When you see a paper crane, what do you think of? A symbol of hope, a delicate craft, The Karate Kid? What you might not see, but is ever present, is the fascinating mathematics underlying it. Origami is increasingly applied to engineering problems, including origami-based stents, deployment of solar arrays in space, architecture, and even furniture design. The topic is actively developing, with recent discoveries at the frontier (e.g., in rigid origami and in curved-crease origami) and an infusion of techniques and algorithms from theoretical computer science. The mathematics is often advanced, but this book instead relies on geometric intuition, making it accessible to readers with only a high school geometry and trigonometry background. Through careful exposition, more than 160 color figures, and 49 exercises all completely solved in an Appendix, the beautiful mathematics leading to stunning origami designs can be appreciated by students, teachers, engineers, and artists alike.

‘Joseph O'Rourke does a great job introducing the myriad ways in which origami can be modeled mathematically to a novice – but interested – reader. This is not an easy task, but Joe handles it with fantastic figures, great examples, clear prose, and even online computer demos. This book is where I will point everyone who asks me how to get started in the field of mathematical origami!'

Thomas Hull - Franklin & Marshall College

‘Joseph O' Rourke lucidly explains the mathematical secrets behind the art and engineering of origami. This book leads math lovers–young and old, amateur and professional-to a great ocean of undiscovered truths, and let them be children once again, enthusiastically at play on the seashore.'

Jin Akiyama - Tokyo University of Science

‘This is a pleasurable ramble through the mathematics of origami, covering patterns from single vertices to complicated arrays that touch on some of the hardest unsolved problems in all of computational mathematics! The style is approachable and conversational, and there are fun exercises, constructions, and guides for further exploration.'

Robert J. Lang - Origami Artist and Consultant

‘The bible of the field of computational origami was written by Demaine and O'Rourke in 2007. Almost 20 years have passed since this comprehensive book was published. Many unsolved problems in the book were solved, and at the same time, many new and fascinating problems were born, which are still being actively researched. This book is a follow-up for beginners by one of the authors of the bible. It provides a very thorough guide for those who are learning this field for the first time. Not only is it thorough, but it also generously presents cutting-edge results on fascinating topics that have developed greatly in the past 20 years. It will be a very reliable and good guide for readers who are new to this field, as well as those who are somewhat familiar with it.'

Ryuhei Uehara - Japan Advanced Institute of Science and Technology