Presenting the new video trailer for AFFORDABLE HOUSING IN NEW YORK

New York City, as expensive as it is progressive, has long had the need for high-quality affordable housing. Affordable Housing in New York, edited by Nicholas Dagen Bloom and Matthew Gordon Lasner, is a richly illustrated, dynamic portrait of an evolving city and the pioneering efforts to make it livable for lower and middle income residents. The book and its photos by David Schalliol was subject of this fabulous New York Times feature this past Sunday. We’re excited to offer you a peek inside, here:


An exclusive trailer for Alice’s Adventures in Wonderland, featuring illustrations by Salvador Dalí

ALICE WAS BEGINNING TO get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations?”

Thus begins Alice’s Adventures in Wonderland, one of the most beloved classics of children’s literature. Commemorating the 150th anniversary of its publication, this illustrated edition of Alice’s Adventures in Wonderland, edited by Lewis Carroll expert Mark Burstein, features rarely seen illustrations by Salvador Dalí. In the introduction, Burstein discusses Dalí’s connections with Carroll, the nature of wonderland, and his treatment of the towering (though sometimes shrinking) figure of Alice.

Take an exclusive peek inside the curiously mathematical world into which Alice famously falls, here:

Ai Weiwei free to travel overseas

Today The Guardian reported that Ai Weiwei is free to travel overseas once again. One of China’s most prolific artists and controversial figures, his art and social media use has championed free speech and human rights, even as he was banned from leaving China. Weiwei has ties to Princeton, where his Circle of Animals/Zodiac Heads is currently on display at the University’s Scudder Plaza through December 4, 2016. His book, Ai Weiwei-isms, a collection of quotes reflecting his thoughts on key aspects of his art, politics, and life, was published with Princeton University Press in 2012.

From The Guardian:

More than four years after he was banned from leaving his native China, artist Ai Weiwei is free to travel again after Beijing authorities returned his passport.

“When I got it back I felt my heart was at peace,” the artist told the Guardian on Wednesday afternoon, just hours after police handed him back the travel document and informed him he was free to go overseas.

“I feel pleased. This was something that needed to be done,” added Ai, who has long been a vocal critic of China’s leaders. “I was quite frustrated when my right to travel was taken away but now I feel much more positive about my condition.

“I think they should have given it back some time ago – and maybe after so many years they understand me better.”

The artist posted a celebratory Instagram message alongside a photograph of himself posing with the document. “Today I got my passport,” it read.

Ai said his first trip would be to Germany, where his six-year-old son has been living since last year.

Read the rest here.

Weiwei wrote in Ai Weiwei-isms, “Once you’ve tasted freedom, it stays in your heart and no one can take it. Then, you can be more powerful than a whole country.”

Wishing him well in his travels.

Weekly Wanderlust: New York City

Home to Basquiat, micro-apartments, and some of the best rooftop bars and restaurants, New York City is the melting pot of America, a city whose attractions will continue to unfold for as much time as you have to spend there. You may already plan to visit Rockefeller Center and Top of the Rock Observation Deck, with its iconic skating rink and opportunity to peer into NBC Studios; spend a day browsing the renowned Greek, Roman, Egyptian, and Islamic exhibits at The Metropolitan Museum of Art, or get your Alice in Wonderland fix at Alice’s Tea Cup. But be sure to leave plenty of unscheduled time for off-the-beaten-track destinations and neighborhoods as well.

NYC picture

One World Trade Center

Whether your interests lean artsy, sociological, or completely open-ended, you can get a taste of the Big Apple before your visit with books that chronicle everything from the city’s rich past, to the idiosyncratic art scene and hidden neighborhoods.

Basquiat_Notebooks_S15 Brooklyn-born Jean-Michel Basquiat (1960-88) was one of the most important artists of the 1980s. A key figure in the New York art scene, he inventively explored the interplay between words and images throughout his career, first as a member of SAMO, a graffiti group active on the Lower East Side in the late 1970s, and then as a painter acclaimed for his unmistakable Neoexpressionist style. From 1980 to 1987, he filled numerous working notebooks with drawings and handwritten texts. This facsimile edition reproduces the pages of eight of these fascinating and rarely seen notebooks for the first time.
 j8758  Which is more important to New York City’s economy, the gleaming corporate office–or the grungy rock club that launches the best new bands? If you said “office,” think again. In The Warhol Economy, Elizabeth Currid argues that creative industries like fashion, art, and music drive the economy of New York as much as–if not more than–finance, real estate, and law. And these creative industries are fueled by the social life that whirls around the clubs, galleries, music venues, and fashion shows where creative people meet, network, exchange ideas, pass judgments, and set the trends that shape popular culture.
j10396 Once known for slum-like conditions in its immigrant and working-class neighborhoods, New York City’s downtown now features luxury housing, chic boutiques and hotels, and, most notably, a vibrant nightlife culture. While a burgeoning bar scene can be viewed as a positive sign of urban transformation, tensions lurk beneath, reflecting the social conflicts within postindustrial cities. Upscaling Downtown examines the perspectives and actions of disparate social groups who have been affected by or played a role in the nightlife of the Lower East Side, East Village, and Bowery. Using the social world of bars as windows into understanding urban development, Richard Ocejo argues that the gentrifying neighborhoods of postindustrial cities are increasingly influenced by upscale commercial projects, causing significant conflicts for the people involved.
j10060 As a child growing up in Manhattan, William Helmreich played a game with his father called “Last Stop.” They would pick a subway line, ride it to its final destination, and explore the neighborhood. Decades later, his love for exploring the city is as strong as ever. Putting his feet to the test, he decided that the only way to truly understand New York was to walk virtually every block of all five boroughs—an astonishing 6,000 miles. His journey took him to every corner of Manhattan, Brooklyn, Queens, the Bronx, and Staten Island. Helmreich spoke with hundreds of New Yorkers from every part of the globe and all walks of life. He finds that to be a New Yorker is to struggle to understand the place and to make a life that is as highly local as it is dynamically cosmopolitan.

