Archives for September 2015

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.

Facebook YEAR OF BOOKS live Q&A with authors of “Portfolios of the Poor”

Collins jacketPortfolios of the Poor: How the World’s Poor Live on $2 a Day by Daryl Collins, Jonathan Morduch, Stuart Rutherford & Orlanda Ruthven is a recent choice by Mark Zuckerberg for his Year of Books project. An unusual investigation of the staggering problem of global poverty, the authors conducted year-long interviews with impoverished villagers and slum dwellers in Bangladesh, India, and South Africa. This morning the authors are taking part in a live Q&A on the Year of Books Facebook page to share the surprising and systematic methods these families used to survive on an income that is, for many, unimaginably small.

Mark Zuckerberg announced the book’s selection on his personal Facebook page with the following thoughts:

It’s mind-blowing that almost half the world — almost 3 billion people — live on $2.50 a day or less. More than one billion people live on $1 a day or less.

This book explains how these families invest their money to best support themselves.

I hope reading this provides some insight into ways we can all work to support them better as well.

You can follow the discussion here.

Paul Krugman hosting free discussion at Cooper Union with authors of THRIVE

Thrive jacketTonight, Nobel-prize-winning economist Paul Krugman will host a free public discussion at Cooper Union with Richard Layard & David M. Clark, co-authors of Thrive: How Better Mental Health Care Transforms Lives and Saves Money. Richard Layard discussed the book with Tom Keene on Bloomberg Surveillance here, and both authors answered some questions on mental health policy for the PUP blog here.

Mental illness is a leading cause of suffering in the modern world. In sheer numbers, it afflicts at least 20 percent of people in developed countries. It reduces life expectancy as much as smoking does, accounts for nearly half of all disability claims, is behind half of all worker sick days, and affects educational achievement and income. There are effective tools for alleviating mental illness, but most sufferers remain untreated or undertreated. What should be done to change this? In Thrive, Richard Layard and David Clark argue for fresh policy approaches to how we think about and deal with mental illness, and they explore effective solutions to its miseries and injustices.

Richard Layard is one of the world’s leading labor economists and a member of the House of Lords. He is the author of Happiness: Lessons from a New Science, which has been translated into twenty languages.

David M. Clark is professor of psychology at the University of Oxford. Layard and Clark were the main drivers behind the UK’s Improving Access to Psychological Therapies program.

Paul Krugman is an author and economist who teaches at Princeton, the London School of Economics and elsewhere. He won the 2008 Nobel Prize in economics. He is also an Op-Ed columnist for the New York Times.

September 29, 2015 @ 6:30 pm – 9:30 pm
The Great Hall
Foundation Building
7 E 7th St, New York, NY 10003

Please RSVP here.

An interview with Paul Wignall: How life on earth survived mass extinctions

Wignall jacketAs scientists ponder NASA’s recent announcement about the likelihood of water and the possibility of life, or extinct life on Mars, Paul Wignall, professor of palaeoenvironments at the University of Leeds, explores a calamitous period of environmental crisis in Earth’s own history. Wignall has been investigating mass extinctions for more than twenty-five years, a scientific quest that has taken him to dozens of countries around the world. Recently he took the time to answer some questions about his new book, The Worst of Times: How Life on Earth Survived Eighty Million Years of Extinctions.

So why was this the worst of times and what died?

PW: For 80 million years, there was a whole series of mass extinctions; it was the most intense period of catastrophes the world has ever known. These extinctions included the end-Permian mass extinction, the worst disaster of all time. All life on earth was affected, from plankton in the oceans to forests on land. Coral reefs were repeatedly decimated, and land animals, dominated by primitive reptiles and amphibians, lost huge numbers of species.

What was responsible for all of these catastrophes?

PW: There is a giant smoking gun for every one of these mass extinctions: vast fields of lava called flood basalts. The problem is how to link their eruption to extinction. The key is understanding the role of volcanic gas emissions. Some of these gases, such as carbon dioxide, are very familiar to us today, and their climatic effects, especially global warming, seem to have been severe.

