In January, we will publish Beautiful Geometry by Eli Maor and with illustrations by Eugen Jost. The book is equal parts beauty and mathematics and we were grateful that both authors took time to answer some questions for our readers. We hope you enjoy this interview.
Look for a slideshow previewing the art from the book in the New Year.
PUP: How did this book come to be? Where did you get the idea to create a book of Beautiful Geometry?
Eli Maor [EM]: It all started some five years ago through a mutual acquaintance of us by the name Reny Montandon, who made me aware of Eugen’s beautiful geometric artwork. Then, in 2010, I met Eugen in the Swiss town of Aarau, where I was invited to give a talk at their famous Cantonal high school where young Albert Einstein spent two of his happiest years. We instantly bonded, and soon thereafter decided to work together on Beautiful Geometry. To our deep regret, Reny passed away just a few months before we completed the project that he helped to launch. We dearly miss him.
PUP: Eugen, where did your interest in geometric artwork come from? Are you a mathematician or an artist by training?
Eugen Jost [EJ]: I have always loved geometry. For me, mathematics is an endless field in which I can play as an artist and I am also intrigued by questions that arise between geometry and philosophy.
For example, Euclid said: A point is that which has no part, and a line is what has no widths. Which raises questions for me as an artist—Do geometric objects really exist, then? Are the points and lines that I produce really geometric objects? With Adobe Illustrator you can transform points and lines into “non-existing” paths, which brings you deeper into geometry.
But, ultimately, it’s not my first purpose to illustrate mathematics—I just want to play and I’m happy if the onlooker of my pictures starts to play as well.
PUP: How did you decide which equations to include?
EM: It’s not so much about equations as about theorems. Of course, geometry has hundreds, if not thousands of theorems to choose from, so we had to be selective. We didn’t have any particular rules to guide us; sometimes Eugen would choose a particular theorem for which he had some artistic design in mind; in other cases we based our selection on theorems with an interesting history, or just for their simplicity. But we always had the artistic point of view in mind: our goal was to showcase the beauty of geometry and make it known to a wide public.
EJ: I think I’m seldom looking for mathematical topics with which I can make pictures. It’s the other way round: mathematics and geometry comes to me. A medieval town, the soles of your shoes, wheel rims, textile printing, patterns in pine cones: wherever I look I see mathematics and beautiful geometry. Euclid’s books among many others provide me with ideas, too.
Often I develop ideas when I’m walking in the woods with our dog or I’m scribbling in my small black diaries while I’m sitting in trains. At home I transform the sketches into pictures.
PUP: Were there any theorems you didn’t get to include but would have liked to?
EM: Yes, there were many theorems we would have liked to have included, but for practical reasons we decided to limit ourselves to about 50 chapters. That leaves us plenty of subjects for Beautiful Geometry II
PUP: Are there some theorems that simply didn’t lend themselves to artistic depictions?
EJ: In our book you won’t find many three-dimensional objects depicted. In my art I tend to create flat objects (circles, triangles, squares …) on surfaces and three-dimensional objects in space. Therefore we avoided theorems that have to do with space—with few exceptions.
PUP: Can you describe the layout of the book? How is it organized?
EM: We followed a more or less chronological sequence, but occasionally we grouped together subjects that are logically related to one another, so as to make the flow of ideas easier to follow.
PUP: The collaboration between you and Eugen Jost reminds me of a lyricist and musician—how did the two of you work together? Did you write and he created art alongside or did he have art already done and you wrote for it?
EM: Yes, that comparison between a lyricist (in opera we call it librettist) and a musician is very apt. As I mentioned earlier, we didn’t have a rigid guideline to follow; we just played with many ideas and decided which ones to include. We exchanged over a thousand emails between us (yes, Eugen actually counted them!) and often talked on Skype, so this aspect of our collaboration was easy. I can’t imagine having done that twenty years ago…
EJ: Communicating with Eli was big fun. He has so many stories to tell and very few of them are restricted to geometry. In 2012, when we thought our manuscript was finished, Eli and his wife came to Switzerland. For many days we travelled and hiked around lakes, cities and mountains with our manuscripts in our book sacks. We discussed all the chapters at great length. In some chapters, the relationship between Eli’s text and my pictures is very close and the art helps readers understand the text. In others, the connection is looser. Readers are invited to get the idea of a picture more or less independently—sort of like solving a riddle.
PUP: Most of the art in the book are original pieces by Eugen, right? Where did you find the other illustrations?
EM: Most of the artwork is Eugen’s work. He also took excellent photographs of sites with interesting geometric patterns or a historical significance related to our book. I have in mind, for example, his image of the famous headstone on the grave of Jakob Bernoulii in the town of Basel, Switzerland, which has the wrong spiral engraved on it—a linear spiral instead of a logarithmic one!
PUP: That is fascinating and also hints at What Eugen mentioned earlier–math and the beauty of math is hidden in plain sight, all around us. Are there other sites that stand out to you, Eugen?
