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.

3-28 Diaconis_MagicalApril 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:

  1. Put the top card on the bottom of the packet.
  2. Turn the current top card face up and place it back on the top of the pack.
  3. Now cut the cards by putting any amount you like on the bottom of the pack.
  4. Take off the top two cards (keeping them together) and turn them over and place them back on top.
  5. Cut the cards again and then turn the top two over and place them back on top.
  6. Give the cards another cut and turn the top two over together and put them back on top.
  7. Give the cards a final cut.
  8. Now turn the top card over and put it on the bottom of the pack.
  9. Put the current top card on the bottom of the pack without turning it over.
  10. Finally, turn the top card over and place it back on top of the pack.
  11. Spread out the cards in your pack. Three will be facing one way and one in the opposite way.
  12. 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 Math-Pickover Magic Squares!

 

Quick Questions for Tim Chartier, author of Math Bytes

Tim Chartier, Photo  courtesy Davidson CollegeTim Chartier is author of Math Bytes: Google Bombs, Chocolate-Covered 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 “back-up” 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 math-based 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 full-time 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 award-winning 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:

bookjacket

Math Bytes
Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing
Tim Chartier

“A magnificent and curious romp through a wonderful array of mathematical topics and applications: maze creation, Google’s PageRank algorithm, doodling, the traveling salesman problem, math on The Simpsons, Fermat’s Last Theorem, viral tweets, fractals, and so much more. Buy this book and feed your brain.”–Clifford A. Pickover, author of The Math Book

Math Bytes is a playful and inviting collection of interesting mathematical examples and applications, sometimes in surprising places. Many of these applications are unique or put a new spin on things. The link to computing helps make many of the topics tangible to a general audience.”–Matt Lane, creator of the Math Goes Pop! Blog

 

The Mystery of Snowfall Explained

photo
The view from my backyard.

We’re still in the midst of a blizzard — indeed Princeton University Press is actually closed for the day as we huddle down at home with 8-10 inches and counting on the ground. But what better time to share this opinion piece from Ian Roulstone, meteorologist and author of Invisible in the Storm. While we’re waiting for an opportunity to get out there and shovel, we hope you will enjoy this article about the strange world of snowfall prediction and why it can’t be more accurate:

A snow-covered landscape is one of the classic images showcasing the beauty of weather on Earth. We are awed by the grandeur of white-capped mountains and the almost magical quality of snow-covered trees. We are also frustrated when the tempests of winter reach far and wide, striking as they have done this year in America’s southern states.

When it comes to forecasting the likelihood of a blizzard, the weather anchors know what to say. But when asked to predict how much snow will actually accumulate, they will give estimates. Why?

Read the rest of the story at CNN.com.

Celebrating the genius of Alan Turing

Considered by many to be the father of computer science, Alan Turing is remembered today for his many contributions to the study of computers, artificial intelligence, and code breaking. On December 24, Queen Elizabeth II officially pardoned the late British mathematician and the action recalled attention to his groundbreaking work as well as his personal life. In 1952, Turing was charged with homosexuality, which was considered a criminal act in England at the time. Two years later, he took his own life. Today, mathematicians and computer scientists celebrate Turing’s broad contributions to his field.

For more on the life and work of Turing, check out these resources:

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 Princeton University Press recently re-released Andrew Hodges’s biography of Alan Turing: Alan Turing: The Enigma — The Centenary Edition.

It is only a slight exaggeration to say that the British mathematician Alan Turing (1912-1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades–all before his suicide at age forty-one. This classic biography of the founder of computer science, reissued on the centenary of his birth with a substantial new preface by the author, is the definitive account of an extraordinary mind and life. A gripping story of mathematics, computers, cryptography, and homosexual persecution, Andrew Hodges’s acclaimed book captures both the inner and outer drama of Turing’s life.

Read Chapter One of the book here.

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Turing earned his Ph.D. in mathematics from Princeton in 1938. Watch the video below to hear Andrew Appel (chair of the department of computer science at Princeton) discuss Turing’s legacy.

Andrew Appel on Alan Turing’s legacy
(Princeton School of Engineering and Applied Science)

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Appel, a Princeton graduate, is the editor of another recent release by Princeton University Press, Alan Turing’s Systems of Logic: The Princeton Thesis.

