarity between this new statement and Fermat’s Last Theorem. In fact, Fermat’s Last Theorem is a specific case of Beal’s Conjecture. If the exponents are made the same (and greater than 2), we come back to Fermat’s Last Theorem and we know that the equation \displaystyle A^x+B^x=C^x has no integer solutions (as proven by Andrew Wiles).

However, when the exponents are not equaled, the equation \displaystyle A^x+B^y=C^z does have integer solutions. To illustrate, note that \displaystyle 3^3+6^3=3^5. Here, A, B and C share the common prime factor 3. Another example: \displaystyle 7^6+7^7=98^3. Now A, B and C share the common prime factor 7. Beal’s Conjecture is essentially saying that if the exponents x, y and z are greater than 2, the equation \displaystyle A^x+B^y=C^z has integer solutions only when A, B and C share a common prime factor (just like in the 2 examples just given).

Andrew Beal, the one who proposed the Beal’s Conjecture in 1993, is a Texas billionaire banker who is also an enthusiastic amateur mathematician. After unsuccessful attempts at resolving his conjecture, he put up prize money to spur interest – $5,000 in 1997 and $100,000 in year 2000. He upped the prize money just recently to $1 million in June 2013.

Just in case there is any reader who still thinks the prize is a joke or a hoax, keep in mind that Andrew Beal is a prominent banker with financial worth in the billions. In fact, the $1 million prize money is already in safe keeping at the American Mathematical Society (AMS). Here’s is more information about the Beal’s Prize. Mr. Beal would also like to use this prize to draw young people into the wonderful world of mathematics. According to AMS, “one of Andrew Beal’s goals is to inspire young people to think about the equation, think about winning the offered prize, and in the process become more interested in the field of mathematics“.

Beal’s conjecture is no doubt a significant math problem because of its connection with Fermat’s Last Theorem. The serious prize money attached to it will give it exposure and generate some buzz. It is no doubt a difficult problem, clearly not a one-weekend project and clearly not a problem that can be solved by just using elementary math techniques (but applied in a clever way). It took over 350 years to resolve Fermat’s Last Theorem. So it will likely take years before we see any progress on Beal’s Conjecture (the conjecture is only 16 years old).

Math problems like Beal’s Conjecture are not a fool’s errand. Many new mathematical ideas and methods will often be created in the process of resolving monumental problems such as Fermat’s Last Theorem and hopefully in the case of Beal’s Conjecture, thus pushing forward the frontier of mathematics. No doubt this is one of the goals of Mr. Andrew Beal in setting up the Beal’s prize.