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Princeton University Press, 2022, 228 pages, US$ 35.00

According to legend, when the great Archimedes was killed by a Roman soldier around 212 BCE, he was busy drawing geometric diagrams. Among the numerous later testimonies of the tragic event, the writer Valerius Maximus (fl. 1st century CE) stated in his collection of moralizing anecdotes Factorum ac dictorum memorabilium (Deeds and sayings worth remembering) that Archimedes appealed to his attacker with his last words, “Noli, inquit, obsecro, istum disturbare” (Do not, he said, I beg you not to destroy this), referring to the drawings on his precious abacus. Of course, Archimedes spoke Greek in a Dorian dialect, and not Latin, but these are the words that have been passed on to us. “Do not disturb my circles,” is another version of the famous saying. It is to my knowledge the first time a mathematician reportedly cried “Do not erase.”

In Greco-Roman antiquity, the abacus was a board covered with dust or fine sand, allowing the user to draw and to calculate with the tip of a stick or a pen. In the Middle Ages, the abacus was also identified with a counting frame with beads (cf. Liber Abaci by Leonardo of Pisa aka Fibonacci). The modern blackboard seems to have evolved from smaller slate boards, where a small piece of chalk was applied for writing and a sponge for erasing. According to some reports, the chalkboard as we know it made its entrance into the classroom in 1801. But there are also reports that Euler, having become blind in the 1770s, used to draft his equations with big letters on a blackboard (schwarze Tafel) in his office for his assistants to copy [1, p. 13]. It was probably a writing slate, but its function was evidently to communicate mathematical ideas.

Until quite recently, chalkboards (used henceforth as a synonym for blackboard) dominated the walls in classrooms and lecture halls from primary schools to universities. Lectures in mathematics given on the chalkboard are compelling. Starting from an empty board, the lecturer gradually fills the surface with symbols and diagrams in real time. The audience has no alternative but to follow the exposition closely, because the writing is due to disappear at some point during the lecture, to be replaced by new images. The cruelest lecturers erase their writing all the time, in order to keep their audience focused. In primary schools, students were at times asked to come up to the front of the room one by one to answer questions at the blackboard without a textbook. Such an exciting situation could easily lead to a negative association with the blackboard (for a history of the use of blackboards in American schools, see, e.g., [2]).


New technology is now making its headway in pedagogy at every level, allowing the use of smartboards and video screens for multimedia education. Technology certainly has its merits, but it is doubtful whether it will ever overtake the merits of the chalkboard, especially in mathematics and other theoretical subjects.

For mathematical scientists, the chalkboard, primitive and ascetic as it is, has retained its charm in visualizing and communicating ideas. Whiteboards and video projectors dazzle at their best, but just like Occam’s razor, the simplest and most purposeful of solutions prevails. The benefits of chalkboards are obvious: no need for electricity, no technical problems, no worn out or smelly whiteboard markers, ease of installation and access, and speed of erasure. Moreover, the chalkboard is silent and patient, which is a prerequisite for the creative process.

My late friend Osmo Pekonen, former Reviews editor of the Mathematical Intelligencer, visited a number of leading research institutes of mathematics around the world, notably those at Bures-sur-Yvette, Cambridge, Mumbai, Princeton, and Stockholm. From all of these hubs, the main task of which is to foster thinking, he reported the same observation: blackboard and chalk were used everywhere, not only in the offices and lecture halls, but sometimes in corridors, in lifts, and in bathrooms. When I was employed in industrial research, I too had a chalkboard installed in my office, possibly the only one in the whole building.

Jessica Wynne is an art photographer and professor at the Fashion Institute of Technology in New York. As a daughter of two boarding school teachers, she literally grew up surrounded by chalkboards. Over the years she became close to two of her neighbors at her summer residence, Amie Wilkinson and Benson Farb, mathematics professors at the University of Chicago. Without mathematical training, she could not understand the meaning of their work, not even when she asked Farb to explain it to a layperson. But what she could appreciate as an artist was the visual impression of the scribbles in his notebook.