Though less well known than his other work, Turing’s 1938 Princeton PhD thesis, “Systems of Logic Based on Ordinals,” which includes his notion of an oracle machine, has had a lasting influence on computer science and mathematics. This book presents a facsimile of the original typescript of the thesis along with essays by Andrew Appel and Solomon Feferman that explain its still-unfolding significance.

Preview the book by reading Chapter One.

 

Appel_AlanTuring's_F12Hodges_Alan Turing_F12

Q&A: Eli Maor and Eugen Jost reveal the surprising inspirations and process of Beautiful Geometry

In k10065[1]January, we will publish Beautiful Geometry by Eli Maor and with illustrations by Eugen Jost. The book is equal parts beauty and mathematics and we were grateful that both authors took time to answer some questions for our readers. We hope you enjoy this interview.

Look for a slideshow previewing the art from the book in the New Year.

PUP: How did this book come to be? Where did you get the idea to create a book of Beautiful Geometry?

Eli Maor [EM]: It all started some five years ago through a mutual acquaintance of us by the name Reny Montandon, who made me aware of Eugen’s beautiful geometric artwork. Then, in 2010, I met Eugen in the Swiss town of Aarau, where I was invited to give a talk at their famous Cantonal high school where young Albert Einstein spent two of his happiest years. We instantly bonded, and soon thereafter decided to work together on Beautiful Geometry. To our deep regret, Reny passed away just a few months before we completed the project that he helped to launch. We dearly miss him.

PUP: Eugen, where did your interest in geometric artwork come from? Are you a mathematician or an artist by training?

Eugen Jost [EJ]: I have always loved geometry. For me, mathematics is an endless field in which I can play as an artist and I am also intrigued by questions that arise between geometry and philosophy.

For example, Euclid said: A point is that which has no part, and a line is what has no widths. Which raises questions for me as an artist—Do geometric objects really exist, then? Are the points and lines that I produce really geometric objects? With Adobe Illustrator you can transform points and lines into “non-existing” paths, which brings you deeper into geometry.

But, ultimately, it’s not my first purpose to illustrate mathematics—I just want to play and I’m happy if the onlooker of my pictures starts to play as well.

PUP: How did you decide which equations to include?

EM: It’s not so much about equations as about theorems. Of course, geometry has hundreds, if not thousands of theorems to choose from, so we had to be selective. We didn’t have any particular rules to guide us; sometimes Eugen would choose a particular theorem for which he had some artistic design in mind; in other cases we based our selection on theorems with an interesting history, or just for their simplicity. But we always had the artistic point of view in mind: our goal was to showcase the beauty of geometry and make it known to a wide public.

EJ: I think I’m seldom looking for mathematical topics with which I can make pictures. It’s the other way round: mathematics and geometry comes to me. A medieval town, the soles of your shoes, wheel rims, textile printing, patterns in pine cones: wherever I look I see mathematics and beautiful geometry. Euclid’s books among many others provide me with ideas, too.

Often I develop ideas when I’m walking in the woods with our dog or I’m scribbling in my small black diaries while I’m sitting in trains. At home I transform the sketches into pictures.

PUP: Were there any theorems you didn’t get to include but would have liked to?

EM: Yes, there were many theorems we would have liked to have included, but for practical reasons we decided to limit ourselves to about 50 chapters. That leaves us plenty of subjects for Beautiful Geometry II :-)

PUP: Are there some theorems that simply didn’t lend themselves to artistic depictions?

EJ: In our book you won’t find many three-dimensional objects depicted. In my art I tend to create flat objects (circles, triangles, squares …) on surfaces and three-dimensional objects in space. Therefore we avoided theorems that have to do with space—with few exceptions.

PUP: Can you describe the layout of the book? How is it organized?

EM: We followed a more or less chronological sequence, but occasionally we grouped together subjects that are logically related to one another, so as to make the flow of ideas easier to follow.

PUP: The collaboration between you and Eugen Jost reminds me of a lyricist and musician—how did the two of you work together? Did you write and he created art alongside or did he have art already done and you wrote for it?

EM: Yes, that comparison between a lyricist (in opera we call it librettist) and a musician is very apt. As I mentioned earlier, we didn’t have a rigid guideline to follow; we just played with many ideas and decided which ones to include. We exchanged over a thousand emails between us (yes, Eugen actually counted them!) and often talked on Skype, so this aspect of our collaboration was easy. I can’t imagine having done that twenty years ago…

EJ: Communicating with Eli was big fun. He has so many stories to tell and very few of them are restricted to geometry. In 2012, when we thought our manuscript was finished, Eli and his wife came to Switzerland. For many days we travelled and hiked around lakes, cities and mountains with our manuscripts in our book sacks. We discussed all the chapters at great length. In some chapters, the relationship between Eli’s text and my pictures is very close and the art helps readers understand the text. In others, the connection is looser. Readers are invited to get the idea of a picture more or less independently—sort of like solving a riddle.