Why did these catastrophes stop happening?

PW: This is the $64,000 dollar question at the core of The Worst of Times. It seems to be because of a supercontinent. For 80 million years, all continents were united into a single entity called Pangea. This world was extremely bad at coping with rapid global warming because the usual feedbacks involved in removing gases from the atmosphere were not functioning very well. Since then, Pangea has broken up into the familiar multi-continent world of today, and flood basalt eruptions have not triggered any more mass extinctions.

What were the survivors like?

PW: Very tough and often very successful. It takes a lot to survive the world’s worst disasters, and many of the common plants and animals of today can trace their origin back to this time. For example, mollusks such as clams and snails were around before this worst of times, and their survival marks the start of their dominance in today’s oceans.

Are there any lessons we can apply to modern day environmental worries?

PW: Yes and no. Rapid global warming features in all of the mass extinctions of the past, which should obviously give us cause for concern. On the plus side, we no longer live in a supercontinent world. Flood basalt eruptions of the recent geological past have triggered short-lived phases of warming, but they have not tipped the world over the brink.

Paul Wignall at Otto Fiord at Cape St Andrew.

Paul Wignall conducting field research at Otto Fiord at Cape St Andrew.

Does this have anything to do with the dinosaurs?

PW: Sort of. Dinosaurs first appear towards the end of this series of calamities and to a great extent they owed their success to the elimination of their competitors, which allowed them to flourish and dominate the land for 140 million years. As we know, their reign was brought to an abrupt halt by a giant meteorite strike – a very different catastrophe to the earlier ones.

What would you say to those who want to know how you can claim knowledge of what happened so long ago?

PW: Geologists have a lot of ways to interpret past worlds. The clues lie in rocks, so mass extinction research first requires finding rocks of the right age. Then, once samples have been collected, analysis of fossils tells us the level where the extinctions happened. This level can then be analyzed to find out what the conditions were like. It’s like taking a sample of mud from the bottom of the ocean and then using it reconstruct environmental conditions. However, not everything gets “fossilized” in ocean sediments. For example, it is very hard to work out what past temperatures were like, and ocean acidity levels are even harder to determine. This leaves plenty of scope for debate, and The Worst of Times looks at some of these on-going scientific clashes.

Read chapter 1 here.

The Bees in Your Backyard – a slideshow

Bees are in decline, bringing many to embrace their value and think twice before decimating a hive. Even urban beekeeping has experienced an explosion in popularity. But the sheer number and variations that exist in the species can be confusing for novice (and seasoned) bee enthusiasts alike.

The Bees in Your Backyard by Joseph S. Wilson and Olivia Messinger Carril provides an engaging introduction to the roughly 4,000 different bee species found in the United States and Canada, dispelling common myths about bees while offering essential tips for telling them apart in the field. The authors are bee and wasp experts, and between them they have been studying these often misunderstood pollinators for more than three decades. The book contains over 900 stunningly detailed color photos, a few of which we’re excited to share with you here:

[portfolio_slideshow size=large id=36624 autoplay=false random=false centered=true carousel=false navstyle_graphical navpos=top pagerstyle=thumbs pagerpos=bottom]

Bird Fact Friday – When is the best time to go bird watching in Botswana?

From page 17 of Birds of Botswana:

The best time to bird watch in Botswana is during the wet summer months. In addition to the 595 bird species currently on record, numerous migrants boost the population. Many of these birds breed during the summer and are conspicuous in their nuptial finery. A birder’s paradise!

Birds of Botswana
Peter Hancock & Ingrid Weiersbye
Sample Entry

BotswanaHere is the ultimate field guide to Botswana’s stunningly diverse birdlife. Covering all 597 species recorded to date, Birds of Botswana features more than 1,200 superb color illustrations, detailed species accounts, seasonality and breeding bars, and a color distribution map for each species. Drawing on the latest regional and national data, the book highlights the best birding areas in Botswana, provides helpful tips on where and when to see key species, and depicts special races and morphs specific to Botswana. This is the first birding guide written by a Botswana-based ornithologist and the only one dedicated specifically to Botswana.