Wynne started photographing chalkboards on a trip to India with her students. In Jaipur, she visited an elementary school, where she found chalkboards everywhere, filled with lessons in Hindi, a language she did not understand. While she was studying the photographs at home, an analogy with the writing of her mathematician friends sprang to mind. She found it striking that unlike many other scientists, mathematicians continue to work on chalkboards rather than on computers. A project, quite unique in its scope, started to take shape. She began visiting centers for mathematical research all over the world. The list of places includes Ann Arbor, Berkeley, Berlin, Boston, Cambridge, Chicago, Evanston, Grenoble, Houston, Lausanne, Los Angeles, Lyon, Nashville, New Haven, New York, Paris, Princeton, Providence, Rio de Janeiro, Seattle, and Zurich. She asked the mathematicians she met to write whatever they wanted on their chalkboards and then photographed the outcome in a real environment, with erasure marks and all forms of light and reflections from windows. Some of the boards are covered with older writings, worthy of keeping for years, some are fresh products of creativity from interaction with fellow mathematicians, some represented merely a geometric object of research, as if for contemplation. The last board documented in the book is, perhaps defiantly, a tabula rasa, all writing wiped out.


Wynne’s book contains one hundred and ten photographs of chalkboards of contemporary mathematicians, pure as well as applied, from well-known prize-winning professors to gifted students with their discoveries yet to come. The images on the boards represent disparate fields of mathematics such as algebra, category theory, combinatorics, dynamical systems and ergodic theory, finance, geometry, group theory, knot theory, measure theory, noncommutative geometry, number theory, quantum computing, statistics, and various domains of theoretical physics.

Opposite each photograph is the mathematician’s personal description of the illustrations on the chalkboard and his or her relation to the topic and to mathematics in general. A recurrent philosophical theme is the question whether mathematics is invented or discovered, or perhaps both. It is evident from the written testimonies that the relationship of the mathematicians to the chalkboard is deeply emotional, even intimate. One mathematician says that he persuaded his wife to have a blackboard in their bedroom, but after six months, tired of the dust on their sheets, they had it removed as useless. In fact, there is also a case of chalk allergy reported in the book. Even blackboards can have their limits.

The positive experiences are nevertheless predominant. Simion Filip says that the blackboard does not have the restrictive linear quality of a written text, and it allows the user to organize the material according to its natural, spatial characteristics. Compared with writing on a page, the blackboard invites wider gestures and larger symbols. Dimitri Shlyakhtenko thinks that blackboards are just the right size to enable you to present the correct amount of information that a colleague or student can follow. Nicholas G. Vlamis describes the chalkboard as the glue that holds together the mathematical community and its rituals. The afterword by Alec Wilkinson, a literary journalist and an uncle of the Amie Wilkinson mentioned earlier, attempts to unfold what mathematics is all about with the help of the photographs. Being a latecomer to mathematics, Wilkinson seems to capture its idea in a way most mathematicians might well agree upon but could hardly define comprehensively themselves. Paradoxically, the goal of mathematics is to express deep mysteries to a maximum degree of certainty and clarity as necessary truths. According to Wilkinson, the blackboards of Wynne’s book fall into four categories: they represent unsolved problems, explanations, narratives, and speculations. For him, the formulas and drawings seem to vibrate, as if with a life of their own.

For Wynne, the goal was never to decipher the meaning of the writings on the boards she photographed, but rather to capture an instant impression. Her feeling about the results is ambivalent: on the one hand, she is attracted by the abstract beauty of the symbols and illustrations, while on the other hand, she is repelled by things she cannot grasp. She sees Greek letters, punctuation, notation, sinuous scrawls and loops, evocative of a patterned tapestry or calligraphic scroll, sometimes a pared-down shape or orderly sequence of lines. She perceives mathematics imagery as a form of visual art and links it to the timeless lineage of artistry and writing, from cave paintings to hieroglyphs to graffiti. Her art explores the aspects that communication between mathematicians shares with visual artists throughout time: a kind of transcendental exchange.

For mathematicians, who are used to regarding their science as an art whose beauty springs from the logic and internal consistency of a theory, this is a really unusual book to digest, but quite fun and interesting to read. The mathematicians contributing to the book seem genuinely excited by this rare opportunity to explain what they are doing and why. The book is a beautiful tribute to a timeless and unsurpassed medium for communicating abstract ideas, but also a worthy document of the mathematical culture of our time and the way mathematicians perceive their profession and their role as mediators between the abstract and the concrete.