PUP: Most of the art in the book are original pieces by Eugen, right? Where did you find the other illustrations?

EM: Most of the artwork is Eugen’s work. He also took excellent photographs of sites with interesting geometric patterns or a historical significance related to our book. I have in mind, for example, his image of the famous headstone on the grave of Jakob Bernoulii in the town of Basel, Switzerland, which has the wrong spiral engraved on it—a linear spiral instead of a logarithmic one!

PUP: That is fascinating and also hints at What Eugen mentioned earlier–math and the beauty of math is hidden in plain sight, all around us. Are there other sites that stand out to you, Eugen?

EJ: Yesterday I went to Zurich. While I was walking through the streets I tried to find answers to your question. Within half an hour I found over a dozen examples of the mathematics that surrounds us:

  • The clock face of the church St. Peter is the biggest one in Europe. This type of clock face links our concept of time with ancient Babylonians who invented a time system based on the numbers 12, 24, 12×30, 3600.
  • In the Bahnhofstrasse of Zurich there is a sculpture by Max Bill in which many big cuboids form a wonderful ensemble. Max Bill was the outstanding artist of the so called “Zürcher Konkrete”; his oeuvre is full of mathematics.
  • I saw a fountain and the jet of water formed wonderful parabolas in the air. Where the water entered the pool, it produced concentric circles.
  • There are literally hundreds of ellipses on any street in Zurich, or any other town for that matter. Every wheel you see is an ellipse—unless you look at it at a precise angle of 90 degrees.
  • Even under our feet, you can find mathematics. Manhole covers very often have wonderful patterns that you can interpret mathematically.

Mathematics and Beautiful Geometry is everywhere around us—we just have to open our eyes.

PUP: What equation does the artwork on the cover of the book illustrate? Can you give us a quick “reading” of it?

EM: The front cover shows the Sierpinski Triangle, named after Polish mathematician Waclaw Sierpinski (1882-1969). It is a bizarre construction, a triangle-like shape that has zero area but an infinite perimeter. This is but one of many fractal-type patterns that have become popular thanks to the ability to create them with modern computers, often adding dazzling color to make them into exquisite works of art.

PUP: Eugen, what is your process to create a piece of art like this?

EJ: Unfortunately, the Sierpinski Triangle was not my own idea, but I was awe-struck by the idea of a shape with an infinitely little area and an infinite perimeter, so I started to think about how it could be depicted. Like most of the pictures in the book, I created this piece of art on a computer. At the same time, I was immersed in other mathematical ideas like the Menger sponge, the Hilbert curve, the Koch snowflake. Of course Sierpinski himself and countless others must have sketched similar triangles, but that was the challenge for our book: to take theorems and to transform them into independent pieces of art that transcend mere geometric drawings.

PUP: Eli, do you have a favorite piece of art in the book?

EM: Truth be told, every piece of art in our book is my favorite! But if I must choose one, I’ll go for the logarithmic spiral that Eugen realized so beautifully in Plate 34.1; it is named spira mirabilis (“The miraculous spiral”), the name that Bernoulli himself used to describe his favorite curve.

PUP: We asked the mathematician to pick his favorite piece of art, so it is only fair that we should ask the artist to pick his favorite theorem. Do you have one, Eugen?

EJ: Being Eli’s first reader for the last three years has been a joy because he tells history and stories in our book. I like the chapter on the surprising theorem of George Alexander Pick, in part because of the biographical details. Eli describes how Pick bonded with Albert Einstein in Prague—imagine Einstein and Pick playing the violin and the viola together! Sadly, Pick ended his life in the concentration camp at Theresienstadt, but he left behind this wonderful contribution to mathematics.

PUP: Are there any particularly surprising pieces of art in the book that might have a good backstory or illustrate a particularly memorable equation?

