Solving last week’s L.A. Math challenge

LA MathWe’re back with the conclusion to last week’s LA Math challenge, The Case of the Vanishing Greenbacks, (taken from chapter 2 of the book). After the conclusion of the story, we’ll talk a little more with the author, Jim Stein. Don’t forget to check out the fantastic trailer for LA Math here.

Forty‑eight hours later I was bleary‑eyed from lack of sleep. I had made no discernible progress. As far as I could tell, both Stevens and Blaisdell were completely on the up‑and‑up.   Either I was losing my touch, or one (or both) of them were wasting their talents, doctoring books for penny‑ante amounts.   Then I remembered the envelope Pete had sealed. Maybe he’d actually seen something that I hadn’t.

I went over to the main house, to find Pete hunkered down happily watching a baseball game. I waited for a commercial break, and then managed to get his attention.

“I’m ready to take a look in the envelope, Pete.”

“Have you figured out who the guilty party is?”

“Frankly, no. To be honest, it’s got me stumped.” I moved to the mantel and unsealed the envelope. The writing was on the other side of the piece of paper. I turned it over. The name Pete had written on it was “Garrett Ryan and the City Council”!

I nearly dropped the piece of paper. Whatever I had been expecting, it certainly wasn’t this. “What in heaven’s name makes you think Ryan and the City Council embezzled the money, Pete?”

“I didn’t say I thought they did. I just think they’re responsible for the missing funds.”

I shook my head. “I don’t get it. How can they be responsible for the missing funds if they didn’t embezzle them?”

“They’re probably just guilty of innumeracy. It’s pretty common.”

“I give up. What’s innumeracy?”

“Innumeracy is the arithmetical equivalent of illiteracy. In this instance, it consists of failing to realize how percentages behave.” A pitching change was taking place, so Pete turned back to me. “An increase in 20% of the tax base will not compensate for a reduction of 20% in each individual’s taxes.   Percentages involve multiplication and division, not addition and subtraction. A gain of 20 dollars will compensate for a loss of 20 dollars, but that’s because you’re dealing with adding and subtracting. It’s not the same with percentages, because the base upon which you figure the percentages varies from calculation to calculation.”

“You may be right, Pete, but how can we tell?”

Pete grabbed a calculator. “Didn’t you say that each faction was out $198,000?”

I checked my figures. “Yeah, that’s the amount.”

Pete punched a few numbers into the calculator. “Call Ryan and see if there were 99,000 taxpayers in the last census. If there were, I’ll show you where the money went.”

I got on the phone to Ryan the next morning. He confirmed that the tax base in the previous census was indeed 99,000. I told Pete that it looked like he had been right, but I wanted to see the numbers to prove it.

Pete got out a piece of paper. “I think you can see where the money went if you simply do a little multiplication. The taxes collected in the previous census were $100 for each of 99,000 individuals, or $9,900,000. An increase of 20% in the population results in 118,800 individuals. If each pays $80 (that’s the 20% reduction from $100), the total taxes collected will be $9,504,000, or $396,000 less than was collected after the previous census. Half of $396,000 is $198,000.”

I was convinced. “There are going to be some awfully red faces down in Linda Vista. I’d like to see the press conference when they finally announce it.” I went back to the guesthouse, called Allen, and filled him in. He was delighted, and said that the check would be in the mail.   As I’ve said before, when Allen says it, he means it. Another advantage of having Allen make the arrangements is that I didn’t have to worry about collecting the fee, which is something I’ve never been very good at.

I wondered exactly how they were going to break the news to the citizens of Linda Vista that they had to pony up another $396,000, but as it was only about $3.34 per taxpayer I didn’t think they’d have too much trouble. Thanks to a combination of Ryan’s frugality and population increase, the tax assessment would still be lower than it was after the previous census, and how many government agencies do you know that actually reduce taxes? I quickly calculated that if they assessed everyone $3.42 they could not only cover the shortage, but Allen’s fee as well. I considered suggesting it to Ryan, but then I thought that Ryan probably wasn’t real interested in hearing from someone who had made him look like a bungler.

My conscience was bothering me, and I don’t like that. I thought about it, and finally came up with a compromise I found acceptable. I went back to the main house.

Pete was watching another baseball game. The Dodgers fouled up an attempted squeeze into an inning‑ending double play. Pete groaned. “It could be a long season,” he sighed.

“It’s early in the year.” I handed him a piece of paper. “Maybe this will console you.”

“What’s this?” He was examining my check for $1,750. “Your rent’s paid up.”

“It’s not for the rent, Pete. It’s your share of my fee.”

“Fee? What fee?”

“That embezzling case in Orange County. It was worth $3,500 to me to come up with the correct answer. I feel you’re entitled to half of it. You crunched the numbers, but I had the contacts and did the legwork.”

Pete looked at the check. “It seems like a lot of money for very little work. Tell you what. I’ll take $250, and credit the rest towards your rent.”

A landlord with a conscience! Maybe I should notify the Guinness Book of Records. “Seems more than fair to me.”

Pete tucked the check in the pocket of his shirt. “Tell me, Freddy, is it always this easy, doing investigations?”

I summoned up a wry laugh. “You’ve got to be kidding. So far, I’ve asked you two questions that just turned out to be right down your alley. I’ve sometimes spent months on a case, and come up dry. That can make the bottom line look pretty sick. What’s it like in your line of work?”

“I don’t really have a line of work. I have this house and some money in the bank. I can rent out the guesthouse and make enough to live on. People know I’m pretty good at certain problems, and sometimes they hire me. If it looks like it might be interesting, I’ll work on it.” He paused. “Of course, if they offer me a ridiculous amount of money, I’ll work on it even if it’s not interesting. Hey, we’re in a recession.”

“I’ll keep that in mind.”   I turned to leave the room. Pete’s voice stopped me.

“Haven’t you forgotten something?”

I turned around. “I give up. What?”

“We had a bet. You owe me five bucks.”

I fished a five out of my wallet and handed it over. He nodded with satisfaction as he stuffed it in the same pocket as the check, and then turned his attention back to the game.

What made you include this particular idea in the book?

JS: The story features one of the most common misunderstandings about percentages.  There are innumerable mistakes made because people assume that percentages work the same way as regular quantities.  But they don’t — if a store lowers the cost of an item by 30% and then by another 20%, the cost of the item hasn’t been lowered by 50% — although many people make the mistake of assuming that it has.  I’m hoping that the story is sufficiently memorable that if a reader is confronted by a 30% discount followed by a 20% discount, they’ll think “Wasn’t there something like that in The Case of the Vanishing Greenbacks?

There are 14 stories in the book, and each features a mathematical point, injected into the story in a similar fashion as the one above.  I think the stories are fun to read, and if someone reads the book and remembers just a few of the points, well, I’ve done a whole lot better than when I was teaching liberal arts math the way it is usually done.

James D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include LA Math, Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian).

Try your hand at solving an L.A. Math mystery

If you caught the rather incredible trailer for L.A. Math, you know it’s not your typical scholarly math book. Romance, crime, and mathematics don’t often go hand in hand, but emeritus professor in the Department of Mathematics at California State University Jim Stein cooked up the idea for an unconventional literary math book that would speak to students in his liberal arts math class. The end result is an entertaining, backdoor approach to practical mathematics knowledge, ranging from percentages and probability to set theory, statistics, and the mathematics of elections. Recently, Stein spoke to us about writing L.A. Math. Not only that, he left us with a mathematical mystery to solve.

