Click the image above to read Story #176 from Bill T. Jones’s Story/Time: The Life of an Idea.
Click the image above to read Story #176 from Bill T. Jones’s Story/Time: The Life of an Idea.
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!
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
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 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 (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
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.
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.
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.
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
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.
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.
Plate 40.2 shows a Star of David-like design made of 21 line parabolas.
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.
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.
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.
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.
Click here to sample selections from the book.
Be among the first to browse and download our new art and architecture catalog!
Of particular interest is T. J. Clark’s Picasso and Truth: From Cubism to Guernica. Was Picasso the artist of the twentieth century? In Picasso and Truth, Clark uses his inimitable skills as art historian and writer to answer this question and reshape our understanding of Picasso’s achievement. Supported by more than 200 images, Clark’s new approach to the central figure of modern art focuses on Picasso after the First World War: his galumphing nudes of the early 1920s, the incandescent Guitar and Mandolin on a Table from 1924, Three Dancers done a year later, the hair-raising Painter and Model from 1927, the monsters and voracious bathers that follow, and finally—summing up but also saying farewell to the age of Cubism—the great mural Guernica.
Also be sure to note Daniel Arasse’s Take a Closer Look. What happens when we look at a painting? What do we think about? What do we imagine? How can we explain, even to ourselves, what we see or think we see? And how can art historians interpret with any seriousness what they observe? In six engaging, short narrative “fictions,” each richly illustrated in color, Arasse, one of the most brilliant art historians of our time, cleverly and gracefully guides readers through a variety of adventures in seeing, from Velázquez to Titian, Bruegel to Tintoretto.
And don’t miss out on Monica Bohm-Duchen’s Art and the Second World War, the first book in English to provide a comprehensive and detailed international overview of the complex and often disturbing relationship between war and the fine arts during this crucial period of modern history. This generously illustrated volume starts by examining the art produced in reaction to the Spanish Civil War (often viewed as “the first battle of World War II”), and then looks at painting, sculpture, prints, and drawing in each of the major combatant nations, including Japan and China. Breathtaking in scope, this scholarly yet accessible publication places wartime art within its broader cultural, political, and military contexts while never losing sight of the power and significance of the individual image and the individual artist.
More of our leading titles in art and architecture can be found in the catalog. You may also sign up with ease to be notified of forthcoming titles at http://press.princeton.edu/subscribe/. Your e-mail address will remain confidential!
If you’re heading to the College Association’s annual meeting in Chicago, IL, February 12th-15th, come visit us at booth 304 and meet our new Executive Editor for Art and Architecture, Michelle Komie. For updates and information on our new and forthcoming titles throughout the meeting, follow #CAA2014 and @PrincetonUnivPress on Twitter. See you there!
When I first approached the editor of The Finch and Pea about possibly reviewing The Unfeathered Bird, he suggested I send three copies and he would ask his colleagues to assist him with a new experiment — a review in three parts. The review(s) have just now published and they were well worth the wait. Calling upon experts in three areas — art; ornithology; and, well, book-reading and curiosity–The Finch and Pea has created a lovely, intertwined reading experience that (fortunately) is also positive about the book being reviewed.
“Daddy, what is this book about?”
“It’s a book about birds. It shows you the insides of birds so we can learn how they work.”
In the “layers” portion of the review (though it really is the curiosity, good-parenting, reading part of the review) Josh Witten describes his 4 year-old catching a first glimpse of The Unfeathered Bird. Subsequent conversations ranged over ostriches at the zoo, penguins, finches, robins, and every other bird a 4-year old might want to discuss. But, as Witten describes:
A book like The Unfeathered Bird is more than pretty pictures and informative prose. It is a resource – a bridge – to knowledge and curiosity. What let’s that hummingbird hover at your feeder? Page 80. How does that vulture find the roadkill? Page X. Our lives are filled with everyday events that make us wonder, “How does that work?”; and we so rarely get the answers. What could be more compelling than those creatures that have mastered the air?
Next up, Michelle Banks approaches the book from an artist’s perspective, which initially makes her a bit skeptical:
I approached this book as a visual artist and a decidedly non-expert reader, and I will admit an initial bias against it. I love color. I was convinced that a coffee-table book of birds drawn without their feathers was like producing a book on ice cream that featured only the cones.
