by Tim Chartier
[This article is cross posted from The Huffington Post] As the last days of April unfold, we head into May and the end of the school year. Many classes focus on testing and final grades. Teachers often must focus and ready their students for endoftheyear testing. Math classes will be asked problem after problem and question after question. In all those classrooms, a thought probably, if not often, races through someone’s mind. Yes, the thought… the one that makes pencils heavier, word problems harder and students wish they were somewhere, anywhere but where they are. There are a lot of ways that thought turns into a question. A common one: “Why study math?” So let’s go and ask, particularly given that we are in April, which is Math Awareness Month. For some, math may be something to beware of rather than be aware of. In fact, that’s precisely the point of the month. Math has many applications, from theoretical to applied. Mathematicians continue to expand the boundaries of what we know mathematically. With the publication of each new issue of a journal, the field of math grows. NBA teams use mathematics to gain a competitive edge over their opponent. Will the better team with better mathematics win? It definitely helped the Oakland As in 2002 with the math that became known as Moneyball. Every day, credit card numbers are encrypted to allow for secure online transactions. Developing methods of encryption that simply cannot be broken with a faster computer comes from mathematics. Studying math enables one to appreciate and possibly understand its applications. Yet one does not need to study math just so the techniques can be used in theoretical or applied settings. Mathematics teaches a way of thinking. Returning to basketball, mathematical formulas won’t pop off the court. Someone must derive them and study them to ensure their usefulness. It can take time to gain such insight. The process toward such understanding is what probably draws many mathematicians to their field. I like to think of it as a path of wonder. For example, I’ve periodically been contacted by ESPN’s Sport Science program to aid in their analysis. They call when they are stuck. When the problem is first presented, my first thought is, “I have no idea how to do this.” And yes, every time I have found a way. Part of this stems from my awareness of that path of mathematical wonder. You don’t have to simply know the answer to a math problem to solve it. In fact, math is usually more interesting when you don’t know how to solve a problem. Would a jigsaw puzzle be fun if it had only two or three pieces? You never know exactly how to fit a 1,000piece puzzle together when you start, and you won’t always try to fit connecting pieces. It’s a puzzle, so you explore and experiment. Math can be the same way. As such, there is a certain sense of mystery to math. You step into a question and simply stand in the unknown. Then you begin to explore, looking for pieces that fit together. This type of thinking is helpful for life, as it offers its unknowns. In life, you may be forced to stand in the unknown. What questions do you want to explore, and what pieces do you want to try to fit together? Some math ideas are developed through a similar process of exploration. For example, about 10 years ago, I learned how Robert Bosch, Adrianne Herman and Craig Kaplan were creating pictures like the one that I made (after learning their ideas) below. The image above is a portrait of Martin Gardner, who we’ll return to momentarily. Later, it occurred to me that I could make mazes with these images if I used a math formula developed by Leonhard Euler, who lived in the 1700s. Seeing that I could fit these two ideas together — one about a decade old and another hundreds of years old — enabled me to create mazes for my book Math Bytes. Returning again to the NBA, here is such a maze: Click here if you’d like a larger version of the maze. This creative edge of math engages me. It makes teaching math every day at Davidson College a great job. And it makes answering that question “Why learn math?” a question I look forward to being asked. But does this sound like the mathematics you know? If not, then you might want to spend some of these last days of April exploring the Mathematics Awareness Month website. The theme for April 2014 is Mathematics, Magic and Mystery. Each day of the month an engaging idea of mathematics has been unfolded. See the ones already shared and await those yet to come. Learn secrets of mental math, mathematics of juggling, optical illusions, and many more interesting ideas and the math behind them! Want to dig deeper? Note that the theme was chosen as 2014 marks what would have been the 100th birthday of Martin Gardner. Simply put, he engaged millions in his mathematical writing and made mathematicians and children alike aware of the wonders and mysteries of math. So be aware of math! It has many applications, from magic to sports to the theoretical to the historical. I often tell my students in class that if you don’t like math, it may simply be that you haven’t discovered the area of math that fits the way you think! Be careful of sampling from only one part of the math buffet and walking away. A great place to sample many engaging ideas of math is every April with Math Awareness Month. This April, you can learn math and soon engage friends and family with ideas in the mystery and magic of mathematics! So why study math? It develops your mathematical sense, which enables you to see life through that lens. In the process, you hone your ability to think in ways that can make you more aware of life itself. So enjoy these last days of April and be aware of math! Follow Tim Chartier on Twitter: www.twitter.com/timchartier 
Exclusive content from Princeton University Press Click on these thumbnails to open larger JPGs Who are these mathematicians? Leave your guesses in the comments. 