L.A. Math is definitely an unusual book.  Brian Clegg described it by saying “It’s as if Ellery Queen, with the help of P. G. Wodehouse, spiced up a collection of detective tales with a generous handful of practical mathematics.”  How did you happen to write it?

JS: I absolutely loved it when he described it that way, because I was brought up on Ellery Queen.  For younger readers, Ellery Queen was one of the greatest literary detectives of the first half of the twentieth century, specializing in classic Sherlock Holmes type cases.  The Ellery Queen stories were written by the team of Manfred Dannay and Frederick Lee — and my mother actually dated one of them!

LA MathThe two other mystery writers who influenced me were Agatha Christie and Rex Stout.  Rex Stout wrote a series featuring Nero Wolfe and Archie Goodwin; the books are presumably written by Archie Goodwin describing their cases, so I used that as the model for Freddy Carmichael.  The relationship between Archie and Nero also served, somewhat, as a parallel for the relationship between Freddy and Pete.  Nero and Pete both have addictions — Nero wants to spend his time eating elaborate cuisine and raising orchids, and Pete wants to spend his time watching and betting on sports.  It’s up to Archie and Freddy to prod them into taking cases.

How does Agatha Christie enter the picture?

JS: I’d taught liberal arts mathematics — math for poets — maybe ten times with temporary success but no retention.  Students would learn what was necessary to pass the course, and a year later they’d forgotten all of it.  That’s not surprising, because the typical liberal arts math course has no context that’s relevant for them.  They’re not math-oriented.  I know I had several history courses discussing the Battle of Azincourt, but I don’t remember anything about it because it has no context for me.

Agatha Christie’s best-known detective is Hercule Poirot, and one day I was in a library reading a collection of short stories she had written entitled The Labors of Hercules.  Christie had a background in the classics, and did something absolutely brilliant — she constructed a series of twelve detective stories featuring Hercule Poirot, each of which was modeled, in one way or another, around the Twelve Labors of Hercules in classical mythology.  I thought to myself — why don’t I do something like that for topics in liberal arts math?  Maybe the students would remember a few of the ideas because they’d have the context of a story from which to remember it.

Could you give an example?

JS: How about this?  Why don’t we take a story from the book, and present it the way Ellery Queen would have.  Ellery Queen always played fair with the reader, giving him or her all the clues, and after all the clues had been presented, EQ would write a paragraph entitled “Challenge to the Reader”.  EQ would tell the reader “Now you have all the clues.  Can you figure out whodunit?” — or words to that effect.

OK, here’s what we’ll do.  We’ll take The Case of the Vanishing Greenbacks, Chapter 2 in L.A. Math, and present the story up to the crucial point.  Then we’ll let the reader try to figure out whodunit, and finish the story next week.

Chapter 2 – The Case of the Vanishing Greenbacks

   The phone rang just as I stepped out of the shower. It was Allen.

“Freddy, are you available for an embezzlement case?”

My biggest success had been in an embezzlement case involving a Wall Street firm specializing in bond trading. Allen had given me a whopping bonus for that one, which was one of the reasons I could afford to take it easy in L.A. I had done well in a couple of other similar cases, and had gotten the reputation of being the go-to guy in embezzlement cases. It never hurts to have a reputation for being good at something. Besides, you don’t see many guys in my line of work who can read balance sheets.

I’ve always felt it’s important to keep the cash flow positive, and the truth was that I was available for a jaywalking case if it would help the aforementioned cash flow. But it never hurts to play a little hard-to-get.

“I can probably clear my calendar if it looks interesting.”

Allen paused for a moment, either to collect his thoughts or to take a bite of one of those big greasy pastrami sandwiches he loves. “I’m pretty sure you’ll find it interesting. It’s stumped some people in L.A., and I told them I had a good man out there. BTW, that’s you.”

It’s nice to be well thought of – especially by someone in a position to send you business. I knew that Allen’s firm, though headquartered in New York, had arrangements with other firms in other cities. I didn’t really care about the details as long as the check cleared – which it always had.

“I’m certainly willing to listen. What’s the arrangement?”

“Consulting and contingency fee. Fifty‑fifty split.”

That was our usual arrangement. Burkitt Investigations got a guaranteed fee, plus a bonus for solving the case. Allen and I split it down the middle.

“OK, Allen, fill me in.”

“Ever heard of Linda Vista, Freddy?”

Temporary blank. Movie star? Socialite? Then I had it. Linda Vista was a town somewhere in Orange County with a big art community.

For those of you not up on California politics, Orange County is a bastion of conservatism. You have Orange County to thank, or blame, for Richard Nixon and Ronald Reagan. But Linda Vista, which my fragmentary Spanish translates as “pretty view”, was different from your basic Orange County bastion.

The vista in Linda Vista was sufficiently linda that it had attracted a thriving artistic community.   There were plenty of artists in Linda Vista, and most of them were liberals.

As a result, Linda Vista was highly polarized. The moderates were few and far between. On the left, you had the artists, with their funky bungalows and workshops. On the right, you had the stockbrokers and real-estate moguls, living in gated communities so they wouldn’t have to have any contact with the riff-raff, except for the tradesmen delivering or repairing stuff. However, there were enough artists and hangers-on to acquire political clout – after all, it’s still one man-one vote in a democracy, rather than one dollar-one vote. Pitched battles had raged over practically every issue from A (abortion) to Z (zoning), and many of these battles had made state and even national news.

That’s all I knew about Linda Vista, other than not to try to drive down there at rush hour, which turned one hour on the 405 to more than twice that. The obvious question was: what kind of a contingency case had they got? So I asked it.

Allen filled me in. “The city is out a bunch of bucks, and each side is accusing the other of fraud and embezzlement. Because of the split in the political situation, the City Manager gave half the budget to the conservatives, and the other half to the liberals, letting each determine how to spend its half. Both sides claim to have been shortchanged.”

Allen paused to catch his breath. “I’ve got a friend who works in the City Manager’s office. I told him I had a good man out there who’d done a lot of first‑class work in embezzlement cases. Want to take a look at it?”

“Sure. How much time should I put in before I throw in the towel?” In other words, how much is the consulting fee?

“As much as you like.” In other words, since Allen’s meter wasn’t running, feel free to burn some midnight oil. “The consulting fee is $3,000, upped to ten if you figure it out and get proof.” You don’t have to be an expert at division to realize that I was guaranteed a minimum of $1,500 for the time I put in, and $5,000 if I doped it out. You also don’t have to be an expert at division to realize that Allen was getting the same amount for making a phone call. I decided to be reincarnated as an employer rather than an employee.

Allen gave me a brief description of the protagonists, and I spent a good portion of the evening with a pot of coffee and my computer, getting some background information on them. I’ll say one thing for the Information Age; it’s a lot easier to run a background check on people than it used to be. What with search engines and social networks, you save a lot on gas money and shoe leather.

The next morning I waited until after rush hour, and made the trek to Linda Vista. The City Hall was located in a section of town where the vista was a long way from linda, unless strip malls filled with 7‑11s and fast-food stores constitute your idea of attractive scenery. I found a place to park, straightened my coat and tie, and prepared for the interviews.