EM: Again, this puts me in a difficult position – to choose among so many interesting subjects that are covered in our book. But if I have to pick my choice, I’d go for Plate 26.1, entitled PI = 3. The title refers to what has been called the “Biblical value of PI” and refers to a verse in the Bible (I Kings vii 23) which, if taken literally, would lead to a value of PI = 3. Our plate shows this value surrounded by the famous verse in its original Hebrew.

PUP: Writing a book is a long process filled with countless hours of hard work. Do any moments from this period stand out in particular?

EJ: I remember sitting in a boat on the lake of Thun with Eli and his wife Dalia in the spring of 2012. Eli and I were pondering on the chapter doubling the cube. As I mentioned before, I do not like to draw three-dimensional objects on a flat surface, so I didn’t want to depict a traditional cube. I was playing instead with the unrolling of two cubes, one having the double volume of the other. Eli was not sure this would work, but on the boat we thoroughly discussed it and all of a sudden Eli said, “Eugen, you have sold me on it.” I hadn’t heard that expression before. I then had a queasy conscience because I didn’t know whether I should have been flexible enough to leave my own idea for a better one. Ultimately, the art came out quite well and really illustrates the Delian problem.

A chance encounter gave me a sense of the broad appeal of the book. I was sitting at a table with a highly trained engineer and I told him about our book. I then tried to explain the theorem of Morley: In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. His response was “You don’t suppose that you could solve any statical problem with this, do you?”

PUP: Math is often quite visual, but where did the idea of making it both visual and beautiful come from?

EM: We are not the first, of course, to point out the visual beauty of many geometric theorems or patterns, but usually these gems are depicted in stark, black-and-white designs of lines and curves. Adding colors to these designs – and sometimes a bit of humor and imagination – makes all the difference between strictly mathematical beauty and a true work of art. This is what has really inspired us in writing our book.

PUP: Who do you hope reads Beautiful Geometry?

EM: We aim at a broad audience of students, teachers and instructors at all levels, and above all, laypersons who enjoy the beauty of patterns and are not afraid of a simple math equation here and there. We hope not to disappoint them!

Hurricane Sandy and Global Warnings, an original article by Ian Roulstone and John Norbury

Hurricane Sandy and Global Warnings

Ian Roulstone and John Norbury

There are many heroes in the story of Hurricane Sandy, but we arguably owe the greatest debt of gratitude to mathematicians who wrangle massive amounts of data to improve the accuracy of our weather predictions. Two devastating storms, decades apart, provide a fantastic snapshot of how weather prediction has improved thanks to the introduction of computational mathematics over the last century.

Just over 75 years ago, on September 9th 1938 above the warm tropical waters near the Cape Verde islands, a storm gathered. As the weather system intensified, it was ushered westward by the prevailing larger-scale ridge of high pressure over the Atlantic. By the 16th the storm had become a hurricane, and the captain of a Brazilian freighter caught sight of the tempest northeast of Puerto Rico. He radioed the U.S. Weather Bureau to warn them of the impending danger – no satellites or sophisticated computer models to help the forecasters in those days.

A deep trough of low pressure over Appalachia forced the storm northward, avoiding the Bahamas and Florida, and towards the north-eastern seaboard of the United States. The forecasters were relying on real-time reports of the storm’s progress, but it advanced at an incredible pace, moving northward at nearly 70mph. By the time the Weather Bureau realised it was on a collision course with Long Island it was too late. The death toll from the Great New England Hurricane approached 600, with over 700 injured, and the damage was estimated at $308 million – or around $4.8 billion at today’s prices.

History very nearly repeated itself on October 29 and 30th last year, when Hurricane Sandy slammed into New Jersey. Meteorologists referred to Superstorm Sandy as a “multi-hazard event”, with major damage resulting from wind gusts, from high seas, from a tidal surge, from heavy rain, and even from driving snow. The number of fatalities in the U.S., attributed either directly or indirectly to Hurricane Sandy, were around 160: a tragedy, but mercifully fewer than the number killed by the Great Hurricane of 1938.

It is almost certain that timely warnings averted greater catastrophe last year. Unlike the storm of 1938, which caught forecasters by surprise, one of the most remarkable features of the forecast of Hurricane Sandy from the European Centre for Medium-Range Weather Forecasts (ECMWF) was the prediction made on October 21st, 36 hours before Sandy even formed, of a one-in-four chance of a severe storm, centred on New York, on October 30th.