Don’t Beware of Math… Be Aware of It!
THIS IS MATH: Beautiful Geometry
Since this is still April, I will direct you back to the Math Awareness Month Calendar to the window marked The Beautiful Geometry of Crop Circles. You can use a compass and ruler to make beautiful geometric patterns and you can use other media as well. Many of you probably have already done this using a Spirograph.
To find out more about the connection between art and geometry, I will point you to Beautiful Geometry. Eli Maor, who is a mathematician, and Eugen Jost, who is an artist, teamed up to illustrate 51 geometric proofs and assorted mathematical curiosities.
Let’s start with one that most people know about—the Pythagorean theorem or a^{2 }+ b^{2} = c^{2}. No one knows exactly how many proofs there are but Elisha Loomis wrote a book that includes 367 of them. The following illustration is a graphical statement of the theorem that if you draw a square on each of the three sides of a triangle, you will find that the sum of the areas of the two small squares equals the area of the big one.
If you look at the colorful figure below by Eugen Jost, you will see something similar, but much more interesting to look at. The figure above is a 30, 60, 90 degree triangle whereas the one below is a 45, 45, 90 degree triangle.
25 + 25 = 49, Eugen Jost, Beautiful Geometry
Using the Pythagorean formula, we know that 5^{2} + 5^{2 } should equal 7^{2}. Now this means that
5^{2 } + 5^{2} = 7^{2}
25 + 25 = 49
I think we all know that is just not true, yet we know that the formula is correct. What is going on here? It seems that the artist is having a bit of fun with us. Mathematics must be precise but art is not bound by the laws of mathematics. See if you can figure out what happened here.
Where’s the Math?
We know that there are at least 367 different proofs for the Pythagorean theorem but the most famous of them is Euclid’s proof. Eli Maor will walk you through it below, and, he will not try to trick you.
Important Note: We are going to assume you agree that all triangles with the same base and top vertices that lie on a line parallel to the base have the same area. Euclid proved this in book I of the Elements (Proposition 38).
Before he gets to the heart of the proof, Euclid proves a lemma (a preliminary result): the square built on one side of a right triangle has the same area as the rectangle formed by the hypotenuse and the projection of that side on the hypotenuse. The figure above shows a right triangle ACB with its right angle at C. Consider the square ACHG built on side AC. Project this side on the hypotenuse AB, giving you segment AD. Now construct AF perpendicular to AB and equal to it in length. Euclid’s lemma says that area ACHG = area AFED.
To show this, divide AFED into two halves by the diagonal FD. By I 38, area FAD = area FAC, the two triangles having a common base AF and vertices D and C that lie on a line parallel to AF. Likewise, divide ACHG into two halves by diagonal GC. Again by I 38, area AGB = area AGC, AG serving as a common base and vertices B and C lying on a line parallel to it. But area FAD = 1⁄2 area AFED, and area AGC = 1⁄2 area ACHG. Thus, if we could only show that area FAC = area BAG, we would be done.
It is here that Euclid produces his trump card: triangles FAC and BAG are congruent because they have two pairs of equal sides (AF = AB and AG = AC) and equal angles ∠FAC and ∠BAG (each consisting of a right angle and the common angle ∠BAC). And as congruent triangles, they have the same area.
Now, what is true for one side of the right triangle is also true of the other side: area BMNC = area BDEK. Thus, area ACHG + area BMNC = area AFED + area BDEK = area AFKB: the Pythagorean theorem.
THIS IS MATH: Magic Squares, Circles, and Stars
If you have been following the opening of the windows in the Mathematical Awareness Month Poster, you might want to go back to window #1 and review Magic Squares. If you haven’t been there yet, please take a look at it. You will learn how to amaze your friends with your magical math abilities.
Magic squares come in many types, shapes, and sizes. Below you will see a magic square, a magic circle, and a magic star. If you would like to see hundreds more, you might want to check out The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions by Clifford Pickover.
Normal Magic Squares
This is a thirdorder normal magic square where all of the rows, columns, and diagonals add to 15.
Is this the only solution to this magic square? Can you find others?
You could also have a 4 x 4 square or a 5 x 5 square and so on. How big of a square can you solve?