I was scheduled to have three of them. I had been hoping to arrange for longer interviews, but everyone’s in a rush nowadays, and I was getting a quarter-hour with each, tops. They’d all been interviewed previously – Allen had mentioned that this case had stumped others – and people are generally less than enthusiastic about being asked the same questions again. And again. The first interview was with Everett Blaisdell, conservative city councilman, who would explain why the conservatives happened to be short. The next was with Melanie Stevens, liberal city councilwoman, ditto. The last interview would be with Garrett Ryan, City Manager.

I have a bad habit. My opinion of members of groups tends to be formed by the members of those groups that I have seen before. Consequently, I was expecting the conservative Everett Blaisdell to look like a typical paunchy southern senator with big jowls. So I was a little surprised to discover that Everett Blaisdell was a forty-ish African-American who looked like he had spent years twenty through thirty as an NBA point guard.

He got right down to business. “I want you to know,” he barked, “that everything that we have done with our budget allocation has been strictly by the book. Our expenses have been completely documented.” He handed me a folder full of ledger sheets and photos of checks, which I glanced at and stashed in my briefcase.

Blaisdell was clearly angry. “The business community is the heart of Linda Vista, and it is ridiculous to suggest that it would act in a manner detrimental to its citizens. We are $198,000 short in our budget.”

You don’t expect NBA point guards to get out of breath too easily, considering the time they have to go up and down the court, but maybe Blaisdell wasn’t in shape. He paused, giving me a chance to get a question in edgewise. “Just what do you think has happened, Mr. Blaisdell?” I inquired mildly.

“I know what has happened. Melanie Stevens and her radical crowd have managed to get hold of that money. They want $200,000 to fund a work of so‑called art which I, and every right‑thinking citizen of Linda Vista, find totally offensive. It’s mighty suspicious that the missing funds, $198,000, almost precisely cover the projected cost of the statue.”

I was curious. “If you don’t mind my asking, exactly what is this statue?”

Blaisdell’s blood pressure was going up. “They are going to build a scale replica of the Statue of Liberty and submerge it in Coca‑Cola. You may know that Coca‑Cola is acidic, and it will eventually dissolve metal. They say that this so‑called dynamic representational art represents the destruction of our civil liberties by over‑commercialization. Well, let me tell you, we’ll fight it.”

He looked at his watch. “Sorry, I’ve got another appointment. When you find out what those scum have done with the money, let me know.” He walked me to his door.

It took a few minutes to locate Melanie Stevens’ office, as it was in a different wing of the building, possibly to minimize confrontations between her and Blaisdell. It was a bad day for stereotypes. My mental picture of Melanie Stevens, ultra‑liberal, was that of a long-haired hippie refugee from the ’60s. The real Melanie Stevens was a pert gray‑haired grandmother who looked like she had been interrupted while baking cookies for her grandchildren. She, too, was evidently on a tight schedule, for she said, “Sorry, I can only give you about ten minutes, but I’ve made copies of all our expenses.” More ledger sheets and photos of checks went into my briefcase.

“Let me tell you, Mr. Carmichael, that we could have used that $198,000. We planned to use it for a free clinic. I know exactly what has happened. Blaisdell has doctored the books. I’m sure glad that Ryan had the guts to ask you to look into it.”

“Blaisdell seems to think that your people are responsible for the missing funds,” I observed.

She snorted. “That’s just typical of what they do. Whenever they’re in the wrong, they lie and accuse the other side of lying. They rip off the community, and channel money into PACs. Political action committees. Or worse. Blaisdell knows he faces a stiff battle for re-election, and I wouldn’t be the least bit surprised to find that money turning up in his campaign fund.”

“He seems to think that you are going to use the funds for an art project, rather than a free clinic,” I remarked.

“He’s just blowing smoke. He knows quite well that the statue will be funded through private subscription.” She looked at her watch. “Let me know when you pin the loss on them.”

I left Stevens’ office for the last interview, with Garrett Ryan, whose anxious expression made it clear that he was not a happy camper. “Have you got any ideas yet?” he asked.

I shook my head. “I’ve just talked to Blaisdell and Stevens. They’ve each handed me files containing what they consider to be complete documentation. They’ve each given me a story asserting their own innocence, and blaming the other. I take it that the missing amount is $198,000?”

Now it was Ryan’s turn to shake his head. “No, each side says that it is missing $198,000. Quite a coincidence. And I’ll tell you, Mr. Carmichael, despite the animosity between them, I think that they are both honorable individuals. I find it difficult to believe that either would rip the city off.”

I focused on Ryan’s coincidence. “It’s funny that they are both short exactly the same amount. Perhaps you could tell me a little more about the budgetary process.”

“It’s really quite simple. Each resident of Linda Vista is taxed a fixed amount. Any complicated tax scheme would just result in a full employment act for accountants. The previous census resulted in a $100 assessment per individual. The population of Linda Vista increased by 20% since the last census. We didn’t need any increase in operating expenditures; under my guidance we’ve done a fiscally conservative and frugal job of running the city. As a result, the Council voted to reduce everybody’s taxes by 20%. Needless to say, this was a very popular move.”

“I’ll bet it was. Did everyone pay their taxes, Mr. Ryan?”

“Everybody. We’re very proud of that ‑‑ a 100% collection rate. Despite what you may have heard, the citizens of Linda Vista are very civic‑minded. Liberals and conservatives alike.”

I’ve spent enough time with balance sheets to know that accuracy is extremely important. “Was this population increase exactly 20%, or is that merely an approximate figure?”

Ryan consulted a sheet of paper. “Exactly 20%. I have a sheet of printout that gives information to four decimal places, so I can be quite sure of that.”

Just then a phone rang. Ryan picked it up, and engaged in some political doubletalk. After a few minutes he replaced the receiver. “Sorry, Mr. Carmichael. I’m behind schedule. Let me know if you make any progress.”   We shook hands, and I left.

A couple of hours later, I got home, having stopped for a bite but still avoiding rush-hour traffic. Pete noticed my presence, and asked, “So how’d things go in Linda Vista, Freddy?”

“I had a pretty interesting day. Want to hear about it?”

He nodded. I took about fifteen minutes to describe the problem and the cast of characters. “It looks like I’ll have to spend a day or so looking over the books.”

Pete shook his head. “It seems pretty clear to me.”

I’d seen it before — everybody’s a detective. Amateurs always think they know who the guilty party is, because it fits in with their preconceptions. I didn’t know whether Pete had cast Blaisdell in the role of a political fat-cat out to line his campaign war chest, or whether he was a conservative who saw Melanie Stevens as a radical troublemaker. Anyway, you’ve got to learn not to jump to conclusions in my line of work.

“You can’t do it like that, Pete. You’ve got to trace down the paper trails. I’ve done this lots of times.”

Pete grabbed a piece of paper, scribbled something on it, and sealed it in an envelope. “Five dollars will get you twenty that the name of the guilty party is inside this envelope.”

Pete needed taking down a peg. Maybe two pegs. Besides, I liked getting four‑to‑one odds on what was obviously an even‑ money proposition. “You’ve got a bet,” I said. We wrote our names on the envelope, and Pete put it on the table next to the HDTV.