ECMWF routinely produce two types of forecast for 10 days ahead. As they state in a recent newsletter “The ECMWF global medium-range forecast comprises a high-resolution forecast (HRES) and an ensemble of lower-resolution forecasts (ENS)”, and it was the ENS that helped forewarn of Sandy.

To calculate a forecast we use supercomputers to solve seven equations for the seven basic variables that describe weather: wind speed and direction (3 variables), pressure, temperature, air density, and humidity. The equations governing weather are highly nonlinear. This means that the ‘cause and effect’ relationships between the basic variables can become ferociously complex. To deal with the potential loss of predictability, forecasters study not one, but many forecasts, called an ensemble. Each member of the ensemble is started from a slightly different initial state. These different initial states reflect our ignorance of exactly how weather systems form. If the forecasts predict similar outcomes, we can be reasonably confident, but if they produce very different scenarios, then the situation is more problematic.

In the figure below the ensemble of forecasts for Sandy, starting at midday on October 23rd indicates the high probability of the ‘left turn’ and the most probable landfall – information that helped save lives. The inset at top right shows the strike probability chart that highlights the region around New York within which there is 25% chance of a severe storm by midnight on October 30th. This forecast was computed from an earlier ensemble starting at midday on October 21st and gave forecasters the vital “heads-up” of severe weather striking a highly populated area.

Forecasting Superstorm Sandy: The ensemble of forecasts covering the 10 days from the formation of the cyclone on October 23; the dotted black line is the actual track of the storm. Top right inset shows the storm strike probability from midday October 21. Bottom right inset shows the ensemble predictions of Sandy’s central pressure. © ECMWF

Forecasting Superstorm Sandy: The ensemble of forecasts covering the 10 days from the formation of the cyclone on October 23; the dotted black line is the actual track of the storm. Top right inset shows the storm strike probability from midday October 21. Bottom right inset shows the ensemble predictions of Sandy’s central pressure. © ECMWF

The science of weather and climate prediction was utterly transformed in the second half of the 20th Century by high-performance computing. But in order to fully exploit the computational power, and the information gathered by weather satellites and weather radar, we need mathematics.  As we explained in our article in Scientific American [hyperlink] math quantified the choreography of Hurricane Sandy. And to account for the ever-present uncertainties in the science of weather forecasting, math delivers the tools to analyse the predictions and to highlight the dangers.

Lives were saved because of the quality of our weather forecasts, which are made possible by an international group of mathematicians and weather prediction centers. The math that helps us quantify uncertainty in weather forecasting is being used to quantify uncertainty in climate prediction. It is easy to underestimate the value of this research, but investing in this science is vital if we are to stave off future billions in damages.

 


For further insights into the math behind weather and climate prediction, see Roulstone and Norbury’s new book Invisible in the Storm: The Role of Mathematics in Understanding Weather.

 

Climate Change: a Movie and the Math by Ian Roulstone and John Norbury

Climate Change: a Movie and the Math

By Ian Roulstone and John Norbury

Next week the Intergovernmental Panel on Climate Change (IPCC) will release the first of three reports that constitute their Fifth Assessment Report on climate change. This first report, The Physical Science Basis, will cover a huge range of topics from the carbon cycle to extreme weather. But climate prediction also relies heavily on mathematics, which is used to quantify uncertainties and improve the models.

The role of math is illustrated by a remarkable video of our ever-changing weather. Last month the National Oceanic and Atmospheric Administration (NOAA) decommissioned Geostationary Operational Environmental Satellite 12 (GOES-12), which monitored our weather for the past 10 years from its isolated vantage point 36,000 kilometers above America and the Atlantic Ocean.

GOES-12 had seen it all – from wildfires, volcanic ash, and landscape parched by drought, to Hurricanes Ike, Katrina and Sandy, and the blizzards that gripped the central United States in the winter of 2009-10. NOAA created a video – 187 seconds and 3641 images – one snapshot from each day of its operational life, which amounts to 10 years’ weather flashing before our eyes in just over 3 minutes. It’s dramatic and amazing:

In Scientific American, Evelyn Lamb commented on how this video highlights “a tension between the unpredictability of the weather and its repetitiveness”. Even after a few seconds it becomes clear that the patterns revealed by clouds differ from one part of the globe to another. Great towering cumulonimbus bubble up and unleash thunderstorms in tropical regions every day, while in more temperate mid-latitudes, the ubiquitous low pressure systems whirl across the Atlantic carrying their warm and cold fronts to Europe. The occasional hurricane, spawned in the tropics, careers towards the United States (Hurricane Sandy can be seen at about 2’50’’). But the mayhem is orchestrated: the cyclones almost seem like a train of ripples or waves, following preferred tracks, and the towering storms are confined largely to the tropics.