Portable and easy to use, Birds of Botswana is the essential travel companion for anyone visiting this remarkable country.

(Stanley) Fish Food for Thought, Part 2: Reflections on Liberal Arts Education

Welcome to Part 2 of PUP’s Stanley Fish series, Fish Food for Thought. All selections are excerpted from Fish’s new book, Think Again.


Fish Food for Thought

Part 2: Reflections on Liberal Arts Education

7.1 Why We Built the Ivory Tower

May 1, 2006

Fish on the difference between the academic and advocacy worlds.

In short, don’t cross the boundary between academic work and partisan advocacy, whether the advocacy is yours or someone else’s. Marx famously said that our job is not to interpret the world, but to change it. In the academy, however, it is exactly the reverse: our job is not to change the world, but to interpret it. (301)

7.4 Devoid of Content

May 31, 2005

Fish on teaching language structure, not content, in the classroom.

Students who take so-called courses in writing . . . are learning how to marshal arguments in ways that will improve their compositional skills. In fact, they will be learning nothing they couldn’t have learned better by sitting around in a dorm room. . . . They will certainly not be learning anything about how language works; and without a knowledge of how language works, they will be unable to either spot the formal breakdown of someone else’s language or to prevent the formal breakdown of their own. (313)

7.6 Will the Humanities Save Us?

January 6, 2008

Fish on the purpose of humanities courses.

To the question, ‘Of what use are the humanities?’, the only honest answer is none whatsoever. And it is an answer that brings honor to its subject. Justification, after all, confers value on an activity from a perspective outside its performance. An activity that cannot be justified is an activity that refuses to regard itself as instrumental to some larger good. The humanities are their own good. There is nothing more to say, and anything that is said . . . diminishes the object of its supposed praise.(323)

7.7 The Uses of the Humanities

January 13, 2008

Fish on why he teaches humanities subjects.

Why do I do it? . . . I don’t do it because it inspires me to do other things, like change my religion or go out and work for the poor. If I had to say, I’d say that I do it because I get something like an athletic satisfaction from the experience of trying to figure out how a remarkable verbal feat has been achieved. . . . [I]t is like solving a puzzle—but the greater satisfaction is the opportunity to marvel at what a few people are able to do with the language we all use. (324325)

7.10 Deep in the Heart of TexasThink Again jacket

June 21, 2010

Fish on recognizing a quality education.

But sometimes (although not always) effective teaching involves the deliberate inducing of confusion, the withholding of clarity, the refusal to provide answers. . . . And sometimes that disappointment, while extremely annoying at the moment, is the sign that you’ve just been the beneficiary of a great course, although you may not realize it for decades. (340)


What does the Bible really say about infertility?

Moss jacket“If fertility is a blessing, then infertility ought to be a curse—so goes the logic of Genesis 1 and the creation story” write Candida Moss and Joel Baden, authors of Reconceiving Infertility, in their recent Daily Beast piece. In the secular view, infertility is a medical condition for which there is logical recourse: fertility treatment, adoption, or the decision to remain childless and pursue other means of fulfillment. But from ancient times to today, fertility through a biblical lens has often appeared as a sign of blessedness and moral uprightness, while infertility has been associated with sin and moral failing.

This week, the pope’s message carries the promise of many things: compassion for immigrants, vigilance about global warming, and redemption for those who have become alienated from the Catholic church because of its stance on divorce and other lifestyle choices. And yet, as Baden and Moss note in The Daily Beast:

Beyond the obvious—faceless corporations, greed, capitalistic exploitation, and so on—there is another group that Francis thinks is selfish: childless couples. In fact, during his tenure Francis has directly described those who choose not to have children as “selfish” and as obsessed with material things. He regularly uses sterility as a pejorative metaphor and fruitfulness as the primary image for that which flourishes. In so doing, he appears unaware of how this language alienates those without children and empowers others to negatively judge them.