“Whenever you’re ready, we’ll unseal the envelope.” I headed back to the guesthouse for a session with the books.

Challenge to the Reader: You have all the clues. Can you name the party responsible for the missing greenbacks? We’ll give you until the next blog to figure it out, when we’ll present the conclusion to the story.

New Physics & Astrophysics Catalog

We invite you to browse our Physics & Astrophysics 2016 catalog:

 

Interacademy Partnership Check out Doing Global Science, an introductory guide to responsible science in our globalized society. Written by a committee of leading scientists from all over the world, this text is required reading for anyone involved in scientific inquiry.
Thorne Modern Classical Physics is a graduate-level text and reference book for first-year students that covers statistical physics, optics, elastodynamics, fluid mechanics, plasma physics, and special and general relativity and cosmology.
Maoz

A. Zee has contributed another new title to our In a Nutshell series entitled Group Theory in a Nutshell for Physicists. He takes all the nuts and bolts of a mathematical subject and makes it accessible for physicists. PUP is also publishing the second edition of Astrophysics in a Nutshell by Dan Maoz this season, a work that has become a standard text in courses on astrophysics.

If you would like updates of new titles emailed to you, subscribe to our newsletter.

Finally, PUP will be at the American Physical Society March Meeting in Baltimore from March 14 to March 18.

New Mathematics Catalog

We invite you to browse our Mathematics 2016 catalog:

 

Penrose In his forthcoming book, Roger Penrose makes the case that physicists are just as prone to be influenced by fashion, faith, and fantasy as anyone else. Sometimes, these forces can be positive, he argues, but they often lead researchers astray. Pick up a copy of Fashion, Faith, and Fantasy in the New Physics of the Universe to learn more.
AshGross Interested in numbers? Then Summing It Up by mathematicians Avner Ash and Robert Gross is for you! Ash and Gross have written an accessible book about current mathematical research that can be enjoyed by those with a casual interest and college math majors alike.
Nahin Paul J. Nahin explains how physics can be found in everyday situations in In Praise of Simple Physics. You’ll be surprised at how often you use it!

If you would like to be updated on new titles, subscribe to our newsletter.

Finally, if you’re going to be in Seattle for the Joint Mathematics Meeting from January 6 to January 9, visit PUP at booth #105 or follow it online using #JMM16.

Introducing the mesmerizing new trailer for Mathematics and Art

Looking for a unique coffee table book for someone mathematically or artistically inclined? Mathematics and art are surprisingly similar disciplines, given their distinctively introspective, expressive natures. Even before antiquity, artists have attempted to render mathematical concepts in visual form, and the results have often been spectacular. In a stunning illustrated cultural history that one truly has to see to appreciate, Lynn Gamwell of the School of Visual Arts in New York explores artistic representations from the Enlightenment—including Greek, Islamic, and Asian mathematics—to the modern era, including Aleksandr Rodchenko’s monochrome paintings. Check out her piece on the Guardian’s Adventures in Numberland blog, and the trailer for Mathematics and Art, here:

 

PUP congratulates writers chosen for The Best Writing on Mathematics 2015

Highlighting the finest articles published throughout the entire year, The Best Writing on Mathematics 2015 shines the spotlight on math’s brightest, most creative minds. Edited by Mircea Pitici, the volume is inviting to experienced mathematicians and numbers novices alike.

The Best Writing on Mathematics, in its sixth edition, offers surprising and meaningful insights and perspectives into the highly influential world of mathematics. Colm Mulcahy and Dana Richards express their appreciation and reflections of the significant work of icon Martin Gardner, Toby Walsh creatively uses the popular game Candy Crush as a vehicle to analyze the hardships of solving computational problems, Benoît Rittaud and Albrecht Heeffer investigate and question the true derivation of the pigeonhole principle, Carlo Cellucci considers and defines beauty in mathematics — and that’s just the beginning.

Best Writing on Math 2015

Congratulations to those chosen to be included in The Best Writing in Mathematics 2015!

Interpreting mathematics is not about mathematical truth (or any other truth); it is a personal take on mathematical facts, and in that it can be true or untrue, or it can even be fiction; it is vision, or it is rigorous reasoning, or it is pure speculation, all occasioned by mathematics; it is imagination on a mathematical theme; it goes back several millennia and it is flourishing today, as I hope this series of books lays clear, (xiii)

— Mircea Pitici, Editor

 


Articles and authors selected in The Best Writing on Mathematics 2015

Articles Authors
A Dusty Discipline Michael J. Barany and Donald MacKenzie
How Puzzles Made Us Human Pradeep Mutalik
Let the Games Continue Colm Mulcahy and Dana Richards
Challenging Magic Squares for Magicians Arthur T. Benjamin and Ethan J. Brown
Candy Crush’s Puzzling Mathematics Toby Walsh
Chaos on the Billiard Table Marianne Freiberger
Juggling with Numbers Erik R. Tou
The Quest for Randomness Scott Aaronson
Synthetic Biology, Real Mathematics Dana Mackenzie
At the Far Ends of a New Universal Law Natalie Wolchover
Twisted Math and Beautiful Geometry Eli Maor and Eugen Jost
Kenichi Miura’s Water Wheel, or The Dance of the Shapes of Constant Width Burkard Polster
Dürer: Disguise, Distance, Disagreements, and Diagonals! Annalisa Crannell, Marc Frantz, and Fumiko Futamura
The Quaternion Group as a Symmetry Group Vi Hart and Henry Segerman
The Steiner-Lehmus Angle Bisector Theorem John Conway and Alex Ryba
Key Ideas and Memorability in Proof Gila Hanna and John Mason
The Future of High School Mathematics Jim Fey, Sol Garfunkel, Diane Briars, Andy Isaacs, Henry Pollak, Eric Robinson, Richard Scheaffer, Alan Schoenfeld, Cathy Seeley, Dan Teague, and Zalman Usiskin
Demystifying the Math Myth: Analyzing the Contributing Factors for the Achievement Gap between Chinese and U.S. Students Guili Zhang and Miguel A. Padilla
The Pigeonhole Principle, Two Centuries before Dirichlet Benoît Rittaud and Albrecht Heeffer
A Prehistory of Nim Lisa Rougetet
Gödel, Gentzen, Goodstein: The Magic Sound of a G-String Jan von Plato
Global and Local James Franklin
Mathematical Beauty, Understanding, and Discovery Carlo Cellucci
A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics Mark Balaguer
Writing about Math for the Perplexed and the Traumatized Steven Strogatz
Is Big Data Enough? A Reflection on the Changing Role of Mathematics in Applications Domenico Napoletani, Marco Panza, and Daniele C. Struppa
The Statistical Crisis in Science Andrew Gelman and Eric Loken
Statistics and the Ontario Lottery Retailer Scandal Jeffrey S. Rosenthal
Never Say Never David J. Hand

Mircea Pitici holds a PhD in mathematics education from Cornell University, where he teaches math and writing. He has edited The Best Writing on Mathematics since 2010.

Feynman on the historic debate between Einstein & Bohr

The golden age of quantum theory put many of the greatest minds of the 20th century in contact with some of the most significant scientific and philosophical questions of their era. But it also put these minds in contact with one another in ways that have themselves been a source of curiosity and ongoing scientific debate.

