Concepts in Color: Beautiful Geometry by Eli Maor and Eugen Jost

If you’ve ever thought that mathematics and art don’t mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each.

With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important and beautiful branches of mathematics.

We’ve created this slideshow so that you can sample some of the beautiful images in this book, so please enjoy!

Plate 00
Plate 4
Plate 6
Plate 7
Plate 10
Plate 15.1
Plate 16
Plate 17
Plate 18
Plate 19
Plate 20
Plate 21
Plate 22
Plate 23
Plate 24.2
Plate 26.2
Plate 29.1
Plate 29.2
Plate 30
Plate 33
Plate 34.1
Plate 36
Plate 37
Plate 38
Plate 39
Plate 40.2
Plate 44
Plate 45
Plate 47
Plate 48
Plate 49
Plate 50
Plate 51

Beautiful Geometry by Eli Maior and Eugen Jost

"My artistic life revolves around patterns, numbers, and forms. I love to play with them, interpret them, and metamorphose them in endless variations." --Eugen Jost

Figurative Numbers

Plate 4, Figurative Numbers, is a playful meditation on ways of arranging 49 dots in different patterns of color and shape. Some of these arrangements hint at the number relations we mentioned previously, while others are artistic expressions of what a keen eye can discover in an assembly of dots. Note, in particular, the second panel in the top row: it illustrates the fact that the sum of eight identical triangular numbers, plus 1, is always a perfect square.

Pythagorean Metamorphosis

Pythagorean Metamorphosis shows a series of right triangles (in white) whose proportions change from one frame to the next, starting with the extreme case where one side has zero length and then going through several phases until the other side diminishes to zero.

The (3, 4, 5) Triangle and its Four Circles

The (3, 4, 5) Triangle and its Four Circles shows the (3, 4, 5) triangle (in red) with its incircle and three excircles (in blue), for which r = (3+4-5)/2 = 1, r = (5+3-4)/2 = 2, rb = (5+4-3)/2 = 3, and rc = (5+4+3)/2 = 6.

Mean Constructions

Mean Constructions (no pun intended!), is a color-coded guide showing how to construct all three means from two line segments of given lengths (shown in red and blue). The arithmetic, geometric, and harmonic means are colored in green, yellow, and purple, respectively, while all auxiliary elements are in white.

Prime and Prime Again

Plate 15.1, Prime and Prime Again, shows a curious number sequence: start with the top eight-digit number and keep peeling off the last digits one by one, until only 7 is left. For no apparent reason, each number in this sequence is a prime.

0.999... = 1

Celtic Motif 1

Our illustration (Plate 17) shows an intriguing lace pattern winding its way around 11 dots arranged in three rows; it is based on an old Celtic motif.

Seven Circles a Flower Maketh

Parquet

Plate 19, Parquet, seems at first to show a stack of identical cubes, arranged so that each layer is offset with respect to the one below it, forming the illusion of an infinite, three-dimensional staircase structure. But if you look carefully at the cubes, you will notice that each corner is the center of a regular hexagon.

Girasole

Plate 20, Girasole, shows a series of squares, each of which, when adjoined to its predecessor, forms a rectangle. Starting with a black square of unit length, adjoin to it its white twin, and you get a 2x1 rectangle. Adjoin to it the green square, and you get a 3x2 rectangle. Continuing in this manner, you get rectangles whose dimensions are exactly the Fibonacci numbers. The word Girasole ("turning to the sun" in Italian) refers to the presence of these numbers in the spiral arrangement of the seeds of a sunflower - a truly remarkable example of mathematics at work in nature.

The Golden Ratio

Plate 21 showcases a sample of the many occurrences of the golden ratio in art and nature.

Pentagons and Pentagrams

Homage to Carl Friedrich Gauss

Gauss's achievement is immortalized in his German hometown of Brunswick, where a large statue of him is decorated with an ornamental 17-pointed star (Plate 23 is an artistic rendition of the actual star on the pedestal, which has deteriorated over the years); reportedly the mason in charge of the job thought that a 17-sided polygon would look too much like a circle, so he opted for the star instead.

Celtic Motif 2

Plate 24.2 shows a laced pattern of 50 dots, based on an ancient Celtic motif. Note that the entire array can be crisscrossed with a single interlacing thread; compare this with the similar pattern of 11 dots (Plate 17), where two separate threads were necessary to cover the entire array. As we said before, every number has its own personality.