Judgement of the childless, rooted as it may be in ancient biblical language, has long been a feature of modern life as well. Infertility carries a lingering stigma, and the decision not to procreate, often seen as a calculated choice, has led many to defend their “childless by choice” lifestyles. Yet according to Baden and Moss, biblical views on procreation and infertility were more diverse than we tend to think, particularly when we take into consideration the ancient contexts from which they emerged:

The good news is that the Bible, one of the primary ideological sources for discrimination against women, is in fact more complicated on the issue of infertility than it at first seems. While biological procreation is a perpetual blessing on God’s people, fertility is not always assumed to be the default human state. Certainly by the New Testament, the biblical “family” was less about biology than about a community drawn together by duty and responsibility. Informal adoption, mentorship as family, and concerns for others as a replacement for biological generation are the norm.

Read the rest of The Daily Beast piece here.

Candida Moss is professor of New Testament and Early Christianity at the University of Notre Dame, and is serving as a papal correspondent for CBS this week. Joel S. Baden is professor of Hebrew Bible at Yale Divinity School.

Behind every meal you eat, there is a story

Louise Fresco, president of Wageningen University and Research Centre in the Netherlands and author of Hamburgers in Paradise, talks about that story here:


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.

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

An interview with Eberhard Faber on “Building the Land of Dreams”

Faber jacketNew Orleans, iconic city of Mardi Gras, gumbo and jazz, was once little more than a sleepy outpost at the edge of Spain’s American empire. By the 1820s, with thriving cotton and sugar industries, the city was well on the way to becoming the urban capital of the antebellum South. Looking the ideological struggle, class politics, and powerful personalities that accompanied its transformation, Building the Land of Dreams is the narrative biography of a fascinating city at the most crucial turning point in its history. Recently, Eberhard Faber took the time to answer some questions about his book.

What inspired you to become a historian?

EF: It took a long time for me to become a professional historian; I was a touring musician for almost fifteen years before going back to school to study history. But I was always fascinated by history. As a kid I remember reading William Shirer’s Rise and Fall of the Third Reich. During my music years I remember reading Robert Caro’s The Power Broker in the back of the tour bus. History is simply a way of looking at the world and trying to understand it. I’ve always looked at the world this way.

As far as how I got interested in New Orleans and the South, it happened early in grad school at Princeton; I wrote a research paper for Linda Colley about the short-lived British colony in West Florida formed in 1763, and all the sources that I read pointed to New Orleans as a crucial strategic point in that era. The next year I wrote a paper for John Murrin about the South in the War of 1812, culminating with the Battle of New Orleans. The year after that I moved down to New Orleans for what was to have been a year of research; we got hooked and live here still.

What was the most interesting thing you learned from writing Building the Land of Dreams?

EF: Well, so many things. But perhaps it would be the biographies of the many people who moved to New Orleans in the months following the Louisiana Purchase, from the northern United States but also from across the Atlantic: England, Ireland, and France especially. In the book I call them the “generation of 1804” because they arrived right after the Louisiana Purchase. They were a varied cohort, and they fought amongst each other a lot, but I found their energy, ambition and idealism very appealing. One common characteristic was that they were all sincere believers in the world-changing possibilities of republican rule. They thought this radical experiment that the United States had only recently embarked upon was going to rewrite everything about human history. In New Orleans they ran into a conservative creole planter class that believed in none of those things, and they had a rude awakening of sorts. It’s a fascinating encounter.

What do you think is the book’s most important contribution?

EF: I think it has been a very common assumption that the United States imposed certain changes on Louisiana after 1803. That the course New Orleans and its wider region took, in the early nineteenth century, was an outcome of the policies of Thomas Jefferson and other American statesmen. What Building the Land of Dreams shows is that there were already very powerful entrenched interests in the area and that they, not the United States, ultimately had the power to dictate outcomes. What Jefferson and Madison could do was actually very limited; while the creole elite, on the other hand, initially threatened by republicanism, figured out that it actually gave them tremendous power to design the regimes – of law, of slavery, and race – that they had long wanted under colonial rule.