Richard Feynman and Albert Einstein, two towering geniuses of their time, were both as revered for their scientific contributions as they were beloved for their bursts of wisdom on a wide range of subjects. It’s hard not to wonder just what these men thought of one another. Princeton University Press, which published The Ultimate Quotable Einstein in 2010 publishes The Quotable Feynman this fall. The book includes reflections by Feynman on Einstein, from his memorable mannerisms to his contributions to some of the most heated debates in 20th century science.Feynman quote

Perhaps because of the gap between their career high points, (Einstein died in 1955; Feynman didn’t receive his Nobel Prize until 1965), there are no verified quotes where Einstein alludes to Feynman or his expansive body of work. But Feynman had made observations on the older physicist, several of which revolve around Einstein’s famous 1927 public debate with Niels Bohr on the correctness of  quantum mechanics. Central to the debate was this question: Were electrons, light, and similar entities waves or particles? In some experiments they behaved like the former, and in others, the latter.

In an attempt to resolve the contradictory observations, Einstein proposed a series of “thought experiments”, which Bohr responded to. Bohr essentially took the stance that the very act of measuring alters reality, whereas Einstein insisted that reality exists, independent of the act of measurement. Key to the philosophy of science, the dispute between the two giants is detailed by Bohr in “Discussions with Einstein on Epistemological Problems in Atomic Physics”. Richard Feynman is quoted as commenting on the debate:Feynman quote 2

An Einstein Encyclopedia contains a section on the Einstein-Bohr debates, as well as a wealth of other information on Einstein’s career, family, friends. There is an entire section dedicated to righting the various misconceptions that swirl around the man, and another on his romantic interests (actual, probable, and possible).

In spite of their differences, Bohr and Einstein were friends and shared great respect for each others’ work. Until Einstein’s death 3 decades later, they continued their debates, which became, in essence, a debate about the nature of reality itself.  feynman quote 3

Check out other new Einstein publications this fall, including:

Relativity
An Einstein Encyclopedia
The Road to Relativity

Why Calculus Will Save You from the Zombie Apocalypse

To survive a zombie apocalypse, one will need more than instinct and short term solutions – one will need logic and, most importantly, math. A thought-out plan comprised of sophisticated calculus equations will ensure long-term safety objectives. Thankfully, Zombies and Calculus by Colin Adams colorfully illustrates the critical implementation of calculus components when going head-to-head with zombies. Adams demonstrates how a professor and his students successfully exercise calculus to survive the attacks of zombies who not only disrupt their calculus class (the horror!), but are also out for human flesh.

Here are a few need-to-knows:

Zombies travel approximately at one yard per second – a constant derivative.

A derivative of a function is its rate of change. If a function is changing quickly, its derivate will be high, while if a function is changing slowly, its derivate will be low. Adams explains that we can measure the function’s rate of change through the steepness of the tangent line. zombies and calculus rate of change
Since speed is defined as distance divided by time, one can calculate the speed required to get from Point A to Point B in a specific time, while being able to evade any unwanted visitors (zombies). Keep in mind — speed tends to vary (not for zombies, remember, they travel in a constant derivative!), so the derivate of the function has the potential to increase or decrease. Using these simple formulas, one is able to plan out the distance, time, and speed needed to outrun these deadly predators.

It’s hard to crack a zombie’s skull. It’s easier to knock a zombie unconscious.

As detailed in Zombies and Calculus, the amount of force necessary to crack a human skull is 10,000 newtons (a newton is a measurement for force that equals 1 kilogram meter per second squared). Adams offers an example: if a baseball is going 90 miles per hour (40.2 meters/second), weighs 5 ounces (0.145 kilograms), and comes into contact with a head for .007 seconds, its force can be calculated through:Screen Shot 2015-10-29 at 4.44.31 PMSo since a baseball, with said specifications, can only create approximately 800 newtons, imagine how much force is needed to produce 10,000 newtons! When attacking a zombie with force, do not try to go for the easy kill — rather play strategically by knocking the zombie unconscious with a sudden sharp blow to the head. This will create a dramatic head jerk, causing the brain to get knocked around in the cranial cavity, thus causing a short circuit. The benefit of knocking a zombie unconscious, of course, is additional planning and escape time!

Zombies pursue in a radiodrome path.

Like a dog pursues a rabbit, a zombie pursues its human prey. A zombie will follow its prey’s path at the prey’s given location at that specific instant. In a scene from Zombies and Calculus, (pause to imagine it), a Dean is running towards the safe haven of an academic building in a straight line. However, a zombie is present and begins to pursue the Dean, always having its tangent vector pointing at the Dean. The zombie is going to travel to wherever the Dean is in that current moment. Screen Shot 2015-10-29 at 4.39.23 PM

Since zombies are incapable of developing an efficient plan, the zombie does not run at a diagonal towards the academic building, which would cut-off the dean’s path. Instead of recognizing the Dean’s travel pattern or destination, the zombie is chasing the dean like a dog chasing a bunny’s tail to the rabbit hole. If only the dog knew that its radiodrome procedure was flawed, the dog would be able (with a speed higher than the rabbit) to cut-off the rabbit at its hole and claim victory. If dogs were to catch on, there would probably be fewer bunnies hopping around.

Cold-blooded creatures are unable to regulate their body heat.

Like other cold-blooded creatures, zombies hibernate. A zombie’s body temperature will decrease according to the differential equation that guides the temperature change of an object placed in a space with a different temperature (so for instance, if a zombie with a temperature of 60 degrees is placed a room of 30 degrees.) According to Newton’s Law of Cooling (remember Newton from discussing the measurement ‘newton’ for force?), the temperature of a body’s rate of change is proportional to the difference between the present temperature of that body and the ambient temperature (basically, the temperature of its surroundings). Given as a function of time, the zombie’s temperature (where Tg is the specific location):Screen Shot 2015-10-29 at 4.42.31 PMThe larger the contrast of temperatures, the faster the body temperature will drop. As the characters in the book discover, if there is a zombie apocalypse, it might be time to consider a move to our friendly neighbor to the north, Canada.

 

Zombies and CalculusTo discover more lifesaving tips, fun and entertaining mathematical applications, and learn the fate of the brave calculus professor and his students, read Colin Adam’s  Zombies and Calculus. Just in case the zombie apocalypse does occurs (maybe tomorrow?) it should be comforting to know there’s a mathematical guide to survival on your bookshelf.

Benford’s Law: A curious statistical phenomenon that keeps getting curiouser

Ted Hill, one of the contributors to The Princeton Companion to Applied Mathematics, as well as the coauthor, with Arno Berger, of An Introduction to Benford’s Law, has written a post on this fascinating statistical phenomenon. You’ll be surprised at the rather unexpected places it pops up, from an analysis of Donald Trump’s finances, to earthquake detection.