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!

Q&A with Frank Farris, Author of Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

Frank A. Farris teaches mathematics at Santa Clara University and is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He is also the author of the new Princeton University Press book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. The book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry.

Frank Farris gave Princeton University Press a look at why he wrote Creating Symmetry, where he feels this book will have major contributions, and what comes next.

Before and After: A Peach and a Sierra Stream Become a Pattern, by Frank A Farris

Before and After: A Peach and a Sierra Stream Become a Pattern, by Frank A Farris

What inspired you to get into mathematical writing?
FF: After editing Mathematics Magazine for many years, I developed a passion for communicating mathematics: I didn’t want dry accounts written by anonymous authors; I wanted stories told by people. I wasn’t so interested in problems and puzzles, but in the stories that bring us face to face with the grand structures of mathematics.

Why did you write this book?
FF: Many years ago, I asked the innocent question: What is a wallpaper pattern, really? Creating Symmetry is the story of my dissatisfaction with standard answers and how it led me on a curious journey to a new kind of mathematical art. I took some risks and let my personality show through, while maintaining an honest, mathematically responsible approach. I hope readers enjoy the balance: real math told by a person.

What do you think is the book’s most important contribution?
FF: Most people who see my artwork say they’ve never seen anything like these images and that pleases me immensely. Of course, people have seen wallpaper patterns before, but the unusual construction method I use—wallpaper waves and photographs—gives my patterns an intricacy and rhythm that people wouldn’t create through the usual potato-stamp construction method, where the patterns is made from discrete blocks.

What is your next project?
FF: I am working on a “wallpaper lookbook,” a book for the simple joy of looking at patterns. Creating Symmetry tells people how to make the patterns, and there’s quite a lot of mathematical detail to process. Not everyone who likes my work wants to know all the details, but can still appreciate the “before and after” nature of the images.

Who do you see as the audience for this book?
FF: There are three audiences and they will read the book in different ways. The general reader, who knows some calculus but may be a little rusty, should find a refreshing and challenging way to reconnect with mathematics. Undergraduate mathematics majors will enjoy the book as a summer project or enrichment reading, as it makes surprising connections among topics they may have studied. The professional mathematician will find this light reading—a chance to enjoy the amazing interconnectedness of our field.


Presenting our new trailer for The Notebooks of Jean-Michel Basquiat

Brooklyn born Jean-Michel Basquiat (Dec. 22, 1960–Aug. 12, 1988) was one of the most important and fascinating figures in the 1980s New York art scene. Even today, pop culture references to the artist abound: Basquiat is referenced in Jay Z’s and Frank Ocean’s song “Oceans,” and in Jay Z’s and Kayne West’s 2011 collaborative album, “Watch the Throne,” to name two. He was known early on for his involvement with 1970s New York street art, including the SAMO tag created with Al Diaz, before he developed a successful studio practice indebted to a range of influences, from Neo-Expressionism to African art to jazz. Basquiat’s work explored the interplay between words and images, often touching on culture, race, and class. Of his extraordinary gifts, The New York Times Magazine, which profiled him in a 1985 cover story, wrote, “Not only does he possess a bold sense of color and composition, but, in his best paintings, unlike many of his contemporaries, he maintains a fine balance between seemingly contradictory forces: control and spontaneity, menace and wit, urban imagery and primitivism.”

From 1980 to 1987, Basquiat filled numerous working notebooks with drawings and pictograms of crowns, teepees, and hatch-marked hearts alongside notes, observations, and fragments of poems that reflect his deep interests in comics, street and pop art, and politics. Many of these images and words found their way into his drawings and paintings. We are proud to publish The Notebooks, a facsimile edition that reproduces the pages of eight of Basquiat’s rarely seen working notebooks for the first time. For a look at the pages, check out the new trailer for the book:

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.

Bill Chats: Story/Time: The Life of An Idea with Bill T. Jones and Jedediah Wheeler

k10299What do grande Starbucks coffees and tickets to see Bill Chats: Story/Time: The Life of An Idea with Bill T. Jones and Jedediah Wheeler  on Sunday November 9th at New York Live Arts at 5pm have in common? They’re both $5 dollars, give or take on the coffee. Jones, “one of the most influential and provocative dance artists our our time,” and author of Story/Time, joins Wheeler, Arts and Cultural Programming Executive Director at Montclair State University, to discuss Jones’ new book and the influence John Cage has had on his own work. This special conversation will also conclude with a book signing event, and don’t forget to use the code “STORYTIME” for $5 tickets! To buy tickets, and for more information on the event, click here.

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




#StoryTime with Bill T. Jones — #176



j10299[1]Click the image above to read Story #176 from Bill T. Jones’s Story/Time: The Life of an Idea.

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


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.

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.


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


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.


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.

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

Click here to sample selections from the book.