What is the biggest misunderstanding people have about what you do?

EF: Well, as far as my work in particular, I don’t think enough people know it for any major misunderstandings to have developed. As for the profession, I think there is a very wide gap between what the general public thinks of as “history” and what historians do in colleges and universities. If you go into a book store, many of the history books will be about military history (and at least half of those will be about the Civil War) and of the rest, a good portion will have Presidents on the cover. This leads many people to believe history is mostly about battles and Presidents, whereas in fact the field is so much bigger than that. The fact that it’s not understood is the fault of the field, of course. We need to do a better job of reaching the public and engaging their interest in historical issues.

What is your next project?

EF: I have two. One is a biography of the lawyer, legal reformer, and politician Edward Livingston, who was Mayor of New York and then fled from a scandal in 1803 to New Orleans. He ended up deeply entangled in New Orleans politics and power struggles and plays a major role in Building the Land of Dreams. I read almost all his personal papers in the course of writing the book and would love to focus on him exclusively for my next project.

The other one is totally different: a history of the music industry in the United States since the invention of the gramophone in the 1890s, with an emphasis on the parallel history of the rise of American capitalism.

What would you have been if not a historian?

EF: Well, that’s easy, since I was a musician (guitar player, songwriter, bandleader, arranger, record producer) for fifteen years before turning to history. I still play actively, too, within the limits imposed by writing and teaching. If the question is what would I have been if I was neither a musician nor a historian – well, my original hope was to be a professional baseball player, but at 5’ 5” that was never entirely realistic.

What are you reading right now?

EF: I’m currently reading Walter Johnson’s River of Dark Dreams. It covers some of the same historical ground as my book – even including the word Dreams in the title! – but from a very different attitude and perspective. I find it alternately infuriating and revelatory. Either way it’s certain to become an important part of the discussion on the antebellum South. Other wonderful books I’ve read lately include Sarah Carr’s brilliant exposé on the New Orleans public school system since Katrina, Hope Against Hope; Robert Gordon’s classic history of Stax Records, Respect Yourself; and Greg Iles’ epic Southern mystery novel Natchez Burning.

What was the most influential book you’ve read?

EF: I always come back to Thomas Mann’s The Magic Mountain, an allegorical novel about Europe before the Great War. Although it’s fiction, it contains great intellectual history, and combines themes that are essentially historical with deep exploration of human psychology. The same is true of War and Peace, which also had a profound effect on me: a meditation on the meaning of history and the sources of historical change, inextricably intertwined with such “interior” issues as the nature of human suffering and the attempt to find meaning in the universe. All historians should read these books. They remind us of the spiritual dimension behind the often dry academic debates that tend to cloud the field.

Meanwhile, in the realm of historical scholarship, I could mention so many – Henry Adams, Schlesinger, Hofstadter, Genovese, Gordon Wood, Rhys Isaac, William Cronon, Alan Taylor, my mentors Linda Colley and Sean Wilentz – but the one book that truly did influence me more deeply and permanently than any other was Richard White’s The Middle Ground. I don’t even think I’m very much like White as a historian, temperamentally and aesthetically; he’s a burrower, while I’m a wanderer; he eschews drama completely, while I am simply incapable of living without it. But the method, the dedication, the integrity, the matching of evidence to ideas, the rigor of the concepts, the sense of change over time, in that book, all of that is just so beautiful to me, and it remains a very distant and unreachable benchmark of sorts.

Describe your writing process. How long did it take you to finish your book? Where do you write?