Benford’s Law

The acclaimed business and technology news website Business Insider proudly offers this advice to its readers, in capital letters:

IF YOU ARE PLANNING TO COMMIT FRAUD…LEARN BENFORD’S LAW

The curious statistical phenomenon known as Benford’s Law, first discovered by Newcomb in 1881 and later rediscovered and popularized by Benford in 1938, is currently experiencing an explosion of research activity, especially in fraud detection ranging from tax data and digital images to clinical trial statistics, and from voting returns to macroeconomic data. Complementing these new forensic Benford tools, recent applications also include earthquake detection, analysis of Big Data and of errors in scientific computations, and diagnostic tests for mathematical models. As is common in developing fields, the quality of this research is all over the map, from scholarly and insightful to amusing and outlandish. The most recent Benford article I have seen is an analysis of Donald Trump’s finances, and I will let interested readers have fun judging these Benford articles for themselves. Most may be found on the open access and fully searchable Benford Online Bibliography, which currently references more than 800 articles on Benford’s Law, as well as other resources (books, websites, lectures, etc.).

The First-digit Law

In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford’s Law says that the leading decimal digit is not equally likely to be any one of the nine possible digits 1, 2, …, 9, but rather follows the logarithmic distribution

equationwhere D1 denotes the first significant decimal digit. Many numerical datasets follow this distribution, from mathematical tables like the Fibonacci numbers and powers of 2 to real-life data like the numbers appearing in newspapers, in tax returns, in eBay auctions, and in the meta-dataset of all numbers on the World Wide Web (see Figure 1).

For datasets like these that are close to being Benford, about 30% of the leading (nonzero) decimal digits are 1, about 18% are 2, and the other leading digit proportions decrease exponentially to about 5% that begin with 9.

fig 1
Figure 1. Empirical Evidence of Benford’s Law

The complete form of Benford’s Law also specifies the probabilities of occurrence of the second and higher significant digits, and more generally, the joint distribution of all the significant digits. For instance, the probability that a number has the same first three significant digits as π = 3.141… is

eqn 2(For non-decimal bases b, the analogous law simply replaces decimal logarithms with logarithms base b.)

Robustness of Benford’s Law

Benford’s Law is remarkably robust, which may help explain its ubiquity in both theory and applications. For example, it is the only distribution on significant digits that is scale invariant (e.g., converting from dollars to euros or feet to meters preserves Benford’s Law), and is the only continuous distribution on significant digits that is base-invariant.

As an example of stochastic robustness, if a random variable X satisfies Benford’s Law, then so does XY for all positive Y independent of X; thus in multiplying independent positive random variables, say to model stock prices, if you ever encounter a single Benford’s Law entry, the whole product will obey Benford’s Law. Moreover, if X follows Benford’s Law, then so do 1/X and X2, (and all other non-zero integral powers of X).

Benford’s Law is also robust under both additive and multiplicative errors: If an increasing unbounded sequence of values X obeys Benford’s Law, then so does X + E for every bounded “error” sequence E, and if X is Benford and E is any independent error with |E| < 1, then (1 + E)X is also exactly Benford.

Applications of Benford’s Law

The most widespread application of Benford’s Law currently is its use in detection of fraud. The idea here is simple: if true data of a certain type is known to be close to Benford’s Law, then chi-squared goodness-of-fit tests can be used as a simple “red flag” test for data fabrication or falsification. Whether the tested data are close to Benford’s Law or are not close proves nothing, but a poor fit raises the level of suspicion, at which time independent (non-Benford) tests or monitoring may be applied.

A similar application is being employed to detect changes in natural processes. If the significant digits are close to Benford’s Law when the process is in one particular state, but not when the process is in a different state, then comparison to Benford can help identify when changes in the state of the process occur. Recent studies have reported successful Benford’s Law tests to detect earthquakes, phase transitions in quantum many-body problems, different states of anesthesia, signal modulations in electrophysiological recordings, and output changes in interventional radiology.

Tests for goodness-of-fit to Benford are also useful as a diagnostic tool for assessing the appropriateness of mathematical models. If current and past data obey Benford’s Law, it is reasonable to expect that future data will also obey Benford’s Law. For example, the 1990, 2000, and 2010 census statistics of populations of the some three thousand counties in the United States follow Benford’s Law very closely (see Figure 1), so to evaluate a proposed mathematical model’s prediction of future populations, simply enter current values as input, and then check to see how closely the output of that model agrees with Benford’s Law (see Figure 2).

fig 2
Figure 2. Benford-in-Benford-out Diagnostic Test

The appearance of Benford’s law in real-life scientific computations is now widely accepted, both as an empirical fact (as reported in Knuth’s classic text), and as a mathematical fact (e.g., Newton’s method and related numerical algorithms have recently been shown to follow Benford’s Law). Thus, in those scientific calculations where Benford’s Law is expected to occur, knowledge of the distribution of the output of the algorithm permits better estimates of both round-off and overflow/underflow errors.

Recent Theoretical Developments

Complementing these applications are new theoretical advancements, which are useful in explaining and predicting when Benford analysis is appropriate, and which are also of independent mathematical interest. Recent results include:

  • The outputs of many numerical algorithms, including Newton’s method, obey Benford’s Law.
  • Iterations of most linear functions follow Benford’s Law exactly, and iterations of most functions close to linear, such as f(x) = 2x + ex, also follow Benford’s Law exactly.
  • Continuous functions with exponential or super-exponential growth or decay typically exhibit Benford’s Law behavior, and thus wide classes of initial value problems obey Benford’s Law exactly.
  • Powers and products of very general classes of random variables, including all random variables with densities, approach Benford’s law in the limit (see Figure 3 for the standard uniform case).
  • Many multidimensional systems such as powers of large classes of square matrices and Markov chains, obey Benford’s Law.
  • Large classes of stochastic processes, including geometric Brownian motion and many Levy processes, obey Benford’s Law.
  • If random samples from different randomly selected probability distributions are combined, the resulting meta-sample also typically converges to Benford’s Law. (This may help explain why numbers in the WWW and newspapers and combined financial data have been found to follow Benford’s Law.)

 

Fig 3
Figure 3. Powers of a Uniform Random Variable

The study of Benford’s Law has also at times been entertaining. I’ve been contacted about its use to support various religious philosophies (including evidence of Benford’s Law in the Bible and Quran, and its appearance in tables of the earth’s elements as evidence of Intelligent Design), as well as a website where Eastern European entrepreneurs sold Benford data to people who need it for 25 euros a pop. For me, however, the main attraction has been its wealth of fascinating and challenging mathematical questions.

Ted Hill is Professor Emeritus of Mathematics at the Georgia Institute of Technology, and currently Research Scholar in Residence at the California Polytechnic State University in San Luis Obispo. He is the co-author, with Arno Berger, of An Introduction to Benford’s Law, (Princeton University Press, 2015).

Intro to Benford's Law

 

Introducing the new video trailer for The Princeton Companion to Applied Mathematics

We are pleased to present the new video trailer for The Princeton Companion to Applied Mathematics. Modeled on the popular Princeton Companion to Mathematics, this is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book. Check out the video in which editor Nicholas Higham, Richardson Professor of Applied Mathematics at The University of Manchester, talks about the major ideas covered in this expansive project, which includes nearly 200 entries organized thematically and written by an international team of distinguished contributors.

An interview with Nicholas Higham on The Princeton Companion to Applied Mathematics

Higham jacket

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, new this month. Recently he took the time to answer some questions about the book, and where the field is headed. Read his popular first post on color in mathematics here.

What is Applied Mathematics?

