To quote Rod Serling, “You unlock this door with the key of imagination. Beyond it is another dimension…” Such is the case with Henry Segerman’s new book: it is a portal into a new mathematical world. The book focuses on visualizing key mathematical concepts in topology and geometry: symmetry (Chapter 1), polyhedra (Chapter 2), 4D space (Chapter 3), tilings and curvature (Chapter 4), knots (Chapter 5), surfaces (Chapter 6), and a brief introduction to his favorite prints, which include some fun fractals, Hilbert curves, and puzzles (Chapter 7).
I have great difficulty thinking about Visualizing Mathematics with 3D Printing as “just a book.” The careful choice, quality and effectiveness of the 140+ images in the book is outstanding. What Segerman has developed is much bigger than a book; he has developed a whole platform to complement the book and explore mathematical concepts. Visualizing Mathematics with 3D printing allows the reader to manipulate with a computer or 3D print the objects discussed, making it possible to physically interact with the concepts. The jump in understanding seems comparable to what was achieved at the time by Mandelbrot’s computer generated pictures of fractals: an entirely new understanding of the fractal concept.
Visualizing Mathematics with 3D Printing includes the actual book, with fantastic images; a web site in which each figure from the book can be interacted with, rotated, and examined from every direction; short videos discussing the different mathematical concepts in relation to the figures; and object files that can be downloaded to 3D print.
The narrative of the book encourages the reader’s exploration as opposed to only showing pictures and rendering facts. Already in the first pages, when he is introducing the reader to thinking about symmetry, Segerman encourages the reader to go “Get a cube to look at.”
Given the breath of Segerman’s work, this “book” (the whole platform) can be approached by many audiences with different goals. An enthusiastic independent reader with some degree of mathematical sophistication can enjoy the material, develop lots of intuition, and dive into additional resources when willing to explore the topics more technically. An expert can delight in the exciting hands on (literally) approach to the material.
I believe Visualizing Mathematics with 3D Printing would be a great platform to build exciting inquiry-based activities for a variety of classes, starting with mathematics for the liberal arts. In my opinion every student of geometry or topology should be required to interact with this book and its online resources.
Zdeňka Guadarrama is an associate professor and Chair of the Department of Mathematics at Rockhurst University in Kansas City, MO. She is passionate about mathematics education and outreach, with her work currently focusing on mathematics curriculum development through inquiry, and the intersections of mathematics with other fields, particularly the arts. She is a coauthor of the book Calculus: A guided Inquiry.