EF: I like to write in libraries. I really love a nice library: the sensation that you are being enveloped in quiet and ideas and books, and that your work is going to merge into this enormous sea of scholarship that surrounds you. I wrote most of the first draft of Building the Land of Dreams on the C Floor of Princeton’s Firestone Library, 3 floors below ground level, in a tiny carrel. I wrote most of the revised version at the Community Coffee shop at the corner of Jefferson Ave. and Magazine St. in New Orleans (which is appropriate, because developing Thomas Jefferson’s part in the story was one of the biggest changes in the second draft).

As far as method and process, I think research and preparation is really 80% of the task, the actual writing is the final 20%. I spent lots of time on research; I read Edward Livingston’s papers in their entirety (140-some boxes worth), I read the New Orleans Conseil de Ville records in their entirety from 1803 to 1819, I have read pretty much every piece of secondary literature on early Louisiana ever written. It all goes into a big database (although, life being what it is, there’s always lots of stuff that never makes it into the database, too). By the time I actually start writing I have a very good idea of what I am going to say, including exact phrasing in many cases. The phrases have been building up in my head during showers and long drives for the months prior to writing them down. When I actually get going I write fast, and I write a lot. I have to trim a lot, too, eventually. The final version of Building the Land of Dreams is probably about equal in size to the pile of stuff that got left out and discarded along the way.

The whole project took almost seven years from beginning to end – three of which were spent working on the project pretty much full time, and four of which were spent balancing the writing and research with teaching.

Why did you write this book?

EF: I think I wrote the book because I had the very good fortune to have the institutional backing of Princeton University and Loyola University New Orleans; because I have a wonderfully supportive family; because I have a terrific network of colleagues and peers including a handful of close friends in my grad student cohort, my mentors at Princeton and in New Orleans, and the brother/sisterhood of Louisiana historians; in short because I am a very fortunate person in many ways. Good work doesn’t just spring from the genius of the author, but from very particular social circumstances in which the author is embedded. When I switched from a music career to an academic one, I knew I still wanted to be a creative person. I was lucky to find a great topic I could throw myself into and a great network of supporters to help me towards the finish. And lastly, in the final phase of turning this from a “project” into an actual book, I have also benefited greatly from the support and advice of Brigitta van Rheinberg, Quinn Fusting, and everyone else at Princeton University Press.

How did you come up with the title or jacket?

EF: Building the Land of Dreams – well, the phrase “Land of Dreams” comes from two places. One is William Blake’s poem, from the Pickering Manuscript, written around the time of the Louisiana Purchase. Blake never traveled to New Orleans, but the poem suggests the expanded mental universe of possibilities in the midst of the Age of Revolutions – and those world-changing possibilities were very much a part of the mental landscape of early 19th century New Orleans in the years after 1803. The second source, of course, is Spencer Williams’ “Basin Street Blues,” made famous by Louis Armstrong’s 1926 recording, which led to the “Land of Dreams” becoming one of the Crescent City’s many nicknames – and which, in the line about the banks of the Mississippi being “the place where dark and light folks meet,” also speaks to the central place of race in the city’s history and in my understanding of it.

The book’s jacket is a painting by John Boqueta de Woiserie, A View of New Orleans Taken from the Plantation of Marigny. It was painted in 1803, in celebration of the Louisiana Purchase and the American takeover, and it shows an eagle hovering over New Orleans, with a banner in its beak that reads “Under my wings/everything prospers.” It shows the enormous optimism with which some people, at least, greeted the prospect of American rule; the linked faith in personal liberty and material prosperity; and an unironic faith in the American promise that seems, in this cynical era, all too naïve. The book is the story of the various ways that promise was both betrayed and fulfilled.

