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

150 years ago today, Alice in Wonderland was published

Alice's Adventures in WonderlandJuly 4, 2015 may be about Independence Day in the United States, but in Oxford, it’s about one of the great heroes of fiction, a young girl who followed a white rabbit, met a hookah-smoking caterpillar and asked, “Who are you?” 

In July 1865, 150 years ago, Charles Lutwidge Dodgson, a professor of mathematics and Anglican deacon, published Alice’s Adventures Underground, a story about a little girl who tumbles down a rabbit hole into a world of nonsense, but keeps her wits about her. With this the world was first introduced to Alice (who was inspired by a real child named Alice Liddell) and her pseudonymous creator, Lewis Carroll. To commemorate the anniversary, the rare first edition recently went on display in Oxford. Princeton University Press is honored to publish our own beautiful new edition of Alice’s Adventures in Wonderlandwith rarely seen illustrations by none other than Salvador Dalí.

Of course, Alice doesn’t just have a whimsical adventure full of anthropomorphic creatures. She falls into a world that is curiously logical and mathematical. Carroll expert Mark Burstein discusses Dalí’s connections with Carroll, his treatment of the symbolic figure of Alice, and the mathematical nature of Wonderland. In addition, mathematician Thomas Banchoff reflects on the friendship he shared with Dalí and the mathematical undercurrents in Dalí’s work.

Explore chapter one in full here, view the best illustrations over the years on Brain Pickings, or click here for a list of anniversary-related events. If you’re here in New Jersey, Washington Crossing’s Open Air Theater will be performing Alice in Wonderland in the park today at 11 and tomorrow at 4.

Happy birthday, Alice!

Stumped for a last minute gift idea? Try these books

Buying gift or coffee-table books online can be nerve-wracking when all you have to go on is the cover and maybe, if you’re lucky, a couple of sample pages. What will the book really look like? Will it be gift-y enough? We want to take the uncertainty out of the process for you with these videos that show off three of our sumptuous recommended gift books. These books are all available now to complete your last minute holiday shopping.


Penguins: The Ultimate Guide:

The Bee: A Natural History:

Atlas of Cities:


For additional holiday gift recommendations, please click here.

Ai Weiwei exhibition at Blenheim Palace: Our UK publicity assistant investigates!

Visitors can expect to experience something different this autumn at Blenheim Palace. Tradition meets modernity as the 18th century baroque architecture of Blenheim, the birthplace of wartime British Prime Minister Sir Winston Churchill, is host to an exhibition of the artwork of Chinese artist and dissident Ai Weiwei.Ai weiwei sign

This exciting exhibition is especially relevant to Princeton University Press for two reasons: not only is Blenheim Palace a stone’s throw from Princeton University Press’s European office in Woodstock, Oxfordshire, but Princeton University Press published Ai Weiwei’s ‘Little Black Book’, Weiwei-isms, last year.

Weiwei-isms is a collection of quotes demonstrating Ai Weiwei’s thoughts on key aspects of his art, politics and life, carefully selected by Larry Warsh from articles, tweets and interviews.

“Everything is art. Everything is politics.” — Weiwei-isms

Like Weiwei-isms, the exhibition at Blenheim Palace clearly demonstrates Ai Weiwei’s commitment to art as a powerful political statement, as a means of reacting against injustice, and inspiring others to do the same.

Blenheim chandelier“I want people to see their own power.” — Weiwei-isms

This certainly becomes clear as you enter the exhibition. You are given a leaflet which serves as a guide to Ai’s artwork, dispersed throughout the rooms of the palace. Despite this, none of the artwork is signposted and it becomes the visitor’s responsibility to seek it out and take meaning and inspiration from what they see.

The collection brings together pieces created by the artist over the past 30 years. It is especially impressive given that it was curated remotely, Ai Weiwei having been under house arrest since 2011. The old and new are often brought together, with artefacts from the past being reimagined in novel ways. Take, for example, the Han Dynasty vases transformed beyond recognition by car paint or by being ‘rebranded’ with the Coca Cola logo.

Blenheim zodiacHis ‘Circle of Animals/Zodiac Heads’ (2010), previously displayed at a year-long exhibition at Princeton University, is also at Blenheim. This work is an ironic interpretation of the bronze zodiac head statues that were looted from the Emperor’s summer palace (Yuan Ming Yuan) in Beijing in 1860.

Other highlights include ‘He Xie’ (2012), a work comprised of 2,300 porcelain crabs on the floor of the Red Drawing Room (‘He Xie’, meaning ‘river crabs’, puns on the Chinese phrase for ‘harmony’).

While some pieces are the first thing you see when you walk into a room, other pieces are integrated more subtly into the sumptuous interiors of Blenheim Palace. The Wave Plate (2014) is seamlessly integrated into the lavish table decoration as the centrepiece in the Salon, and a pair of handcuffs made of Huali wood (2012) – a reminder of Ai Weiwei’s current situation – placed suggestively on the bed in Churchill’s birth room might escape your attention due to the large number of visitors moving from room to room, all engrossed in the same treasure hunt as you.