Though after a few days with the book, skepticism is pushed aside:
The cream-colored pages, sepia-tinted pencil drawings, and hand-drawn fonts give the book the look of a timeless classic….The book is full of visual delights. If I had to pick a single image that sums it up, Van Grouw’s rendering of an ostrich skeleton (p 229) is a tour de force, both exquisitely detailed and powerfully dramatic. The Unfeathered Bird is itself a unique specimen. While it’s sure to be treasured by bird-lovers, it has much to offer to readers who don’t know a grebe from a loon.
Lastly, Rebecca Heiss puts her hefty ornithology education credentials to work assessing the avian content of the book — the devil is in the details after all. Early on, Katrina decided to use a rather traditional system to categorize and group birds, a departure that Heiss describes:
Nodding to Linnaeus, the godfather of modern classification systems, van Grouw charges into the meat of her book, pairing species by anatomical features that appear to be common between the species. As it turns out, many of these features actually evolved independently through a process known as convergent evolution. In recent years, we have tended to reject groupings based on morphology in favor of grouping that reflect a species evolutionary history determined by DNA sequence. The old school naturalist in me, celebrates this throwback to the days where morphology was king and features were classified and compared based on functional similarity. Apologies to all my molecularly focused colleagues, but van Grouw’s pairings simply work for a book of this nature. It may be my bias as an organismal biologist, but focusing on functional similarity is the “right” way to organize species when your goal is teach people about the mechanics of birds. It also allows van Grouw to highlight the interesting and confusing aspects of convergent evolution.
To me, the power of this section was represented by the Secretary Bird. An intimidating image of a majestic, tall, and powerful bird, glowering beneath overhanging “eyebrows”, dominates a page while the accompanying text details its unique hunting habits. Those long, powerful legs are not just for show. The Secretary Bird uses them to literally stomp and kick its prey to death. Of course it does. Just look at the picture.
Be among the first to check out our new art and architecture catalog!
Of particular interest is Ai Weiwei’s Weiwei-isms, edited by Larry Warsh. Ai Weiwei is one of China’s most influential and inspiring figures. Artist, architect, curator, and activist, he has been an outspoken critic of the Chinese government’s stance on human rights and democracy. This collection of quotes demonstrates the elegant simplicity of Ai Weiwei’s thoughts on key aspects of his art, politics, and life.
Also be sure to note Robert Geddes’ Fit: An Architect’s Manifesto, a book about architecture and society that seeks to fundamentally change how architects and the public think about the task of design. For further reading on architecture, check out the POINT: Essays on Architecture series, featuring titles such as David Joselit’s trenchant illustrated After Art in which the author describes how art and architecture are being transformed in the age of Google.
We’ll also see you at the College Art Association’s annual conference February 13-16 in New York, NY at booth 107. Random drawings for two signed copies of Weiwei-isms will be held. Stop by for a visit and to enter for a chance to win!
We invite you to be among the first to check out our new literature catalog! http://press.princeton.edu/catalogs/lit13.pdf
Of particular interest is the fourth edition of The Princeton Encyclopedia of Poetry and Poetics, which no reader or writer of poetry will want to be without. Also be sure to note Simon Gikandi’s Slavery and the Culture of Taste, co-winner of the 2011 Modern Language Association’s James Russell Lowell Prize. The catalog also features our Essays in the Arts series including Alexander Nemerov’s stunning Wartime Kiss: Visions of the Moment in the 1940s and Leonard Barkan’s examination of the deliciously ambiguous history of the relationship between words and pictures, Mute Poetry, Speaking Pictures.
Forthcoming titles you’ll want to add to your reading list include the expertly rendered Italo Calvino: Letters, 1941-1985 selected and with an introduction by Michael Wood, Reiner Stach’s riveting Kafka biographies, and Ruth R. Wisse’s fascinating No Joke: Making Jewish Humor.
If you’re interested in hearing more about our literature titles, sign up with ease here: http://press.princeton.edu/subscribe/ Your email address will remain confidential!
We’ll see everyone at the meeting of the Modern Language Association January 3-6 in Boston, MA. Come visit us at booth 508! Be sure to stop by at 4:30 p.m. Friday, January 4th for a celebratory reception with the editors of the fourth edition of The Princeton Encyclopedia of Poetry and Poetics—the most comprehensive and authoritative poetry reference for more than four decades. Wine and cheese will be served!
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