Eberhard L. Faber teaches history and music industry studies at Loyola University, New Orleans. Previously, he spent twelve years leading the New York-based rock band God Street Wine. He blogs on New Orleans history and other topics at

An interview with Louise Fresco on “Hamburgers in Paradise”

Fresco JacketIn Louise Fresco’s new book, Hamburgers in Paradise, the term “Paradise”, in her own words, is “a metaphor that refers to the landscapes on our planet, the collective Garden of Eden which, according to some views, we have irreversibly destroyed through large scale food production.” It is a view shared by many in a world simultaneously plagued by food shortages and GMO fears. In such a climate, is there room for optimism? Fresco looks at our food situation in all its complexity, taking the stance that there is no one perfect way to produce or consume food, and that balance and trade-offs between different goals are central to any long term solution. You can see her TED talk here, and the English subtitled version to a documentary she made about the food industry here. Recently Fresco took some time to answer some questions about her book.

What’s new in this book?

LF: Human history has been one of continuous scarcity. The abundance of food that has emerged for the majority of the world population in the last decades is so unique that we have not yet learnt to deal with it. We are still scared that there will not be enough, and that we will destroy our environment. Scarcity is our default mode, and that of our bodies, hence our difficulties to balance our diets and to reduce our ecological footprint. Abundance is a triumph of science and trade; it allows us to shed our fears of shortages. But the book argues that we require new ways of thinking, to reign in our needs (for example of meat) while producing food sustainably for all, with new methods (for example through recycling or using algae). The book demonstrates in detail that there is not one perfect way to produce and consume food, but that we always have to balance the trade-offs between different goals, such as large scale production (i.e. low food prices) and biodiversity. What is best depends on our goals and our insight in unintended side effects (we may like to see free roaming chickens but they may be more prone to disease that way).

Can you explain the title Hamburgers in Paradise?

LF: The title refers to a thought experiment: if Eve were alive today, what food would she offer Adam as a temptation? Paradise as a metaphor also refers to the landscapes on our planet, the collective Garden of Eden which, according to some views, we have irreversibly destroyed through large scale food production. These semi-conscious images of a pre-Industrial and idealized past are still guiding many of our reactions to modernization. The hamburger, of course, is the iconic food that symbolizes the rise of the modern middle class, from suburban America to places like Moscow or Mumbai, as well as the critical counterforces: slow food, vegetarian and organic products. The hamburger illustrates also the adaptation to new demands: fat and salt contents have been lowered, information on calories and nutritional values are published, wrappings are made of recycled materials and advertising to children is limited. In the most recent twist of history, the hamburger becomes popular once again in upper class restaurants, dipped in liquid Nitrogen, or in a vegetarian reincarnation.

Food is the source of much confusion today, we hear so many, contradictory stories about what we must or mustn’t eat and why. What is the reason for this confusion?

LF: Food and agriculture are the basis human survival. Food conjures up strong feelings, based on individual memories, strict convictions and long traditions, especially in times of rapid modernization. Many people, even in rural areas, are hardly aware of how food is really produced and how it lands on their plates. Nearly all of us rely on others, often far away, to feed us. Ignorance and dependency make us feel vulnerable and worried about food.

But we can also turn this around: behind every meal there is a story, one that is nearly always fascinating and often complex, but always worth telling. Food connects us with the past and the future.

Is there room for optimism?

LF: With current knowledge, we can feed nine or ten billion people quite easily. This doesn’t mean that there is no world food problem. Even if enough food can be produced this is not easy and more production does not mean food reaches people automatically. The current gap between actual and attainable yields is still enormous. At the same time, agricultural research and innovation continue to be needed to tackle specific problems of animal and plant diseases, poor soils and climate variation. The application of existing knowledge is hampered by poor infrastructure, unavailability of irrigation or fertilizer, dysfunctional markets and policy.

Food shortage is more a matter of distribution than just production. Hunger is caused by poverty, so creating employment is essential. The great improvements in agricultural production since the 1970s have benefitted the urban poor more than the rural poor. These improvements involve higher yields, through better agricultural techniques such as irrigation, leading to lower food prices that benefit those who buy food (those living in cities), while farmers selling foods are at a disadvantage. Today more than 850 million people go hungry and perhaps as many as 2 billion may lack balanced nutrition. Most of the hungry live in areas of civil war or frequent natural disasters, so peace and resettlement are priorities.