Blenheim crabsAll in all, the collaboration between Blenheim Palace and Ai Weiwei really does merit a visit. Ai Weiwei’s work is all the more interesting and thought-provoking for being situated in the context of Blenheim Palace and its grounds.

The exhibition at Blenheim Palace highlights the ‘clash’ of the old and new, which is indeed something that is key to much of Ai Weiwei’s work.

“If a nation cannot face its past, it has no future.” — Weiwei-isms

In years to come, the Ai Weiwei exhibition at Blenheim Palace is sure to become part of the artist’s legacy and a poignant reminder of his struggle for justice and truth.

“The art always wins. Anything can happen to me, but the art will stay.” — Weiwei-isms

The exhibition runs until 14th December.

Princeton University Press and Places Journal Launch Places Books

Princeton, NJ, October 8, 2014 – Princeton University Press and Places Journal are excited to announce a new series: Places Books. The series will present smart, lively, peer-reviewed titles on architecture, landscape, and urbanism that are characterized by strong narrative, provocative argument, and engaging prose. Featuring the work of emerging and established scholars alike, Places Books will offer readers a range of the best contemporary writing on the built environment.

Places Books

Interested readers can sign up for a newsletter to learn more about forthcoming books in the series.

Edited by Nancy Levinson and Josh Wallaert and published by Princeton University Press, the books will be developed from Places articles and expanded into compact and accessible paperbacks and e-books with the aim of inciting dialogue across disciplines. According to Nancy Levinson, Editor and Executive Director of Places Journal, “We are thrilled to be collaborating with Princeton University Press. Places Books is an exciting opportunity to bring the very best public scholarship in design to a wider readership.”

The collaboration was conceived as an alternative to lengthy and heavily illustrated scholarly studies in art, architecture, and urbanism. Though the volumes will feature sophisticated design, lavish production values will be set aside to ensure that Places Books are affordable for a wide range of readers. The subjects of the series will be more timely and topical than authors would take on in traditional monographic projects, but investigated at greater length than in journal articles.

Places Books will launch with two titles. Where are the Women Architects?, by architectural historian Despina Stratigakos, will be an insightful exploration of why women have historically been underrepresented in architecture and what’s being done to rectify the imbalance. D.J. Waldie’s The Poetics of Suburbia will use photography and text to establish a new vocabulary for how suburban spaces are discussed, represented, and experienced. According to Michelle Komie, Executive Editor for Art and Architecture at Princeton University Press, “We want Places Books to influence a wider cultural conversation. Our goal is large: to reinvigorate the tradition of the public intellectual in architecture and urbanism.”


About Places Journal

Places is a leading journal of contemporary architecture, landscape, and urbanism, dedicated to harnessing the moral and investigative power of ambitious public scholarship to promote equitable cities and sustainable landscapes. Founded in 1983 by faculty at the Massachusetts Institute of Technology and the University of California, Berkeley, Places was a print journal for twenty-five years before moving fully online in 2009. Places is supported by an international network of academic partners as well as institutional and individual donors, whose collective engagement ensures that the journal’s rich and substantial content remains publicly accessible and free of charge.

About Princeton University Press

Princeton University Press is an independent publisher with close connections, both formal and informal, to Princeton University. As such it has overlapping responsibilities to the University, the academic community, and the reading public. Our fundamental mission is to disseminate scholarship (through print and digital media) both within academia and to society at large.


Julia Haav, Senior Publicist, Princeton University Press

Nancy Levinson, Editor and Executive Director, Places Journal




Book trailer for Atlas of Cities edited by Paul Knox

Princeton University Press senior designer Jason Alejandro created this book trailer for Atlas of Cities edited by Paul Knox. (The catchy song in the background is the aptly named “Weekend in the City” by Silent Partner.)

8-7 Atlas of Cities Atlas of Cities
Edited by Paul Knox


Throwback Thursday #TBT: Erwin Goodenough’s Jewish Symbols in the Greco-Roman Period (1992)

Throwback Thursday: Week 3

Goodenough, Jewish Symbols in the Greco-Roman Period

It’s Thursday again, folks, and you know what that means: time for a Throwback (#TBT)! This week’s #TBT honors Erwin Goodenough’s Jewish Symbols in the Greco-Roman Period (1992), another fundamental text found in the Princeton Legacy Library. Here’s a little bit of information on your favorite relic – both a literal and figurative designation, in this case:

This volume presents the most important portions of Erwin Goodenough’s classic thirteen-volume work, a magisterial attempt to encompass human spiritual history in general through the study of Jewish symbols in particular. Revealing that the Jewish religion of the period was much more varied and complex than the extant Talmudic literature would lead us to believe, Goodenough offered evidence for the existence of a Hellenistic-Jewish mystic mythology far closer to the Qabbalah than to rabbinical Judaism.