Magic Circles
Below you will see a magic circle composed of eight circles of four numbers each and the numbers on each circle all add to 18. The thing that makes this magic circle special is that each number is at the intersection of four circles but no other point is common to the same four circles.
Magic Stars
The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.
So, where’s the math?
Well, you should have noticed already that there are numbers on this page. However, there is more to math than numbers. Let’s add at least one equation.
If we go back to the normal magic square you should know that all these magic squares have the same number of rows and columns, they are n^{2}. The constant that is the same for every column, row, and diagonal is called the magic sum and we will call it M. Now we can figure out what that constant should be. If we use our 3 x 3 square above, we know that n = 3. If we plug our n into the given formula below we will find what our constant has to be.
Since our n = 3, the formula says M = [3 (3^{2} + 1)]/2, which simplifies to 15. For normal magic squares of order n = 4, 5, and 6 the magic constants are, respectively: 34, 65, and 111. What would M be for n = 8? See if you can solve this square. (The figure for the normal square is from Wikipedia.)
Princeton University Press’s bestselling audio books
We’re changing things up a bit. Each week we list the bestselling titles according to BookScan, but today we’re focusing on our audio titles. These are Princeton University Press’s bestselling audio books for the final quarter of 2013. Click through to listen to samples or to add them to your book queue.
 The 5 Elements of Effective Thinking by Edward Burger & Michael Starbird
 Women Don’t Ask by Linda Babcock and Sara Laschever
 Einstein and the Quantum by A. Douglas Stone
 Lost Enlightenment by S. Frederick Starr
 The Founders’ Dilemmas by Noam Wasserman
THIS IS MATH!: Amaze your friends with The Baby Hummer card trick
Welcome to THIS IS MATH! a new series from math editor Vickie Kearn.
This is the first of a series of essays on interesting ways you can use math. You just may not have thought about it before but math is all around us. I hope that you will take away something from each of the forthcoming essays and that you will pass it on to someone you know.
April is Math Awareness Month and the theme this year is Mathematics, Magic, and Mystery. There is a wonderful website where you will find all kinds of videos, puzzles, games, and interesting facts about math. The homepage has a poster with 30 different images. Each day of the month, a new window will open and reveal all of the wonders for that day.
Today I am going to elaborate on something behind window 3 which is about math and card magic. You will find more magic behind another window later this month. This particular trick is from Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks by Persi Diaconis and Ron Graham. It is a great trick and it is easy to learn. You only need any four playing cards. Take a look at the bottom card of your pack of four cards. Now remember this card and follow the directions carefully:
 Put the top card on the bottom of the packet.
 Turn the current top card face up and place it back on the top of the pack.
 Now cut the cards by putting any amount you like on the bottom of the pack.
 Take off the top two cards (keeping them together) and turn them over and place them back on top.
 Cut the cards again and then turn the top two over and place them back on top.
 Give the cards another cut and turn the top two over together and put them back on top.
 Give the cards a final cut.
 Now turn the top card over and put it on the bottom of the pack.
 Put the current top card on the bottom of the pack without turning it over.
 Finally, turn the top card over and place it back on top of the pack.
 Spread out the cards in your pack. Three will be facing one way and one in the opposite way.
 Surprise! Your card will be the one facing the opposite way.
This trick is called the Baby Hummer and was invented by magician Charles Hudson. It is a variation on a trick invented by Bob Hummer.
So where’s the math?
The math behind this trick covers 16 pages in the book mentioned above.
THIS IS MATH! will be back next week with an article on MathPickover Magic Squares!
Celebrate Math Awareness Month with Us
April is Math Awareness month and this year the theme is Mathematics, Magic, and Mystery. To kick off the celebration, visit the Math Awareness web site where you can “open” the days on an advent calendar revealing wonderful math and magic tricks. Today, for example, you can learn a bit about Geometrical Vanishes which make everyday objects appear to … disappear. The videos show how to make everything from dollar bills to chocolate disappear. Tomorrow we’ll start a new series of posts called This Is Math! in which our acquisitions editor for math titles will explain the various ways we encounter math in our everyday lives…and perhaps even add a few tricks of her own!
In the meantime, here’s another Geometrical Vanish courtesy of Tim Chartier, author of Math Bytes:
Play along by printing and cutting out your own set of vanishing PUP Logos. Cut along the solid lines and reverse the top two sections to see a logo magically disappear and reappear. if you have a suggestion for something else you would like to make appear and disappear, leave a comment below and I’ll see if we can get more of these print outs made (keep it clean please!).
