Stack of Books – Mathematical Connections

The bookmark inserted in this book shows the Fibonacci spiral, which uses quarter circles to give an approximation to the golden spiral. Fibonacci is a nickname for Leonardo di Pisa, the 13th century Italian mathematician who played an important role in converting Europe to the Hindu-Arabic numeral notation, as opposed to the Roman numerals still in use then.

The bookmark shows the geometric drawing on a sangaku, illustrating a unique Japanese use of mathematics. Sangaku were mathematical problems hung as offerings in Japanese shrines in the Edo period. The one of which a picture was redrawn for this bookmark was hung in Fukushima prefecture in 1885.

Ostensibly a translation of Laplace’s Traité de mécanique céleste, this book went far beyond a simple translation, adding many mathematical explanations and diagrams that made the content more accessible. It was later used as a textbook for advanced courses at Cambridge University. In a (then anonymous) review of this book, historian of science Wiliam Whewell coined the term scientist, noting that “philosopher” was no longer an adequate term covering those with a profound interest in the natural sciences. Mary Somerville and Caroline Herschel were the very first women to be elected to the Royal Society.

The bookmark shows a sketched portrait of Katherine Johnson, who worked for NASA, computing for the US Space program (the Heavens indeed!); the Katherine Johnson sheet in the Cavalcade features the first page of one of her technical papers.

The Train Tracks of the title are families of geodesic curves on a fixed surface, typically of negative curvature; like train tracks in everyday life, some of them tend to run almost “parallel” for a while, to then diverge and go off in completely different directions. These measured geodesic laminations were first named train tracks by William Thurston; several drawings from his notes are reproduced on the Thurston figures sheet in the Cavalcade, including a playful one of train tracks with a little train steaming along. That drawing was the inspiration for the more elaborate graphic by Conan Wu that is reproduced in a little painting on the back wall of the Bakery.

The book Combinatorics of Train Tracks was one of the favorites of Maryam Mirzakhani, whose portrait is featured on the bookmark inserted in this book.

Coxeter was one of the greatest geometers of the 20th century. Apart from his serious mathematical work, he was also interested in connections with other fields, including music, art and engineering. He had a sustained correspondence with artist M.C. Escher (of whom a “self portrait” is shown on the bookmark), inspiring Escher’s work using hyperbolic tessellations of the disk. Coxeter also used some of Escher’s prints (with his permission) to illustrate his own mathematical work.

Hermann Weyl was another giant of 20th century mathematics, making fundamental contributions to a wide range of fields, including analytic number theory, mathematical physics, group theory and representations, and the foundations of mathematics.

His bookmark features Roger Penrose, another polymath working in a variety of fields, whose contributions to general relativity led to his sharing the 2020 Nobel Prize in Physics, and who is also famous for his discovery of non-periodic tessellations of the plane in which large portions have 5-fold symmetry (which is known not be globally possible for a plane tiling). Like Coxeter, Penrose also interacted with M.C. Escher: the Penrose “tribar” was partially inspired by Escher’s prints of impossible situations, and in turn inspired other work by Escher.

Number Theory is a field of mathematics in which some very hard theorems or conjectures can be formulated in very simple terms (such as the twin prime conjecture — that there are infinitely many pairs of prime numbers that differ by just 2, like 11 and 13 — widely believed to be true, but not proved yet as of this writing, in 2021). For a long time it was also believed to be the “purest” of all mathematical disciplines, without any concrete applications — until it turned out to be absolutely essential for modern cryptography.

The bookmark portrays Andrew Wiles, who proved Fermat’s Last Theorem at the end of the 20th century, another famous number theory conjecture that can be stated in simple terms.

Martin Gardner wrote the column Mathematical Games for the magazine Scientific American for more than 25 years. Many of the topics that came up in these recreational mathematics columns were treated more extensively in the many (more than 100 …) books he wrote over the years. Although he was not an academic mathematical researcher, and did not prove any famous theorems, Gardner inspired many generations of mathematicians; he also introduced mathematicians to each other who went on to highly productive and important research collaborations. His inspirational and bridging role also inspired the Martin Gardner Mathematical Games sheet in the Cavalcade.

Throughout his life, the renowned peripatetic mathematician Paul Erdős, depicted on the bookmark of this Mathemalchemy book, would qualify a particularly beautiful and elegant proof as a “Proof from the Book” — the mythical book in which the most elegant proof of each theorem was given. A book entitled Proofs from the Book was published to honor his memory. But the Mathemalchemy world would not be complete without the presence of the Book itself.

Anni Albers was a leading textile artist in the 20th century, head of the weaving workshop at Bauhaus for a while. Her book On Weaving is a classic on the art and history of weaving. She emphasized the hands-on aspects of weaving and the diversity of possible materials and techniques; she experimented extensively with different design principles.

The bookmark shows a pattern designed by Veronika Irvine, a computer scientist who studies the computational generation of textile structures based on the techniques of bobbin lace; her algorithmic approach has led to the development of new bobbin lace stitches.

David Lewis was an experienced world ocean sailor, who learned early on to appreciate Polynesian culture; his book We, the Navigators explains the traditional navigation techniques used by Polynesian sail masters to sail large distances, which he learned “hands-on” from two traditional navigators with whom he traveled on long-distance trips. These techniques rely on a wide range of observations, including the sun and stars, wind and weather, but also delicate observations of ocean swell behavior. The navigators use a complex mental representation of many of the variables that is essentially mathematical in nature.

The bookmark sketches Mau Piailug, one of the last master navigators in Micronesia, who taught his craft to sailors of other cultures willing to devote the necessary time to learn it.

The Music book alludes to a few of the many ways in which mathematics and music are connected. Its top open page displays a 19th century clever representation of the name BACH by showing a single note on four crossing staves; with their different clefs, the note is read in turns as B, A, C and H. The opposing page shows the score of the Adagio by BACH that accompanied the launch of the Mathemalchemy project on internet. On other pages, one can glimpse Tibetan chants, denoted in ways that have some similarity to new proposed music notations and other mathematical representations used in connection with music.