NH: There isn’t any generally agreed definition, but I rather like Lord Rayleigh’s comment that applied mathematics is about using mathematics to solve real-world problems “neither seeking nor avoiding mathematical difficulties”. This means that in applied mathematics we don’t go out of our way to consider special cases that will never arise in practice, but equally we do not sidestep genuine difficulties.

What is the purpose of The Companion?

NH: The Companion is intended to give an overview of the main areas of applied mathematics, to give examples of particular problems and connections with other areas, and to explain what applied mathematicians do—which includes writing, teaching, and promoting mathematics as well as studying the subject. The coverage of the book is not meant to be exhaustive, but it is certainly very broad and I hope that everyone from undergraduate students and mathematically interested lay readers to professionals in mathematics and related subjects will find it useful.

What is an example of something aspect of applied mathematics that you’ve learned while editing the book?

NH: Applied mathematics is a big subject and so there are many articles on topics outside my particular areas of expertise. A good example concerns applications of computational fluid dynamics in sport. An article by Nicola Parolini and Alfio Quarteroni describes the mathematical modeling of yachts for the America’s cup. The designer wishes to minimize water resistance on the hull and maximize the thrust produced by the sails. Numerical computations allow designs to be simulated without building and testing them. The article also describes mathematical modeling of the hi-tech swimsuits that are now banned from competition. The model enables the benefit of the suits on race times to be estimated.

The Companion is about 1000 pages. How would advise people to read the book.

NH: The book has a logical structure, with eight parts ranging from introductory material in Part I, the main areas of applied mathematics in Part IV (the longest part), through to broader essays in the final part. It is a good idea to start by reading some of the articles in Part I, especially if you are less familiar with the subject. But a perfectly sensible alternative approach is to select articles of interest from the table of contents, read them, and follow cross-references. Or, you can just choose a random article and start reading—or simply follow interesting index entries! We worked very hard on the cross-references and index so an unstructured approach to reading should lead you around the book and allow you to discover a lot of relevant material.

What was the hardest thing about editing The Companion?

NH: The hardest aspect of the project was ensuring that it was completed in a reasonable time-frame. With 165 authors it’s hard to keep track of everything and to to ensure that drafts, revisions, and corrected proofs are delivered on time.

How much of the book did you write?

NH: I wrote about 100 of the 1000 pages. This was great fun, but it was some of the hardest writing I’ve done. The reason is partly that I was sometimes writing about topics that I don’t normally write about. But it was also because Companion articles are quite different from the papers I’m used to writing: they should have a minimal number of equations and formal statements of theorems, lots of diagrams and illustrations, and no citations (just Further Reading at the end of the article).

How did you choose the cover?

NH: We considered many different ideas. But after a lot of thought we settled on the motor boat picture, which captures the important topics of fluid mechanics, waves, and ocean, all of which are covered in the book in a number of articles.

What is the most unexpected article?

NH: Perhaps the article Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter by sociologist of education Heather Mendick. She discusses the way mathematics is represented in numerous TV shows and films.

What would you be doing if you hadn’t become a mathematician?

NH: I’d be playing a Hammond B3 organ in a jazz or blues band. I’m a keen musician and played keyboards semi-professionally for many years, starting in my teens.

How did you go about organizing the book?

NH: I recruited five Associate Editors with expertise in different areas and we met and planned out the eight parts of the book and the articles, along with a list of authors to invite. We looked for authors who are leading international experts in their field and are at the same time excellent expositors. Signing up the 165 authors was quite a long process. We were able to find authors for almost every article, so just a very small number had to be dropped. In some cases the authors suggested changes of content or emphasis that we were happy to agree with.

What range of readers is The Companion aimed at?

NH: The target audience for The Companion is very broad. It includes mathematicians at undergraduate level or above, students, researchers, and professionals in other subjects who use mathematics, and mathematically interested lay readers. Some articles will also be accessible to students studying mathematics at pre-university level.

Why not just seek information online? Why is there a need for a book?

NH: When Princeton University Press asked me to edit The Companion they told me that reference books still have great value. Many people have trouble navigating the vast amount of information available online and so the need for carefully curated thematic reference works, written by high calibre authors, is as great as ever. So PUP’s experience is that print is definitely not dead, and indeed my own experience is that I have many books in PDF form on my computer, but if I want to read them seriously I use a hard copy.

How have you ensured that the book will not go out of date quickly?

NH: This was a major consideration. This was a five and a half year project and we wanted to make sure that the book will still be relevant 10, 20, or 50 years from now. To do that we were careful to choose articles on topics that have proven long-term value and are not likely to be of short-term interest. This is not to say that we don’t cover some relatively new, hot topics. For example, there are articles on compressed sensing (recovering sparse, high-dimensional data from a small number of indirect measurements) and on cloaking (hiding an object from an observer who is using electromagnetic, or other, forms of imaging, as in Harry Potter or Romulan space ships in Star Trek), both of which are areas that have grown tremendously in the last decade.

What sort of overview of applied mathematics does the book give?

NH: Applied mathematics is a huge subject, so we cannot cover everything in 1000 pages. We have tried to include the main areas of research as well as key underlying concepts, key equations, function and laws, as well as lots of example of applied mathematics problems. The examples range from the flight of a golf ball, to robotics, to ranking web pages. We also cover the use of applied mathematics in other disciplines such as engineering, biology, computer science, and physics. Indeed the book also has a significant mathematical physics component.

Where is the field going?

NH: Prior to the 20th century, applied mathematics was driven by problems in astronomy and mechanics. In the 20th century physics became the main driver, with other areas such as biology, chemistry, economics, engineering, and medicine also providing many challenging mathematical problems from the 1950s onwards. With the massive and still growing amounts of data available to us in today’s digital society information, in its many guises, will be an increasingly important influence on applied mathematics in the 21st century.

To what extent does The Companion discuss the history of applied mathematics?

NH: We have an excellent 25-page article in Part I titled The History of Applied Mathematics by historians of mathematics June Barrow-Green and Reinhard Siegmund-Schultze. Many articles contain historical information and anecdotes. So while The Companion looks to the future it also gives an appreciation of the history of the subject.

How do you see the connections between applied mathematics and other disciplines developing?

NH: Applied mathematics is becoming ever more interdisciplinary. Many articles in The Companion illustrate this. For example,

  • various facets of imaging feature in several articles, including those on compressed sensing, medical imaging, radar, and airport baggage screening,
  • the article on max-plus algebras shows how what may seem like an esoteric area of pure mathematics has applications to all kinds of scheduling processes,
  • the article on the spread of infectious diseases shows the value of mathematical models in epidemiology,
  • several articles show how mathematics can be used to help understand the earth’s weather and climate, focusing on topics such as weather prediction, tsunamis, and sea ice.

What are you thoughts on the role of computing in applied mathematics?

NH: Computation has been a growing aspect of applied mathematics ever since the first stored program computer was invented here in Manchester. More and more it is the case that numerical computations and simulations are used in tandem with, or even in place of, the classical analysis that relies just on pen and paper. What I find particularly interesting is that while the needs of mathematics and of science in general have, naturally, influenced the development of computers and programming languages, there have been influences in the other direction. For example, the notation for the ceiling and floor functions that map a real number to the next larger or smaller integer, respectively, was first introduced in the programming language APL.

Of course numerical computations are expressed in terms of algorithms, and algorithms are ubiquitous in applied mathematics, and strongly represented in the book.