There seems to be much concern about Genetically Modified Organisms, is this concern justified?

LF: This is a very complex issue about which it is impossible to generalize and about which there are many misunderstandings. For example, if cows are fed genetically modified soy bean, their milk does not become genetically modified, even if some people fear this. The modified genes and cells do not survive the gut. What the effects and risks are depends very much on what crop or animal, what genes are used for what purpose and where. Certain problems, for example diseases in banana, can only be tackled with biotechnology, a large toolkit which does not necessarily result in GMOs. There are two types of risk, for human and animal health. While we need to continue to monitor the situation, there are no indications that GM crops lead to additional food related risks in human beings or animals. So far, there are no indications of environment effects (such as insect mortality or genes “escaping “), but ecosystems are complex and difficult to monitor. Finally, there is also the issue of intellectual property rights: while a fair reward is needed for the companies developing the biotechnology or GMOs, we must also make sure that farmers and scientists and breeders elsewhere can keep access to varieties or breeds. Here the U.S. and EU legislation and traditions do not coincide.

What about chemical inputs such as fertilizer. Are we not destroying the land?

LF: Plants and animals need food just as we do. These nutrients come nearly exclusively from the soil (and through water, transported from soils elsewhere). Only very few soils can sustain production for long periods and their nutrient reserve needs to be built up through other sources of nutrients. Whilst manure from animals can be used for this, this does not solve the problem, it just means that animals have to graze somewhere from where they take up the nutrients. Almost without exception agriculture requires fertilizer to be sustainable. Fertilizer has a bad name mainly because it has been overused in the past with detrimental effects on surface water, but in itself, if wisely used, it is a blessing. More land is depleted through lack of fertilizer than is affected through its use.

Is fast food the source of all evils?

LF: Fast food is part of a complex process of transformation of society: greater mobility, work pressures, urbanization, diversification through trade, smaller and singe households, greater affluence of young, ubiquitous equipment like microwaves and fridges people all lead to out of home eating and pre-packed meals. As with all foods, it is not the individual item that is “bad” but the pattern. Eating fast food from time to time is acceptable in an otherwise healthy lifestyle. However, fast food often contains too many calories and we should be concerned if there are no other options, In so-called food deserts, neighbourhoods devoid of shops selling vegetables, fast food is often the recourse for single parents.

What type of agriculture is most sustainable?

LF: There is no blueprint for an agricultural model that fits all situations. Agriculture is the art of the location-specific and always depends on soils, climate, geography, culture and economics. Agriculture is forever changing, adapting to new consumer demands and new technology. However, using resources as efficiently as possible is essential to avoid wasting labour, water, land, fertilizer, seeds or animals as well as reducing post harvest losses in the entire value chain. Efficiency is often misunderstood as large-scale and anonymous, but it applies at all scales. No farmer can afford to waste resources, nor can we as humanity. The World will need 50% more calories in 2030.

Would the world be better off if all meat would be prohibited?

LF: No, meat is necessary for certain groups such as pregnant and lactating women, children, the elderly and sick and of course the malnourished. Meat is not only a source of proteins but also of essential nutrients such as iron and certain vitamins (B12). Humans evolved as omnivores; vegetarians, even in India, have always been a small minority. Also, there are areas in the world where nothing else can be produced but grazing land and animals. The growth in demand for meat and fish is expected to increase faster than the growth in population, especially in Asia and Africa.

However, there are major problems associated with meat production: environmental (water, emissions, production and transportation of feed) veterinary public health, human health (diseases associated with high red meat intake and overuse of antibiotics) as well as animal welfare. These can all be solved with adequate research and regulation. Reducing meat consumption through substitution of animal proteins in healthy individuals in affluent societies is part of that.

Louise O. Fresco is president of Wageningen University and Research Centre in the Netherlands. The author of several books, she is a member of the Council of Advisors for the World Food Prize and has worked extensively in developing countries for many years. She lives in Amsterdam.