Click the smaller images above to open full size images.
Quick Questions for Tim Chartier, author of Math Bytes
Tim Chartier is author of Math Bytes: Google Bombs, ChocolateCovered Pi, and Other Cool Bits in Computing. He agreed to be our first victi… interview subject in what will become a regular series. We will ask our authors to answer a series of questions in hopes to uncover details about why they wrote their book, what they do in their day job, and what their writing process is. We hope you enjoy getting to know Tim!
PUP: Why did you write this book?
Tim Chartier: My hope is that readers simply delight in the book. A friend told me the book is full of small mathematical treasures. I have had folks who don’t like math say they want to read it. For me, it is like extending my Davidson College classroom. Come and let’s talk math together. What might we discover and enjoy? Don’t like math? Maybe it is simply you haven’t taken a byte of a mathematical delight that fits your palate!
PUP: Who do you see as the audience for this book?
TC: I wanted this book, at least large segments of it, to read down to middle school. I worked with public school teachers on many of the ideas in this book. They adapted the ideas to their classrooms. And yet, the other day, I was almost late taking my kids to school as I had to pull them from reading my book, a most satisfying reason. In my mime training, Marcel Marceau often said, “Create your piece and let the genius of the audience teach you what you created.” I see this book that way. I wrote a book that I see my students and the many to whom I speak in broad public settings smiling at as they listen. Who all will be in the audience of this book? That’s for me to learn from the readers. I look forward to it.
Don’t like math? Maybe it is simply you haven’t taken a byte of a mathematical delight that fits your palate! 
PUP: What do you think is the book’s most important contribution?
TC: When I describe the book to people, many respond with surprise or even better a comment like, “I wish I had a teacher like you.” My current and former students often note that the book is very much like class. Let’s create and play with ideas and discover how far they can go and, of great interest to me, how fun and whimsical they can be.
PUP: What inspired you to get into your field?
TC: My journey into math came via my endeavors in performing arts. I was performing in mime and puppetry at international levels in college. Math was my “backup” plan. Originally, I was taking math classes as required courses in my studies in computer science. I enjoyed the courses but tended to be fonder of ideas in computer science. I like the creative edge to writing programming. We don’t all program in the same way and I enjoyed the elegance of solutions that could be found. This same idea attracted me to math — when I took mathematical proofs. I remember studying infinity – a topic far from being entirely encompassed by my finite mind. Yet, through a mathematical lens, I could examine the topic and prove aspects of it. Much like when I studied mime with Marcel Marceau, the artistry and creativity of mathematical study is what drew me to the field and kept me hooked through doctoral studies.
PUP: What is the biggest misunderstanding people have about what you do?
TC: Many think mathematics is about numbers. Much of mathematics is about ideas and concepts. My work lies at the boundary of computer science and mathematics. So, my work often models the real world so often mathematics is more about thinking how to use it to glean interesting or new information about our dynamic world. Numbers are interesting and wonderful but so is taking a handful of M&Ms and creating a mathbased mosaic of my son or sitting with my daughter and using chocolate chips to estimate the value of Pi. And, just for the record, the ideas would be interesting even without the use of chocolate but that doesn’t hurt!
PUP: What would you have been if not a mathematician?
TC: Many people think I would have been a fulltime performer. I actually intentionally walked away from that field. I want to be home, have a home, walk through a neighborhood where I know my neighbors. To me, I would have found a field, of some kind, where I could teach. Then, again, I always wanted to be a creative member of the Muppet team – either creating ideas or performing!
I pick projects that I believe aren’t just exciting now, but will be exciting in retrospect. 
PUP: What was the best piece of advice you ever received?
TC: At one time, I was quite ill. It was a scary time with many unknowns. I remember resting in a dark room and wondering if I could improve and get better. I reflected on my life and felt good about where I was, even if I was heading into my final stretch. I remember promising myself that if I ever got better that I would live a life that later — whether it be a decade later or decades and decades later — that I would try to live a life that I could again feel good about whenever I might again be in such a state. I did improve but I pick projects that I believe aren’t just exciting now, but will be exciting in retrospect. This book is easily an example of such a decision.
PUP: Describe your writing process. How long did it take you to finish your book? Where do you write?