Do you have any views on ensuring the correctness of work in applied mathematics?

NH: As the problems we solve become every more complicated, and the computations we perform run for longer and longer, questions about the correctness of our results become more important. Applied mathematicians have always been good at estimating answers, perhaps by an asymptotic analysis, so we usually know roughly what the answer should look like and we may be able to spot gross errors. Several particular aspects of treating correctness are covered in The Companion.

Uncertainty quantification is about understanding how uncertainties in the data of a problem affect the solution. It’s particularly important because often we don’t know the problem data exactly—for example, in analyzing groundwater flow we don’t know the exact structure of what lies under the ground and so have to make statistical assumptions, and we want to know how these impact the computed flows.

A different aspect of correctness concerns the reproducibility of our computations and treats issues such as whether another scientist can reproduce our results and whether a computation on a high-performance computer will produce exactly the same answer when the computation is repeated.

All of these issues are covered in multiple articles in the book.

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. Mark R. Dennis is professor of theoretical physics at the University of Bristol. Paul Glendinning is professor of applied mathematics at The University of Manchester. Paul A. Martin is professor of applied mathematics at the Colorado School of Mines. Fadil Santosa is professor of mathematics at the University of Minnesota. Jared Tanner is professor of the mathematics of information at the University of Oxford.

Patterns are math we love to look at

This piece by Frank Farris was originally published on The Conversation.

Frank A Farris, Santa Clara University

Why do humans love to look at patterns? I can only guess, but I’ve written a whole book about new mathematical ways to make them. In Creating Symmetry, The Artful Mathematics of Wallpaper Patterns, I include a comprehensive set of recipes for turning photographs into patterns. The official definition of “pattern” is cumbersome; but you can think of a pattern as an image that repeats in some way, perhaps when we rotate, perhaps when we jump one unit along.

Here’s a pattern I made, using the logo of The Conversation, along with some strawberries and a lemon:

Repeating forever left and right.
Frank A Farris, CC BY-ND

Mathematicians call this a frieze pattern because it repeats over and over again left and right. Your mind leads you to believe that this pattern repeats indefinitely in either direction; somehow you know how to continue the pattern beyond the frame. You also can see that the pattern along the bottom of the image is the same as the pattern along the top, only flipped and slid over a bit.

When we can do something to a pattern that leaves it unchanged, we call that a symmetry of the pattern. So sliding this pattern sideways just the right amount – let’s call that translation by one unit – is a symmetry of my pattern. The flip-and-slide motion is called a glide reflection, so we say the above pattern has glide symmetry.

A row of A’s has multiple symmetries.
Frank A Farris, CC BY-ND

You can make frieze patterns from rows of letters, as long as you can imagine that the row continues indefinitely left and right. I’ll indicate that idea by …AAAAA…. This row of letters definitely has what we call translational symmetry, since we can slide along the row, one A at a time, and wind up with the same pattern.

What other symmetries does it have? If you use a different font for your A’s, that could mess up the symmetry, but if the legs of the letter A are the same, as above, then this row has reflection symmetry about a vertical axis drawn through the center of each A.

Now here’s where some interesting mathematics comes in: did you notice the reflection axis between the As? It turns out that every frieze pattern with one vertical mirror axis, and hence an infinite row of them (by the translational symmetry shared by all friezes), must necessarily have an additional set of vertical mirror axes exactly halfway between the others. And the mathematical explanation is not too hard.

Suppose a pattern stays the same when you flip it about a mirror axis. And suppose the same pattern is preserved if you slide it one unit to the right. If doing the first motion leaves the pattern alone and doing the second motion also leaves the pattern alone, then doing first one and then the other leaves the pattern alone.

Flipping and then sliding is the same as one big flip.
Frank A Farris, CC BY-ND

You can act this out with your hand: put your right hand face down on a table with the mirror axis through your middle finger. First flip your hand over (the mirror symmetry), then slide it one unit to the right (the translation). Observe that this is exactly the same motion as flipping your hand about an axis half a unit from the first.

That proves it! No one can create a pattern with translational symmetry and mirrors without also creating those intermediate mirror symmetries. This is the essence of the mathematical concept of group: if a pattern has some symmetries, then it must have all the others that arise from combining those.

The surprising thing is that there are only a few different types of frieze symmetry. When I talk about types, I mean that a row of A’s has the same type as a row of V’s. (Look for those intermediate mirror axes!) Mathematicians say that the two groups of symmetries are isomorphic, meaning of the same form.

It turns out there are exactly seven different frieze groups. Surprised? You can probably figure out what they are, with some help. Let me explain how to name them, according to the International Union of Crystallographers.

The naming symbol uses the template prvh, where the p is just a placeholder, the r denotes rotational symmetry (think of a row of N’s), the v marks vertical qualities and the h is for horizontal. The name for the pattern of A’s is p1m1: no rotation, vertical mirror, no horizontal feature beyond translation. They use 1 as a placeholder when that particular kind of symmetry does not occur in the pattern.

What do I mean by horizontal stuff? My introductory frieze was p11g, because there’s glide symmetry in the horizontal directions and no symmetry in the other slots.

Another frieze pattern, this one based on a photo of a persimmon.
Frank A Farris, CC BY-ND

Write down a bunch of rows of letters and see what types of symmetry you can name. Hint: the persimmon pattern above (or that row of N’s) would be named p211. There can’t be a p1g1 because we insist that our frieze has translational symmetry in the horizontal direction. There can’t be a p1mg because if you have the m in the vertical direction and a g in the horizontal, you’re forced (not by me, but by the nature of reality) to have rotational symmetry, which lands you in p2mg.

A p2mg pattern based on some of the same raw materials as our first frieze pattern.
CC BY-ND

It’s hard to make p2mg patterns with letters, so here’s one made from the same lemon and strawberries. I left out the logo, as the words became too distorted. Look for the horizontal glides, vertical mirrors, and centers of twofold rotational symmetry. (Here’s a funny feature: the smiling strawberry faces turn sad when you see them upside down.)

One consequence of the limitation on wallpaper groups is that honeybees cannot make combs with fivefold symmetry.
LHG Creative Photography, CC BY-NC-ND

In my book, I focus more on wallpaper patterns: those that repeat forever along two different axes. I explain how to use mathematical formulas called complex wave forms to construct wallpaper patterns. I prove that every wallpaper group is isomorphic – a mathematical concept meaning of the same form – to one of only 17 prototype groups. Since pattern types limit the possible structures of crystals and even atoms, all results of this type say something deep about the nature of reality.

Ancient Roman mosaic floor in Carranque, Spain.
a_marga, CC BY-SA

Whatever the adaptive reasons for our human love for patterns, we have been making them for a long time. Every decorative tradition includes the same limited set of pattern types, though sometimes there are cultural reasons for breaking symmetry or omitting certain types. Did our visual love for recognizing that “Yes, this is the same as that!” originally have a useful root, perhaps evolving from an advantage in distinguishing edible from poisonous plants, for instance? Or do we just like them? Whyever it is, we still get pleasure from these repetitive patterns tens of thousands of years later.

Frank A Farris, Associate Professor of Mathematics, Santa Clara University. He is the author of Creating Symmetry.

This article was originally published on The Conversation. Read the original article.

 

The Conversation