David M. Hay of Studia Philonia Annual 1 praises the volume, saying that, “[s]ince [Jacob Neusner’s one-volume abridgement] presents the fruits of Goodenough’s decades-long study of ancient Jewish art, climaxed by his study of the third-century synagogue at Dura-Europas, it is probably the best introduction to Goodenough’s mature thought. Neusner contributes a twenty-nine-page foreword that explains the enduring importance of the entire thirteen-volume work.”

And if we’ve peaked your interest with this book, you can find similar materials over in Mythos: The Princeton/Bollingen Series in World Mythology. We hope you’ve enjoyed this edition of Throwback Thursday (#TBT), and we’ll see you next week!

Untranslatable Tuesdays – Politics


To mark the publication of Dictionary of Untranslatables: A Philosophical Lexicon, we are delighted to share a series of playful graphics by our design team which illustrate some of the most interesting terms from the Dictionary. For week seven in the “Untranslatable Tuesdays” series we present politics, policy (excerpted from the full entry by Philippe Raynaud):

In French, the noun politique refers to two orders of reality that English designates as two different words, “policy,” and “politics.” In one sense, which is that of policy, we speak in French of la politique to designate “an individual’s, a group’s, or a government’s conception, program or action, or the action itself” (Aron, Democracy and Totalitarianism): it is in this sense that we speak of politiques of health or education or of Richelieu’s or Bismarck’s politiques in foreign affairs. In another sense, which translates as the English word “politics,” la politiques designates everything that concerns public debate, competition for access to power, and thus the “domain in which various politiques [in the sense of “policy”] compete or oppose each other” (ibid.). This slight difference between French and English does not generally post insurmountable problems, because the context usually suffices to indicate which meaning of politique should be understood, but in certain cases it is nonetheless difficult to render in French all the nuances conveyed by the English term, or, on the contrary, to avoid contamination between the two notions that English distinguishes so clearly. On the basis of an examination of the uses of the two words in political literature in English, we will hypothesize that their respective semantic fields are not unrelated to the way in which scholarly theories (and academic institutions) conceive what French call la politique.



Untranslatable Tuesdays – Media


To mark the publication of Dictionary of Untranslatables: A Philosophical Lexicon, we are delighted to share a series of playful graphics by our design team which illustrate some of the most interesting terms from the Dictionary. For week six in the “Untranslatable Tuesdays” series we present Media/Medium (of communication):

By the beginning of the twentieth century, the recognition of a family resemblance between the various “implements of intercommunication” meant that they could be compared and contrasted in profitable new ways. . . . The term “mass media” found its niche in scholarly articles by such influential American midcentury thinkers as Hadley Cantril, Harold Lasswell, and Paul Lazarsfeld. But European philosophers resisted this tendency. . . . For Sartre, Adorno, and their contemporaries, “mass media” was less an untranslatable than an untouchable sullied by intellectual and institutional associations with American cultural imperialism. . . . This resistance was soon exhausted. . . . Cognates like “multimedia,” “remediation,” and “mediality” proliferate globally. This reflects less the dominance of English than the collective urgency of an intellectual project. (Ben Kafka)


Untranslatable Tuesdays – Gender

Cassin gender image

To mark the publication of Dictionary of Untranslatables: A Philosophical Lexicon, we are delighted to share a series of playful graphics by our design team which illustrate some of the most interesting terms from the Dictionary. For week five in the “Untranslatable Tuesdays” series we present Gender:

FRENCH différence des sexes, identité sexuelle, genre

GERMAN Geschlecht

ITALIAN genere

SPANISH género

Untranslatable Tuesdays – Work


To mark the publication of Dictionary of Untranslatables: A Philosophical Lexicon, we are delighted to share a series of playful graphics by our design team which illustrate some of the most interesting terms from the Dictionary. For  the fourth in the “Untranslatable Tuesdays” series we present Work, with an abridged entry by Pascal David:

FRENCH       travail, oeuvre

GERMAN     Arbeit, Werk

GREEK       ponos, ergon

LATIN         labor, opus

The human activity that falls under the category of “work,” at least in some of its uses, is linked to pain (the French word travail derives from the Latin word for an instrument of torture), to labor (Lat. labor [the load], Eng. “labor”), and to accomplishment, to the notion of putting to work (Gr. ergasomai [ἐϱγάζομαι], Lat. opus, Fr. mise en oeuvre, Eng. “work,” Ger. Werk), which is not necessarily the oppo­site of leisure but can be its partner. With Hegel, work (Ger. Arbeit) becomes a philosophical concept, but it designates self-realization (whether the course of history or the life of God) rather than a reality that is exclusively or even primarily anthropological.

What does work mean to you?

Untranslatable Tuesdays – Polis

Cassin polis graphic

To mark the publication of Dictionary of Untranslatables: A Philosophical Lexicon, we are delighted to share a series of playful graphics by our design team which illustrate some of the most interesting terms from the Dictionary. For  week three in the “Untranslatable Tuesdays” series we present Polis (Greek):


ENGLISH               city-state, state, society, nation

FRENCH                 cité, État, société, nation

What’s your favourite untranslatable word?