## 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 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

where 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.

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

(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).

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.)

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).

## 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.

## Bird Fact Friday – Where do penguins live?

From page 16 of Penguins:

A popular misconception is that all penguins live around the poles. Penguins are actually constrained to the southern hemisphere, but only four species of 18 (or 19, depending on the taxonomy used) form colonies along parts of the Antarctic coastline, remaining at least 1200 km (745 miles) from the South Pole. The entire ‘crested dynasty’ (seven species), live in slightly milder climates, mostly north of the Polar Front, nesting on subantarctic islands. Still others make their homes in Australia, New Zealand, and the Galapagos Islands. Many penguins live in areas so remote that they are rarely observed or photographed.

Penguins: The Ultimate Guide
Tui De Roy, Mark Jones & Julie Cornthwaite

Penguins are perhaps the most beloved birds. On land, their behavior appears so humorous and expressive that we can be excused for attributing to them moods and foibles similar to our own. Few realize how complex and mysterious their private lives truly are, as most of their existence takes place far from our prying eyes, hidden beneath the ocean waves. This stunningly illustrated book provides a unique look at these extraordinary creatures and the cutting-edge science that is helping us to better understand them. Featuring more than 400 breathtaking photos, this is the ultimate guide to all 18 species of penguins, including those with retiring personalities or nocturnal habits that tend to be overlooked and rarely photographed.

A book that no bird enthusiast or armchair naturalist should do without, Penguins includes discussions of penguin conservation, informative species profiles, fascinating penguin facts, and tips on where to see penguins in the wild.

• Covers all 18 species of the world’s penguins
• Features more than 400 photos
• Explores the latest science on penguins and their conservation
• Includes informative species profiles and fascinating penguin facts

## New Brain & Cognitive Science Catalog

We invite you to scroll through the 2016 Brain & Cognitive Science catalog below.

Unable to display PDF

Don’t miss Phishing for Phools! Nobel Prize-winning economists George A. Akerlof and Robert J. Shiller challenge the traditional idea that the free market is inherently benign and show us through numerous stories how sellers can manipulate and deceive us.

In Thrive, Richard Layard and David M. Clark argue that the economic and social advantages of investing in modern psychological therapies more than make up for the cost, and that we cannot afford to ignore an issue that affects at least 20% of people in developed countries.

The Future of the Brain, edited by Gary Marcus and Jeremy Freeman, is a collection of essays that explore the exciting advances that will allow us to understand the brain as we never have before.

Joseph Henrich makes the case that our success as a species can be attributed to our ability to socially connect with each other and benefit from a collective intelligence in The Secret of Our Success.

For these and many more titles in cognitive science, see our catalog above! And be sure to subscribe to our newsletter to get 30% off on select titles through November 13, 2015.

Finally, if you’ll be attending the Society for Neuroscience Annual Meeting in Chicago from October 17-21, visit us at booth 126. You can also join the conversation on Twitter using #SfN15!

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

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.

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.

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

Tonight, 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
USA

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

As 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.

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 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.

## 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:

Image by Bob Peterson

Image by Jaco Visser

Image by Rick Avis

Image by B. Seth Topham

Image by Jillian H. Cowles

Image by Jillian H. Cowles

Image by USDA Bee Biology and Systematics Laboratory

Image by Jillian H. Cowles

## 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

Here 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.

## Patterns are math we love to look at

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

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.

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.

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.

## Bird Fact Friday – Falcons’ need for speed

From page 85 of Hawks from Every Angle:

While in direct pursuit of small birds, their main prey, falcons may reach speeds of more than 80 miles per hour. The Peregrine Falcon can exceed 200 miles per hour in a steep dive!

Hawks from Every Angle: How to Identify Raptors in Flight
Jerry Liguori
Foreword by David A. Sibley

Identifying hawks in flight is a tricky business. Across North America, tens of thousands of people gather every spring and fall at more than one thousand known hawk migration sites–from New Jersey’s Cape May to California’s Golden Gate. Yet, as many discover, a standard field guide, with its emphasis on plumage, is often of little help in identifying those raptors soaring, gliding, or flapping far, far away.

Hawks from Every Angle takes hawk identification to new heights. It offers a fresh approach that literally looks at the birds from every angle, compares and contrasts deceptively similar species, and provides the pictures (and words) needed for identification in the field. Jerry Liguori pinpoints innovative, field-tested identification traits for each species from the various angles that they are seen.

Featuring 339 striking color photos on 68 color plates and 32 black & white photos, Hawks from Every Angle is unique in presenting a host of meticulously crafted pictures for each of the 19 species it covers in detail–the species most common to migration sites throughout the United States and Canada. All aspects of raptor identification are discussed, including plumage, shape, and flight style traits.

For all birders who follow hawk migration and have found themselves wondering if the raptor in the sky matches the one in the guide, Hawks from Every Angle—distilling an expert’s years of experience for the first time into a comprehensive array of truly useful photos and other pointers for each species–is quite simply a must.