EJ: Yesterday I went to Zurich. While I was walking through the streets I tried to find answers to your question. Within half an hour I found over a dozen examples of the mathematics that surrounds us:
- The clock face of the church St. Peter is the biggest one in Europe. This type of clock face links our concept of time with ancient Babylonians who invented a time system based on the numbers 12, 24, 12×30, 3600.
- In the Bahnhofstrasse of Zurich there is a sculpture by Max Bill in which many big cuboids form a wonderful ensemble. Max Bill was the outstanding artist of the so called “Zürcher Konkrete”; his oeuvre is full of mathematics.
- I saw a fountain and the jet of water formed wonderful parabolas in the air. Where the water entered the pool, it produced concentric circles.
- There are literally hundreds of ellipses on any street in Zurich, or any other town for that matter. Every wheel you see is an ellipse—unless you look at it at a precise angle of 90 degrees.
- Even under our feet, you can find mathematics. Manhole covers very often have wonderful patterns that you can interpret mathematically.
Mathematics and Beautiful Geometry is everywhere around us—we just have to open our eyes.
PUP: What equation does the artwork on the cover of the book illustrate? Can you give us a quick “reading” of it?
EM: The front cover shows the Sierpinski Triangle, named after Polish mathematician Waclaw Sierpinski (1882-1969). It is a bizarre construction, a triangle-like shape that has zero area but an infinite perimeter. This is but one of many fractal-type patterns that have become popular thanks to the ability to create them with modern computers, often adding dazzling color to make them into exquisite works of art.
PUP: Eugen, what is your process to create a piece of art like this?
EJ: Unfortunately, the Sierpinski Triangle was not my own idea, but I was awe-struck by the idea of a shape with an infinitely little area and an infinite perimeter, so I started to think about how it could be depicted. Like most of the pictures in the book, I created this piece of art on a computer. At the same time, I was immersed in other mathematical ideas like the Menger sponge, the Hilbert curve, the Koch snowflake. Of course Sierpinski himself and countless others must have sketched similar triangles, but that was the challenge for our book: to take theorems and to transform them into independent pieces of art that transcend mere geometric drawings.
PUP: Eli, do you have a favorite piece of art in the book?
EM: Truth be told, every piece of art in our book is my favorite! But if I must choose one, I’ll go for the logarithmic spiral that Eugen realized so beautifully in Plate 34.1; it is named spira mirabilis (“The miraculous spiral”), the name that Bernoulli himself used to describe his favorite curve.
PUP: We asked the mathematician to pick his favorite piece of art, so it is only fair that we should ask the artist to pick his favorite theorem. Do you have one, Eugen?
EJ: Being Eli’s first reader for the last three years has been a joy because he tells history and stories in our book. I like the chapter on the surprising theorem of George Alexander Pick, in part because of the biographical details. Eli describes how Pick bonded with Albert Einstein in Prague—imagine Einstein and Pick playing the violin and the viola together! Sadly, Pick ended his life in the concentration camp at Theresienstadt, but he left behind this wonderful contribution to mathematics.
PUP: Are there any particularly surprising pieces of art in the book that might have a good backstory or illustrate a particularly memorable equation?
EM: Again, this puts me in a difficult position – to choose among so many interesting subjects that are covered in our book. But if I have to pick my choice, I’d go for Plate 26.1, entitled PI = 3. The title refers to what has been called the “Biblical value of PI” and refers to a verse in the Bible (I Kings vii 23) which, if taken literally, would lead to a value of PI = 3. Our plate shows this value surrounded by the famous verse in its original Hebrew.
PUP: Writing a book is a long process filled with countless hours of hard work. Do any moments from this period stand out in particular?
EJ: I remember sitting in a boat on the lake of Thun with Eli and his wife Dalia in the spring of 2012. Eli and I were pondering on the chapter doubling the cube. As I mentioned before, I do not like to draw three-dimensional objects on a flat surface, so I didn’t want to depict a traditional cube. I was playing instead with the unrolling of two cubes, one having the double volume of the other. Eli was not sure this would work, but on the boat we thoroughly discussed it and all of a sudden Eli said, “Eugen, you have sold me on it.” I hadn’t heard that expression before. I then had a queasy conscience because I didn’t know whether I should have been flexible enough to leave my own idea for a better one. Ultimately, the art came out quite well and really illustrates the Delian problem.
A chance encounter gave me a sense of the broad appeal of the book. I was sitting at a table with a highly trained engineer and I told him about our book. I then tried to explain the theorem of Morley: In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. His response was “You don’t suppose that you could solve any statical problem with this, do you?”
PUP: Math is often quite visual, but where did the idea of making it both visual and beautiful come from?
EM: We are not the first, of course, to point out the visual beauty of many geometric theorems or patterns, but usually these gems are depicted in stark, black-and-white designs of lines and curves. Adding colors to these designs – and sometimes a bit of humor and imagination – makes all the difference between strictly mathematical beauty and a true work of art. This is what has really inspired us in writing our book.
PUP: Who do you hope reads Beautiful Geometry?
EM: We aim at a broad audience of students, teachers and instructors at all levels, and above all, laypersons who enjoy the beauty of patterns and are not afraid of a simple math equation here and there. We hope not to disappoint them!