What if ruled pages grew unruly? In Nonstandard Notebook, as we journey from lines to parabolas to waves, deep questions arise—about form, art, and mathematics. How do we harness the infinite? Why do patterns permeate nature? What are the limitations and possibilities of human vision? Page after page explores these questions and more through provocative and inspirational images, each displayed with the mathematics that generated it. We see how straight lines can form fractal crenelations, how circles can disrupt and unify, and how waves and scaling can form complex landscapes (or even famous faces). Created by mathematicians, educators, and math popularizers Tim Chartier and Amy Langville, and with a foreword from Ben Orlin (bestselling author of Math with Bad Drawings), Nonstandard Notebook shows that rules—both the rules of mathematics and the rules of a notebook—do not mark the end of creativity, but the beginning.

Read on for Q & A with Tim and Amy about their fun, revolutionary mathematical notebook that challenges us to play outside (and with) the lines.    

How did you wind up in your field, and what do you love about it?

Tim: I had a fondness for computer science since learning to program in middle school.  In college, applied mathematics attracted me as my area of application is computer science and the mathematics is inherently connected to the world. Our notebook is applied mathematics, in my mind. Amy and I are applying math to art and specifically a notebook that can inspire us as we write, doodle, or simply look through the pages. 

Amy: Since childhood, mathematics has been my favorite subject. By adolescence, my interests extended to the humanities, particularly writing. By the time graduate school came around, being a mathematics professor seemed a natural way to combine these fields and skills. As professors, Tim and I have each written several books full of words, paragraphs, and chapters about mathematical topics. So it’s neat for us that the Nonstandard Notebook is a book of so few words. That’s one of the beauties of mathematics—its ability to say so much so compactly. In this case, through lines, curves, and equations.

The Nonstandard Notebook is a wonderfully unique blend of mathematics, art . . . and classic composition notebook. Can you tell us a bit about the origins of the project? And do you have a favorite variation in the book?

Amy: The Nonstandard Notebook grew from Deconstruct Calculus Series of activity books that teach calculus in an interactive, personalized way. I started adding mathematical notebook pages in strategic locations throughout each chapter as space for students to reflect, work problems, and take notes. The student response was wonderful, yet it was the response from others, a mix of curiosity and wonder, that made me think extracting these pages into their own book might have universal appeal. Being my longest and closest collaborator and friend, it was Tim’s exuberant reaction and desire to help that made this book a physical reality.

Tim: I still remember the first time I saw Amy’s notebook pages! I was very excited and, even more so, to begin collaborating. I shared the early pages we created with family, students and friends.  

Amy: It’s too hard to pick a favorite page from the book. I know we had fun riffing on a theme. For example, the page titled “halved”—in one draft, we actually had a whole chapter of pages that explored the many ways the lined composition page could be halved. We’d email pages back and forth, generating more and more pages, each one a unique take on the idea of halving the lined page. I remember the excitement of receiving a new page from Tim and sometimes chuckling at his clever and new interpretation of halving because it was so different from any idea I’d had about halving. That was really fun for us so, as teachers, we created a way to let students in on the fun too.

Tim: A few summers ago, I taught a math and art workshop to middle schoolers, which included a section on creating nonstandard notebook pages using Desmos. Within minutes, the room was buzzing with mathematical creativity. Soon, we had a digital gallery of mathematical art. The middle schoolers wandered around, admiring each other’s work, and whenever they were inspired by someone’s creation, they’d rush back to their computers, craft their own page, and eagerly ask to share it. We’ve since developed online tools, available through the book’s website, to help foster these kinds of experiences for others.

Amy: This also gets at another one of the beauties of mathematics—the concept of infinity. We could make (and prove that we could make) infinitely many halved-lined pages 🙂

Tim: I agree, it’s too hard to pick a favorite page from the book as each page felt like its own adventure in coding!

What do you hope readers will take away from the book with regard to math? And how about creative thinking?

Tim and Amy: We hope readers will take away the following message regarding mathematics. Mathematics can be art and art can be mathematical. For creative types such as writers, artists, and designers, instead of a message, we have an invitation. We are eager to see if and how these nonstandard pages influence creation and invite readers to share their inspirations with us through the book’s website.

While you were working on this project, what did you learn that surprised you the most?

Tim: Every page of the book is graphed in python. The mathematics curriculum covers the scaling and transposing of graphs. The notebook is filled with applications of this idea and we became very versed in this process as we produced the images of the notebook.  Transposing a function became one of our biggest tools of the book. 

Amy: We received a firsthand lesson on how tools can make or break a project. Midway through the project, we found ourselves at a technical impasse. Early images were made with several different tools and so line weights and colors didn’t quite match no matter how we fiddled with a tool’s features. We needed one tool with both design precision and functionality as well as powerful mathematical graphing capability. We found that tool almost accidentally. During the first Covid summer, we decided to review a colleague’s online materials for his class, Numerical Methods with Python. That summer is now known as the summer we learned python. By Week 7 of the class, the week on graphing with python, three things became clear: (1) python was our solution, (2) Tim is a coding rockstar, and, as a result, (3) the Nonstandard Notebook could proceed. This book exists because we found the right tool.

Where will your research and writing take you next?

Tim: Ben Orlin alluded to a current project in his foreword. By the time I was in college, I was performing nationally and internationally in puppetry. During my time as the 2022-23 Distinguished Visiting Professor at the National Museum of Mathematics, I worked on filming puppetry sketches related to math. I’m currently working on such videos for early elementary and college curricula. In the early stages of the project, I created several videos for Ben and his daughter. 

Amy: For over a decade and a half, I’ve been consumed by the idea of making calculus more approachable, active, and relatable through the Deconstruct Calculus Project. After so much work, it’s exciting to be nearing the completion of the series. Next year several books in the series will be released, including the graph novel CalcuComix. And in the coming years, the series of seven books will finally be complete.

In addition, Tim and I hope there will be interest in future editions of the Nonstandard Notebook, such as the Dotted Edition and the Graph Paper Edition. For example, the Dotted Edition is our mathematical take on the standard dotted page. Whereas this current Lined Edition uses the mathematical tools of lines, curves, and equations, the Dotted Edition uses the mathematical tools of sequences and series to make beautiful mathematically dotted pages.

What’s the best book you’ve read lately?

Tim: On my desk is Love Triangle by Matt Parker, Once Upon a Prime by Sarah Hart, and Math for English Majors by Ben Orlin. Exciting reading lies ahead!

Amy: Lately, I’m reading a lot of books in Spanish in the hope that one day I’ll be able to read Borges and Neruda in the original language. I’ve also enjoyed the latest MathEd books by Jo Boaler and Peter Liljedahl.

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Tim Chartier is the Joseph R. Morton Professor of Mathematics and Computer Science at Davidson College. His books include Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing and Get in the Game: An Interactive Introduction to Sports Analytics, the latter also published by the University of Chicago Press.

Amy Langville is professor of mathematics at the College of Charleston. Her books include Google’s PageRank and Beyond: The Science of Search Engine Rankings, Who’s #1: The Science of Rating and Ranking, and the Deconstruct Calculus Series. She is also a frequent consultant on applied mathematics projects.

Nonstandard Notebook is available now from our website or your favorite bookseller.