CaptureThis image of water vapour in the atmosphere (taken by GOES-13) reveals the swirling cyclones and the tropical storms. While the detail varies from hour to hour and from day to day, there are recurring patterns. Image courtesy of NEODAAS/University of Dundee.

In fact, this movie is affording us a glimpse of a remarkable world – it is a roller-coaster ride on the ‘weather attractor’.

An ‘attractor’ is a mathematician’s way of representing recurring behavior in complex systems, such as our atmosphere. A familiar illustration of an attractor can be seen in the figure below, and it is named after one of the fathers of chaos, Edward Lorenz.

Capture
The Lorenz attractor: every point within the space delineated by the coordinate axes represents a possible state of a circulating fluid, such as the ascent of warm air and the temperature difference of the warmer rising air to the cooler descending air. The points on the ‘butterfly wings’ are the attractor: they represent the set of states through (or around) which such a system will evolve. Even if the system begins from a state that does not lie on the attractor, it tends towards the states that do. The transition from one wing of the attractor to the other (which might represent a change in the ‘weather’) can be difficult to predict, due to inherent chaos in the system. But the overall pattern captures the repetitiveness.

It is impossible to illustrate the weather attractor for the atmosphere in terms of a simple three-dimensional image: Lorenz’s very simple model of a circulating cell had only three variables. Our modern computer models used in climate prediction have around 100 million variables, so the attractor resides in a space we cannot even begin to visualise. And this is why the movie created by NOAA is so valuable: it gives us a vivid impression of the repetitiveness emerging from otherwise complex, chaotic behaviour.

Weather forecasters try to predict how our atmosphere evolves and how it moves around the attractor – a hugely difficult task that requires us to explore many possible outcomes (called an ensemble of forecasts) when trying to estimate the weather several days ahead. But climate scientists are faced with a very different problem: instead of trying to figure out which point on the 100 million-dimensional attractor represents the weather 100 years from now, they are trying to figure out whether the shape of the attractor is changing. In other words, are the butterfly wings ‘folding’ as the average weather changes? This is a mathematician’s way of quantifying climate change.

If 100 years from now, when a distant successor of GOES-12 is retired, our descendants create a movie of this future weather, will they see the same patterns of recurring behaviour, or will there be more hurricanes? Will the waves of cyclones follow different tracks? And will tropical storms be more intense? Math enables us to “capture the pattern” even though chaos stops us from saying exactly what will happen, and to calculate answers to these questions we have to calculate how the weather attractor is changing.

 


This article is cross-posted with the Huffington Post: http://www.huffingtonpost.com/ian-roulstone/climate-prediction-mathematics_b_3961853.html

For further insights into the math behind weather and climate prediction, see Roulstone and Norbury’s new book Invisible in the Storm: The Role of Mathematics in Understanding Weather.

“I found the first ballistic capture orbit to the moon with a painting,” Ed Belbruno

Ed Belbruno’s life and discoveries are the subject of a new documentary titled Painting the Way to the Moon by Jacob Akira Okada. Belbruno, a trained mathematician, discovered new ways to navigate the universe by taking advantage of gravitational pulls of various celestial bodies. Because of his work, space missions now use less fuel to traverse the stars and planets. And millions of Angry Birds Space fans should also thank Belbruno because his research is what determines the birds’ trajectories around space bodies and through gravitational pulls to eventual pig annihilation.

In the documentary, Belbruno, a brilliant painter in addition to mathematician and space scientist, credits his discovery to a Van Gogh-style painting he made of possible travel routes through space for his inspiration. Enjoy the complete trailer below:

Curious about Belbruno’s research? Please check out these Princeton University Press titles. Fly Me to the Moon is intended for general audiences, while Capture Dynamics and Chaotic Motions in Celestial Mechanics is a specialized textbook.

 

bookjacket

Fly Me to the Moon
An Insider’s Guide to the New Science of Space Travel
Edward Belbruno
With a foreword by Neil deGrasse Tyson

 

bookjacket

Capture Dynamics and Chaotic Motions in Celestial Mechanics
With Applications to the Construction of Low Energy Transfers
Edward Belbruno

In Celebration of Mathematicians

This week San Diego, California is home to the largest mathematics meeting in the world. Hosted by the Mathematical Association of America (MAA) and the American Mathematical Society (AMS), the 2013 Joint Mathematics Meeting is more than just panels and presentations—it is a mass gathering of people who are passionate about mathematics.