Metamorphosis of a Circle

Plate 26.2, Metamorphosis of a Circle, shows four large panels. The panel on the upper left contains nine smaller frames, each with a square (in blue) and a circular disk (in red) centered on it. As the squares decrease in size, the circles expand, yet the sum of their areas remains constant. In the central frame, the square and circle have the same area, thus offering a computer-generated "solution" to the quadrature problem. In the panel on the lower right, the squares and circles reverse their roles, but the sum of their areas ins till constant. The entire sequence is thus a metamorphosis from square to circle and back.

Reflecting Parabola

Ellipses and Hyperbolas

When you throw two stones into a pond, each will create a disturbance that propagates outward from the point of impact in concentric circles. The two systems of circular waves eventually cross each other and form a pattern of ripples, alternating between crests and troughs. Because this interference pattern depends on the phase difference between the two oncoming waves, the ripples invariably form a system of confocal ellipses and hyperbolas, all sharing the same two foci. In this system, no two ellipses ever cross one another, nor do two hyperbolas, but every ellipse crosses every hyperbola at right angles. The two families form an orthogonal system of curves, as we see in plate 29.2.

3/3=4/4

Euler's e

Plate 33, Euler's e, gives the first 203 decimal places of this famous number - accurate enough for most practical applications, but still short of the exact value, which would require an infinite string of nonrepeating digits. In the margins there are several allusions to events that played a role in the history of e and the person most associated with it, Leonhard Euler: an owl ("Eule" in German); the Episcopal crosier on the flag of Euler's birthplace, the city of Basel; the latitude and longitude of Königsberg (now Kaliningrad in Russia), whose seven bridges inspired Euler to solve a famous problem that marked the birth of graph theory; and an assortment of formulas associated with e

Spira Mirabilis

Epicycloids

Plate 36 shows a five-looped epicycloid (in blue) and a prolate epicycloid (in red) similar to Ptolemy's planetary epicycles. In fact, this latter curve closely resembles the path of Venus against the backdrop of the fixed stars, as seen from Earth. This is due to an 8-year cycle during which Earth, Venus, and the Sun will be aligned almost perfectly five times. Surprisingly, 8 Earth years also coincide with 13 Venusian years, locking the two planets in an 8:13 celestial resonance and giving Fibonacci aficionados one more reason to celebrate!

Nine Points and Ten Lines

Our illustration Nine Points and Ten Lines (plate 37) shows the point-by-point construction of Euler's line, beginning with the three points of defining the triangle (marked in blue). The circumference O, the centroid G, and the orthocenter H are marked in green, red, and orange, respectively, and the Euler line, in yellow. We call this a construction without words, where the points and lines speak for themselves.

Inverted Circles

Steiner's Prism

Plate 39 illustrates several Steiner chains, each comprising five circles that touch an outer circle (alternately colored in blue and orange) and an inner black circle. The central panel shows this chain in its inverted, symmetric "ball-bearing" configuration.

Line Design

Plate 40.2 shows a Star of David-like design made of 21 line parabolas.

Gothic Rose

Plate 44, Gothic Rose, shows a rosette, a common motif on stained glass windows like those one can find at numerous places of worship. The circle at the center illustrates a fourfold rotation and reflection symmetry, while five of the remaining circles exhibit threefold rotation symmetries with or without reflection (if you disregard the inner details in some of them). The circle in the 10-o'clock position has the twofold rotation symmetry of the yin-yang icon.

Symmetry

Pick's Theorem

Plate 47 shows a lattice polygon with 28 grid points (in red) and 185 interior points (in yellow). Pick's formula gives us the area of this polygon as A = 185 + 28/2 - 1 = 198 square units.

Morley's Theorem

Variations on a Snowflake Curve

Plate 49 is an artistic interpretation of Koch's curve, starting at the center with an equilateral triangle and a hexagram (Star of David) design but approaching the actual curve as we move toward the periphery.

Sierpinski's Triangle

The Rationals Are Countable!

In a way, [Cantor] accomplished the vision of William Blake's famous verse in Auguries of Innocence:

To see the world in a grain of sand,
And heaven in a wild flower.
Hold infinitely in the palm of your hand,
And eternity in an hour.

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Click here to sample selections from the book.

Q&A: Eli Maor and Eugen Jost reveal the surprising inspirations and process of Beautiful Geometry

In k10065[1]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!