Key Features:

• The essential new approach to identifying hawks in flight
• Innovative, accurate, and field-tested identification traits for each species
• 339 color photos on 68 color plates, 32 black & white photos
• Compares and contrasts species easily confused with one another, and provides the pictures (and words) needed for identification in the field
• Covers in detail 19 species common to migration sites throughout the North America
• Discusses light conditions, how molt can alter the shape of a bird, aberrant plumages, and migration seasons and sites
• User-friendly format

## Nicholas Higham on Mathematics in Color

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, which is out this month. A slightly longer version of this post on color in mathematics can be found on Higham’s blog, and it has been cross posted at John Cook’s blog, The Endeavour. —PUP Blog Editor

Color is a fascinating subject. Important early contributions to our understanding of it came from physicists and mathematicians such as Newton, Young, Grassmann, Maxwell, and Helmholtz. Today, the science of color measurement and description is well established and we rely on it in our daily lives, from when we view images on a computer screen to when we order paint, wallpaper, or a car, of a specified color.

For practical purposes color space, as perceived by humans, is three-dimensional, because our retinas have three different types of cones, which have peak sensitivities at wavelengths corresponding roughly to red, green, and blue. It’s therefore possible to use linear algebra in three dimensions to analyze various aspects of color.

## Metamerism

A good example of the use of linear algebra is to understand metamerism, which is the phenomenon whereby two objects can appear to have the same color but are actually giving off light having different spectral decompositions. This is something we are usually unaware of, but it is welcome in that color output systems (such as televisions and computer monitors) rely on it.

Mathematically, the response of the cones on the retina to light can be modeled as a matrix-vector product $Af$, where $A$ is a 3-by-$n$ matrix and $f$ is an $n$-vector that contains samples of the spectral distribution of the light hitting the retina. The parameter $n$ is a discretization parameter that is typically about 80 in practice. Metamerism corresponds to the fact that $Af_1 = Af_2$ is possible for different vectors $f_1$ and $f_2$. This equation is equivalent to saying that $Ag = 0$ for a nonzero vector $g =f_1-f_2$, or, in other words, that a matrix with fewer rows than columns has a nontrivial null space.

Metamerism is not always welcome. If you have ever printed your photographs on an inkjet printer you may have observed that a print that looked fine when viewed indoors under tungsten lighting can have a color cast when viewed in daylight.

## LAB Space: Separating Color from Luminosity

In digital imaging the term channel refers to the grayscale image representing the values of the pixels in one of the coordinates, most often R, G, or B (for red, green, and blue) in an RGB image. It is sometimes said that an image has ten channels. The number ten is arrived at by combining coordinates from the representation of an image in three different color spaces. RGB supplies three channels, a space called LAB (pronounced “ell-A-B”) provides another three channels, and the last four channels are from CMYK (cyan, magenta, yellow, black), the color space in which all printing is done.

LAB is a rather esoteric color space that separates luminosity (or lightness, the L coordinate) from color (the A and B coordinates). In recent years photographers have realized that LAB can be very useful for image manipulations, allowing certain things to be done much more easily than in RGB. This usage is an example of a technique used all the time by mathematicians: if we can’t solve a problem in a given form then we transform it into another representation of the problem that we can solve.

As an example of the power of LAB space, consider this image of aeroplanes at Schiphol airport.

Original image.

Suppose that KLM are considering changing their livery from blue to pink. How can the image be edited to illustrate how the new livery would look? “Painting in” the new color over the old using the brush tool in image editing software would be a painstaking task (note the many windows to paint around and the darker blue in the shadow area under the tail). The next image was produced in
just a few seconds.

Image converted to LAB space and A channel flipped.

How was it done? The image was converted from RGB to LAB space (which is a nonlinear transformation) and then the coordinates of the A channel were replaced by their negatives. Why did this work? The A channel represents color on a green–magenta axis (and the B channel on a blue–yellow axis). Apart from the blue fuselage, most pixels have a small A component, so reversing the sign of this component doesn’t make much difference to them. But for the blue, which has a negative A component, this flipping of the A channel adds just enough magenta to make the planes pink.

You may recall from earlier this year the infamous photo of a dress that generated a huge amount of interest on the web because some viewers perceived the dress as being blue and black while others saw it as white and gold. A recent paper What Can We Learn from a Dress with Ambiguous Colors? analyzes both the photo and the original dress using LAB coordinates. One reason for using LAB in this context is its device independence, which contrasts with RGB, for which the coordinates have no universally agreed meaning.

## The Princeton Companion to Applied Mathematics

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester, and editor of The Princeton Companion to Applied Mathematics. His article Color Spaces and Digital Imaging in The Princeton Companion to Applied Mathematics gives an introduction to the mathematics of color and the representation and manipulation of digital images. In particular, it emphasizes the role of linear algebra in modeling color and gives more detail on LAB space.