Mathematical Connections in the Stack of Books

Stack of Books

Mathematical Connections

There are basically four categories of books in and near the Stack: historically important books going back to the very roots of mathematical thinking in different cultures, more contemporary mathematics books of which the title, content or author are particularly meaningful for the concepts illustrated in the installation in a serious or playful way, and books that are not mathematical per se but that illustrate the importance of mathematical approaches in completely different areas where not everyone would expect them.

  • Janos Bolyai
  • Sulbas Sutras diagrams
  • Sangaku
  • Roger Penrose's bookmark
  • Ramanujan
  • Paul Erdos
  • Bookmark of Nikolai Lobachevsky for The Elements by Euclid
  • Mau Piailug
  • Maryam Mirzakhani's Bookmark for Mathemalchemy
  • Katherine Johnson's Bookmark for Mathemalchemy
  • John Conway
  • Golden Spiral (representing Fibonacci)
  • Game of Life
  • Escher's bookmark for Beauty of geometry by Coxeter
  • Doris Schattenschneider
  • Dentelle by Veronika Irvine
  • Andrew Wiles

For each book, a bookmark is inserted (sometimes two) giving a sketch portrait of or alluding to a particular person linked to the book.

Historical Roots Books

For each of the 4 Historical Roots books, we wrote the title in the Stack in its original language. Here the titles are translated to English. We give a pointer for each to a Wikipedia entry that has more information; many books on the history of mathematics contain a more-in-depth treatment of their contents and importance.

The Elements, by Euclid

The 13 books that make up the Elements were already famous in Greek antiquity. Two of the 4 historic books to which our Stack pays tribute are much older — Euclid’s Elements date back to “only” about 300 BCE. But it is the oldest large-scale work in which mathematics is treated deductively. More information can be found here.

In this book, two bookmarks are inserted, depicting Nikolai Lobachevsky and János Bolyai, the two 19th century mathematicians who started developing non-Euclidean geometry. There is an interesting anecdote about János Bolyai’s likeness: the portrait that was believed to be his for many years, which was even used as the model for postage stamps in his honor, turned out to be someone else’s. More information about how this was discovered can be found here.

The bookmark shows a sketch of a 3D-bust that is believed to truly depict Bolyai.

Janos Bolyai
Nikolai Lobachevsky
The Compendious Book on Calculation by Completion and Balancing, by al-Khwarizmi

The Completion and Balancing in the title of this book refer to systematic methods to solve equations, e.g. the quadratic equation is solved by completing the square. This book, dating to about 800 CE and the youngest of the 4 in the Stack, is the first to treat as a separate subject what we now call Algebra. The very name Algebra traces its origin to the (original) Arabic name of this book; the name of its author gave rise to the term algorithm. More information is here.

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.

Golden Spiral (representing Fibonacci)
The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a compendium of mathematical knowledge in ancient China, written by several generations of mathematicians, ranging from the 10th century BCE to the 2nd century CE. It proceeds mainly by stating problems and their solutions, with explanations of the method by which the solution was obtained. Some of the algorithms described predate by many centuries their analogs in Europe. More information is here. The Nine Chapters on the Mathematical Art had an enormous influence on the development of mathematics not only in China, but also in Japan and Korea.

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.

Sangaku
Ṡulbasūtras, by Bodhāyana, Āpastamba, Kātyāyana and Manu

The Ṡulbasūtras were composed primarily to assist with the geometry of constructing vedic altars, and contain mathematical discussions of these geometric principles. The oldest, due to Baudhāyana, is thought to have been written around 800 BCE. More information can be found here.

This book has two bookmarks: one is a drawing from one of the Sūtras, and the other is a sketch of the famous mathematician Ramanujan, who attributed his flashes of visceral mathematical inspiration to Namagiri, an incarnation of Lakshmi, the Hindu goddess of good fortune.

Ramanujan
Sulbas Sutras diagrams

More Contemporary Mathematics Books (serious)

Mechanisms of the Heavens, by Mary Somerville

Katherine Johnson

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.

Combinatorics of Train Tracks, by Penner & Harer

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.

Maryam Mirzakhani
The Beauty of Geometry, by H.S.M. Coxeter

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.

M.C. Escher
Symmetry, by Herman Weyl

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.

Roger Penrose
Elliptic Curves: Number Theory and Cryptography, by L. Washington

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.

Andrew Wiles

More Contemporary Mathematical Books (playful)

Mathematical Games, by Martin Gardner

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.

Doris Schattenschneider

The bookmark shows a sketch of Doris Schattschneider, who learned of Marjorie Rice’s work on pentagonal tilings (several of which are illustrated in Mathemalchemy, on the floor of the Bakery, the long pillow with triangular cross-section on the Terrace and the “leaves” on the Great Doodle Page) from Gardner, and then went on to study and showcase it. Rice’s work was itself inspired by Martin Gardner: one of his columns listed several different patterns of tilings of the plane each using only one irregular-pentagon-shaped tile, and he had conjectured that these might be all the possibilities; Rice, intrigued, wondered whether she could find others, and did, after which she sent them to Gardner.

The Book (of Proofs)

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.

Paul Erdős

Non-mathematical Books that show mathematics is everywhere

On Weaving, by Anni Albers

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.

Dentelle by Veronika Irvine
We, the Navigators, by David Lewis

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.

Mau Piailug
Polyhedron Models, by Magnus Wenninger

Magnus Wenninger was a Benedictine monk who taught high school mathematics for most of his career. He became interested in polyhedra in his late 40s, after encountering Coxeter’s  Uniform polyhedra. He set out to build all of them as models, and published his methods in this book. A complete catalog, with pictures, can be found here

The two bookmarks in this book both allude to John H. Conway, who loved polyhedra constructions and had large numbers of them in his office, hanging from the ceiling and lying about. One bookmark just shows Conway’s picture; the other alludes to the Game-of-Life gif created by Randall Munroe in his xkcd webcomics series in memory of Conway, who died from Covid in 2020. Conway’s love of all objects mathematical inspired the name of Matemalchemy’s curio shop.

John Conway
Game of Life
Music Book

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.

Music Book

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