TC: The early core of the book happened at 2 points. First, I was on sabbatical from Davidson College working at the University of Washington where I taught Mathematical Modeling. Some of the ideas of the book drew from my teaching at Davidson and were integrated into that course taught in Seattle. At the end of the term, my wife Tanya said, “You can see your students and hear them responding. Sit now and write a draft. Write quickly and let it flow. Talk to them and get the class to smile.” It was great advice to me. The second stage came with my first reader, my sister Melody. She is not a math lover and is a critical reader of any manuscript. She has a good eye. I asked her to be my first reader. She was stunned. I wanted her to read it as I knew if she enjoyed it, even though there would be parts she wouldn’t understand fully, then I had a draft of the book I wanted to write. She loved it and soon after I dove into the second draft.
PUP: Do you have advice for other authors?
TC: My main advice came from awardwinning author Alan Michael Parker from Davidson College. As I was finishing, what at the time I saw as close to my final draft, Alan said, “Tim, you are the one who will live with this book for a lifetime. Many will read it only once. You have it for the rest of your life. Write your book. Make sure it is your voice. Take your time and know it is you.” His words echoed in me for months. I put the book down for several months and then did a revision in which I saw my reflection in the book’s pages — I had seen my reflection before but never as clearly.
Tim is the author of:

These Two Numbers Make Spring Possible (and the other seasons too)
Happy first day of spring! Princeton University Press is celebrating the coming of the crocuses and daffodils with this mathematical post by Oscar Fernandez, author of Everyday Calculus: Discovering the Hidden Math All around Us.
March 20^{th}. Don’t recognize that date? You should, it’s the official start of spring! I won’t blame you for not knowing, because after the unusually cold winter we’ve had it’s easy to forget that higher temperatures are coming. But why March 20^{th}, and not the 21^{st} or the 19^{th}? And while we’re at it, why are there even seasons at all?
The answer has to do with 2 numbers. Don’t worry, they’re simple numbers (not like pi [1]). Stick around and I’ll show you some neat graphs to help you understand where they come from, and hopefully entertain you in the process too.
The first star of this show is the number 92 million. No, it’s not the current Powerball jackpot; it’s also not the number of times a teenager texts per day. To appreciate its significance, have a look at our first chart:
Figure 1: The average surface temperature (on the vertical axis) of the planets in our solar system sorted by their distance from our sun (the horizontal axis). From left to right: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto (technically, Pluto lost its planet status in 2006).
That first planet on the left is Mercury. It’s about 36 million miles away from the sun and has an average surface temperature of 333^{o}. (Bring LOTS of sunscreen.) Fourth down the line is the red planet, Mars. At a distance of about 141 million miles from the sun, Mars’ average temperature is 85^{o}. (Bring LOTS of hot chocolate.) We could keep going, but the general trend is clear: planets farther away from the sun have lower average temperatures.[2]
If neither 333^{o} nor 85^{o} sound inviting, I’ve got just the place for you: Earth! At a cool 59^{o} this planet is … drumroll please … 92 million miles from the sun.
We actually got lucky here. You see, it turns out that a planet’s temperature T is related to its distance r from the sun by the formula , where k is a number that depends on certain properties of the planet. I’ve graphed this curve in Figure 1. Notice that all the planets (except for the pesky Venus) closely follow the curve. But there’s more here than meets the eye. Specifically, the T formula predicts that a 1% change in distance will result in a 0.5% change in temperature.[3] For example, were Earth just 3% closer to the sun—about 89 million miles away instead of 92 million—the average temperature would be about 1.5^{o} higher. To put that in perspective, note that at the end of the last ice age average temperatures were only 5^{o} to 9^{o} cooler than today.[4]
So our distance from the sun gets us more reasonable temperatures than Mercury and Mars have, but where do the seasons come from? That’s where our second number comes in: 23.4.
Imagine yourself in a park sitting in front of a bonfire. You’re standing close enough to feel the heat but not close enough to feel the burn. Now lean in. Your head is now hotter than your toes; this tilt has produced a temperature difference between your “northern hemisphere” and your “southern hemisphere.” This “tilt effect” is exactly what happens as Earth orbits the sun. More specifically, our planet is tilted about 23.4^{o} from its vertical axis (Figure 2).
Figure 2. Earth is tilted about 23.4^{o} from the plane of orbit with the sun (called the ecliptic plane).