Mathematicians come from diverse backgrounds, maintain varying interests, and have their own unique journeys. In Fascinating Mathematical People: Interviews and Memoirs, Fern Hunt describes what it was like to be among the first black women to earn a PhD in mathematics, Harold Bacon makes trips to Alcatraz to help a prisoner learn calculus, and Thomas Banchoff, who first became interested in the fourth dimension while reading a Captain Marvel comic, relates his fascinating friendship with Salvador Dalí and their shared passion for art, mathematics, and the profound connection between the two. But whether they view mathematics as reason, art, or something else, all mathematicians are in search of truth.

This week is not only an endeavor in furthering the pursuit of knowledge, but a celebration of the gifted mathematical intellectuals who shape society, culture, and our awareness and understanding of ourselves and the world in which we live. Browse our website or latest mathematics catalog to see more by and about mathematicians, such as Paul J. Nahin’s The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age. If you’re at the Joint Mathematics Meeting, you may even visit us at booth 311. As Underwood Dudley wrote in “What Is Mathematics For?” included in The Best Writing on Mathematics: 2011 (The Best Writing on Mathematics: 2012 also available.), “What mathematics education is for is not for jobs. It is to teach the race to reason,” and we’ve all got room to learn.

New Mathematics Catalog!

Be among the first to check out our new mathematics catalog!
http://press.princeton.edu/catalogs/math13.pdf

Of particular interest are Alexander J. Hahn’s eye-opening Mathematical Excursions to the World’s Great Buildings, Glen Van Brummelen’s rich Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, and Dana Mackenzie’s lucid The Universe in Zero Words: The Story of Mathematics as Told through Equations. Also be sure to check out our textbooks, including Anne Greenbaum and Timothy P. Chartier’s Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals.

The selection of critical, cutting-edge titles abounds, so if you’re interested in learning more about our other mathematics books, browse our catalog. You may also sign up to stay current on our publishing endeavors with ease here: http://press.princeton.edu/subscribe/ Your email address will remain confidential!

We’ll also see you at the Joint Mathematics Meeting January 9-12 in San Diego, CA at booth 311! The following book signings will be held at our booth:

Wednesday, January 9
2:00 p.m.-3:00 p.m., Glen Van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

Thursday, January 10
1:30 p.m.-2:30 p.m., Alexander J. Hahn, Mathematical Excursions to the World’s Great Buildings
3:00 p.m.-4:00 p.m., Michael Starbird, The 5 Elements of Effective Thinking

Friday, January 11
11:00 a.m.-12:00 p.m., Persi Diaconis and Ron Graham, Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks
1:00 p.m.-2:00 p.m., Dana Mackenzie, The Universe in Zero Words: The Story of Mathematics as Told through Equations
3:00 p.m.-4:00 p.m., Siobhan Roberts, Wind Wizard: Alan G. Davenport and the Art of Wind Engineering

Also, stop booth 311 to chat about March Mathness! We’re aiming to double last year’s six participating schools with a goal of twelve in 2013, providing entertainment for math and basketball aficionados alike! Find out more here in the meantime: http://blog.press.princeton.edu/march-mathness/

Tim Chartier’s Bracketology 101 Webinar — the Math You Need to Pick the Winners

[Update for 2014 -- check out Tim's new tips: http://blog.press.princeton.edu/2014/03/17/top-tips-for-2014-march-madness-brackets-from-tim-chartier/]

Check out Tim Chartier’s webinars on the applications of linear algebra for more helpful hints on how to fill out your bracket.

Tim looks at how famous bracketologists have fared in the past (tip — Obama did OK in the 80th percentile, Dwyane Wade, not so much in the 43rd percentile) and provides step by step analysis of the math behind the Colley method and others.  He also guides viewers through a free java program that is available for download on his professional web site at Davidson College.

