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

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 – 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 – Kitsch


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. This second week in the “Untranslatable Tuesdays” series we present Kitsch (German):

ENGLISH      junk art, garish art, kitsch

The word Kitsch is German in origin and had previously been translated into French as art de pacotille (junk art) or art tape-á-l’oeil (garish art), but the original term has now become firmly established in all European languages. Used as an adjective, kitsch or kitschy qualifies cultural products intended for the masses and appreciated by them….As a kind of debased popularization, it offers a decadent model that is all the more alluring for being so easily accessible. This is, at least, what its detractors say.

Series Design Series, part 1 – The Jung Paperbacks

In part 1 of the new Series Design Series of blog posts, we speak with Maria Lindenfeldar, Art Director for Princeton University Press, about the series design for The Jung Paperbacks. With this series, PUP has undertaken to repackage our extensive backlist of Jung books and present them as what they are — a significant and cohesive portion of our publishing program. Unlike a “new” series, these books all existed with in different forms for years before they were re-purposed in this series — a distinction that differentiates this design initiative from other series. Scroll down below the image for Maria’s thoughts on the design process and the unique challenges posed by this series.


Image Map

View this image as a larger PDF: The Jung Paperbacks (pdf)

The Design Overview: This was a repackaging project. As the organizing motif, we used mandalas created by one of Jung’s patients, a repeating circle theme and almost identical typography.  For “Introduction to Jungian Psychology,” we kept most elements consistent, replacing the mandala with an engaging portrait of Jung and reducing the palette to black and gold.

Q: When you approach a project like this, does the original book cover from the earlier edition sway the design at all?

Maria: In this case, the original covers influenced us very little. The books had come from various spin-off series we had done, and only a few of them looked similar to one another. From the outset, we knew that we wanted the new editions to look more like a set. Because we came up with the mandala idea fairly early in the process, we did not investigate using any of the art on the previous covers.

This is not always the case with redesigns. In some instances, the original design provides a direct inspiration for the new project. I’m working on some series designs right now, and my first instinct is to dig into our own past for a touchstone.

Q: The mandalas are beautiful and colorful, but also provocative with snakes slithering around and through the patterns. How did you come on the idea to include mandalas and how many mandalas did you have to choose from? Did you give some thought to matching mandala to subject?

Maria: The first time that we spoke about the project, I suggested the mandalas to Kathleen Lynch (the fantastic designer we used for this series). I had seen several of them reproduced in a color insert of one of our previously published volumes. We were able to get further examples from the Jung foundation. Kathleen narrowed the choices and paired the images with the titles. She and I didn’t discuss matching mandalas to particular books, but Kathleen is very much a “thinking designer,” and I wouldn’t be surprised if there were deliberate choices made.

Q: Each cover features three circles, intersecting – one the mandala, one the title of the book, and one the cover. The impression I get is that title, author, and mandala must be looked at for their individual parts, but also as a single, combined graphic. Was this a deliberate choice or just a happy accident?

Maria: A deliberate choice. It’s part of the gestalt that we developed for the series design, and it’s why the design works so well. All of the elements click into place.

Q: Why did you decide to modify the design for Introduction to Jungian Psychology?

Maria: We always thought of “The Introduction” as the mother ship with the other books as its satellites. We wanted there to be a very strong family resemblance, but we didn’t want the hierarchy to be flat. By altering the color scheme and replacing the mandala with the portrait, we hoped to say, “This is related but not identical.” Also, by using black and gold, we hoped to suggest that it was more elemental or foundational than the others.


Explore the mandalas used on these covers by clicking on any of the thumbnails below or above.