Because of its tilt, as the Earth orbits the sun sometimes the Northern Hemisphere tilts toward the sun—roughly MarchSeptember—and other times it tilts away from the sun—roughly SeptemberMarch (Figure 3).[5]
Figure 3. Earth’s tilt points toward the sun between midMarch and midSeptember, and points away from it the remaining months of the year. The four marked dates describe how this “tilt effect” changes the number of daylight hours throughout the year. Assuming you live in the Northern Hemisphere, days are longest during the summer solstice (shorter nights) and shortest during the winter solstice (longer nights). During the equinoxes, daytime and nighttime are about the same length.
Now that you know how two numbers—92 million and 23.4—explain the seasons, let’s get back to spring in particular. As Figure 3 shows, there are two days each year when Earth’s tilt neither points toward nor away from the sun. Those two days, called the equinoxes, divide the warmer months from the colder ones. And that’s exactly what happened on March 20^{th}: we passed the spring equinox.
Before you go, I have a little confession to make. It’s not entirely true that just two numbers explain the seasons. Distance to the sun and Earth’s tilt are arguably the most important factors, but other factors—like our atmosphere—are also important. But that would’ve made the title a lot longer. And anyway, I would’ve ended up explaining those factors using more numbers. The takeaway: math is powerful, and the more you learn the better you’ll understand just about anything.[6]
[1] The ratio of a circle’s circumference to its diameter, pi is a neverending, never repeating number. It is approximately 3.14.
[2] Venus is the exception. Its thick atmosphere prevents the planet from cooling.
[3] Here’s the explanation for the mathematically inclined. In calculus, changes in a function are described by the function’s derivative; the derivative of T is . This tells us that for a small change dr in r the temperature change dT is . Relative changes are ratios of small changes in a quantity to its original value. Thus, the relative change in temperature, dT/T, is
which is minus 0.5 times the relative change in distance, dr/r. The minus sign says that the temperature decreases as r increases, confirming the results of Figure 1.
[5] Just like in our thought experiment, the Southern Hemisphere’s seasons are swapped with our own; when one is cold the other is warm and vice versa.
[6] One last thing, I promise. Here are two links that animate Figure 3:
Interactive seasons animation, from McGrawHill.
Animation showing how the number of daylight hours change, from Mathisfun.com.
Need help filling out your brackets? Watch these free videos from Tim Chartier
Still rushing to fill out your brackets for the NCAA tournament? This free online course from mathematician Tim Chartier, author of Math Bytes, might help.
In this course, you will learn three popular rating methods two of which are also used by the Bowl Championship Series, the organization that determines which college football teams are invited to which bowl games. The first method is simple winning percentage. The other two methods are the Colley Method and the Massey Method, each of which computes a ranking by solving a system of linear equations. We also learn how to adapt the methods to take late season momentum into account. This allows you to create your very own mathematicallyproduced brackets for March Madness by writing your own code or using the software provided with this course.
From this course, you will learn math driven methods that have led Dr. Chartier and his students to place in the top 97% of 4.6 million brackets submitted to ESPN!
Explore Tim Chartier’s March MATHness lectures:
Pi Day: Where did π come from anyway?
This article is extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols. For more Pi Day features from Princeton University Press, please click here.
When one sees π in an equation, the savvy reader automatically knows that something circular is lurking behind. So the symbol (a relatively modern one, of course) does not fool the mathematician who is familiar with its many disguises that unintentionally drag along in the mind to play into imagination long after the symbol was read.
Here is another disguise of π: Consider a river flowing in uniformly erodible sand under the influence of a gentle slope. Theory predicts that over time the river’s actual length divided by the straightline distance between its beginning and end will tend toward π. If you guessed that the circle might be a cause, you would be right.
The physicist Eugene Wigner gives an apt story in his celebrated essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” A statistician tries to explain the meaning of the symbols in a reprint about population trends that used the Gaussian distribution. “And what is this symbol here?” the friend asked.
“Oh,” said the statistician. “This is pi.”
“What is that?”
“The ratio of the circumference of the circle to its diameter.”
“Well, now, surely the population has nothing to do with the circumference of the circle.”
Wigner’s point in telling this story is to show us that mathematical concepts turn up in surprisingly unexpected circumstances such as river lengths and population trends. Of course, he was more concerned with understanding the reasons for the unexpected connections between mathematics and the physical world, but his story also points to the question of why such concepts turn up in unexpected ways within pure mathematics itself.