 


Bracketology 101: http://davidson.mediasite.mcnc.org/mcnc/SilverlightPlayer/Default.aspx?peid=bbac5c0483ad4659bea8adfc7258e3051d

Advanced Bracketology: http://davidson.mediasite.mcnc.org/mcnc/SilverlightPlayer/Default.aspx?peid=600e3c4c27e741edbc34cbf89a510b7a1d

You can find a link to the Java code for ranking on Tim’s blog: http://sites.davidson.edu/mathmovement/bracketology-101/

Listen to Tim’s talk on March Mathness at the MAA Distinguished Lecture Series: http://www.maa.org/dist-lecture/past-lectures.html

Tim Chartier is an Associate Professor of mathematics at Davidson College. His ability to communicate math both in and beyond the classroom were recognized with the Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member from the Mathematical Association of America. His research and scholarship were recognized with an Alfred P. Sloan Research Fellowship. Tim serves on the Editorial Board for Math Horizons, a mathematics magazine of the Mathematical Association of America. He also serves as chair of the Advisory Council for the Museum of Mathematics. Tim has been a resource for a variety of media inquiries which includes fielding mathematical questions for the Sports Science program on ESPN. He is co-author with Anne Greenbaum of Numerical Methods: Design, Analysis and Computer Implementation of Algorithms. His latest book, Math Bytes, includes a section on bracketology as well as other fun math and computing endeavors.

March Mathness — The Massey Method

In this post for March Mathness, Kenneth Massey whose popular ratings (http://masseyratings.com) help rank the BCS teams each year, offers an overview of what goes into filling out his brackets for March Madness.

 

 


Advice on filling out March Madness brackets from Kenneth Massey

 

I’m a college basketball fan, but to be honest, I don’t watch many games during the regular season. Since my personal expertise is a function of ESPN highlights and commentary, I’ve learned to trust the math more than my own feelings about a matchup.

All season I compile a monster list of the various computer rankings for college basketball: http://www.masseyratings.com/cb/compare.htm

By following the results, I have a pretty good idea about which teams are over/under rated by the media and which ones are coming on strong or fading into tournament time.

Once the pairings are announced, I usually fill out a bracket based on my limited first-hand knowledge of the teams, the impressions I have from following the rankings, and maybe some “gut” intuitions. For that particular bracket, I don’t do any additional analysis, or even look at the numbers–I just rely on what’s already accumulated in my brain.

Now let me describe how I fill out my more analytical brackets. I have two different strategies, but both of them start with estimating the probabilities that each particular team will advance past each round.
In this post, I will describe that process. In a later post, I’ll describe how I use those probabilities to actually fill out my bracket picks.

I’ve been doing computer ratings for years, and have experimented with many mathematical models, one of which is described in Who’s # 1?. The model I currently use, and post on masseyratings.com, is proprietary, but I will list some of the pertinent aspects of it.

1) Margin of victory matters, but in an intelligent way. There are diminishing returns for blowouts, and adjustments are made for the pace of the game. For example 60-45 may be more impressive than 100-80.

2) Winning is rewarded, especially on the road. Even if the margin is small, a team gets a bump by winning games against good competition.

3) Schedule strength is implicit in all the equations. Everything is measured relative to the opponent, so there is higher reward and less risk for playing tough opponents.

4) The model has a decaying memory of early season games. The team in March is different from the team in November.

5) Games between mis-matched opponents are not as important as games between well-matched opponents. There is a lot more information in a #18 vs #23 matchup than there is when #18 plays #230.

6) My model produces offensive and defensive ratings for each team, as well as homefield advantage estimates. From these, it is possible to predict the distribution of final scores for a hypothetical matchup between any two teams.

After the ratings are computed, I use conditional probability to effectively account for every possible scenario of how the bracket could “play out”. For example, if team X makes it to the Sweet 16, who are they likely to face? According to the seedings, some teams have easier paths of advancement. I can compute the probability that each team advances past a given round, the expected number of rounds a team will win, and ultimately each team’s probability of winning the championship.

The great thing about probabilities is that you are never “wrong”. For example, last year my calculations showed that UConn had an 86% chance of winning the first round, a 54% chance of advancing to Sweet 16, a 29% chance of advancing to Elite 8, 12% chance of advancing to Final 4, 5% of playing in the championship game, and a 2.3% chance of winning it all.

By the nature of randomness, it is not really surprising that underdogs occasionally win. Even a dominant #1 overall seed rarely has more than a 25% chance of winning the entire tournament. That’s what makes the event so exciting–nobody knows what will happen.

After all the probabilities are computed, I proceed to fill in my picks. Don’t I just pick the teams with the highest probabilities? Not exactly. I’ll address that in a subsequent post.