The Good Symbol
The first appearance of the symbol π came in 1706. William Jones (how many of us have ever heard of him?) used the Greek letter π to denote the ratio of the circumference to the diameter of a circle. How simple. “No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.” But for the next thirty years, it was not used again until Euler used it in his correspondence with Stirling.
We could accuse π of not being a real symbol. It is, after all, just the first letter of the word “periphery.” True, but like i, it evokes notions that might not surface with symbols carrying too much baggage. Certain questions such as “what is i^{i}?” might pass our thoughts without a contemplating pause. Pure mathematics asks such questions because it is not just engaged with symbolic definitions and rules, but with how far the boundaries can be pushed by asking questions that everyday words could ignore. You might think that i^{i} makes no sense, that it’s nothing at all, or maybe a complex number. Surprise: it turns out to be a real number!
It seems that number has a far broader meaning than it once had when we first started counting sheep in the meadow. We have extended the idea to include collections of conceptual things that include the usual members of the number family that still obey the rules of numerical operations. Like many of the words we use, number has a far broader meaning than it once had.
Think only of the socalled imaginary quantities with which mathematicians long operated, and from which they even obtained important results ere they were in a position to assign to them a perfectly determinate and withal visualizable meaning.
It is not the job of mathematics to stick with earthly relevance. Yet the world seems to eventually pick up on mathematics abstractions and generalizations and apply them to something relevant to Earth’s existence. Almost a whole century passed with mathematicians using imaginary exponents while a new concept germinated. And then, from the symbol i that once stood for that onetime peculiar abhorrence √−1, there emerged a new notion: that magnitude, direction, rotation may be embodied in the symbol itself. It is as if symbols have some intelligence of their own.
What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand. It must be a guide to our own intelligence. Here is Mach again:
In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand.
The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence—an impression which the great Euler confessed he often could not get rid of.
A single symbol can tell a whole story.
There was no single moment when x^{n} was first used to indicate the nth power of x. A half century separated Bombelli’s , from Descartes’s x^{n}. It may seem like a clearcut idea to us, but the idea of symbolically labeling the number of copies of x in the product was a huge step forward. The reader no longer had to count the number of x’s, which paused contemplation, interrupted the smoothness of reading, and hindered any broad insights of associations and similarities that could extend ideas. The laws x^{n}x^{m} = x^{n+m} and (x^{n})^{m} = x^{nm}, where n and m are integers, were almost immediately suggested from the indexing symbol. Not far behind was the idea to let x^{½} denote √x, inspired by extending the law x^{n}x^{m} = x^{n+m} to include fractions, so x^{½} x^{½} = x^{1}.
Further speculation on what n^{x} might be would surely have inspired questions such as what x might be for a given y in an equation such as y = 10^{x}. Answer that and we would have a way of performing multiplication by addition. But Napier, the inventor of logarithms, already knew the answer long before mathematics had any symbols at all!
Symbols acquire meanings that they originally didn’t have. But symbolic representation has, likewise, the disadvantage that the object represented is very easily lost sight of, and that operations are continued with the symbols to which frequently no object whatever corresponds.
Ernst Mach once again:
A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other.
Top Tips for 2014 March Madness Brackets from Tim Chartier
With a $1 billion dollar payday on the line, we predict there will be more people filling out March Madness brackets this year than ever before, so it isn’t surprising that everyone is looking to mathematician Tim Chartier for tips and tricks on how to pick the winners. Tim has been using math to fill out March Madness brackets with his students for years and his new book Math Bytes will have an entire section devoted to best tips and tricks. In the meantime, we invite you to check out these tips from an interview at iCrunchData News.
ICrunchData: What are a few variables that are used that are out of the ordinary?
Chartier: “In terms of past years, it helps if you look at scores in buckets. For instance, you decide close games are within 3 points and count those as ties. Medium wins are 4 to 10 points and could as 6 points and anything bigger is an 11 point win. That’s worked really well in some cases and reduces some of the noise of scores.”
“Here is another that comes out of our most current research. This year’s tournament will enable us to test it in brackets. We tried it on conference tournaments and it had good success. We use statistics (specifically Dean Oliver’s 4 Factors) and look at that as a point, in this case in 4D space. Then we find another team that has a point in the fourth dimension closest to that team’s point. This means they play similarly. Suddenly, we can begin to look at who similar teams win and lose against.”