Secret World of Flexagons

Scott Sherman is a leading software designer focused on building tools for seeing and understanding information, from computer-aided-design to diagramming to data visualization. He was a researcher and software engineering architect at Tableau, a data visualization company, where he built the first version of many essential tools including dashboards and highlighting. At Microsoft, he designed and built the layout engine for Microsoft Office’s SmartArt feature. Scott has been fascinated with flexagons since he first learned about them from Martin Gardner’s writings and has figured out techniques for generating a huge variety of new flexagons and flexes. He has also applied his software skills to create programs for exploring flexagon dynamics. He has given talks on various topics in recreational mathematics, including flexagon theory, at the Gathering for Gardner and Georgia Southern University. With Dr. Elran and Ann Schwartz, he co-led the 2019 Neil Shore Flexagon Workshop at the Davidson Institute, the educational arm of the Weizmann Institute of Science in Rehovot, Israel. He also is the author of an online, interactive book called Explorable Flexagons.Yossi Elran is a British-Israeli recreational mathematician at the Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science in Rehovot, Israel, and teaches various courses at the Western Galilee College in Israel. He holds a PhD in theoretical quantum chemistry and has done post-doctoral research on decoherence, one of the main challenges of quantum computing. Yossi frequently journeys nationally and globally, delivering engaging and captivating presentations to diverse audiences ranging from schools and universities to festivals and other events on various subjects, including recreational math, astronomy, quantum mechanics, the history and philosophy of science, and creativity. He has written many papers, is the author of Lewis Carroll’s Cats and Rats and Other Puzzles with Interesting Tails, and Archimedes’ Stomach and Other Puzzles You’ll Love to Digest, and co-author of The Paper Puzzle Book. He is the creator of four popular "Future Learn" MOOC’s on recreational math, two of them on flexagons, and two Ted-Ed puzzle videos that have reached nearly 20 million views. Yossi is a longtime member of the Gathering 4 Gardner Foundation, whose mission is to stimulate curiosity and the playful exchange of ideas and critical thinking in recreational math, magic, science, literature, and puzzles to preserve and extend the legacy of writer and polymath Martin Gardner.Ann Schwartz has created more than a dozen new flexagons. She has presented a new flexagon at the biennial Gathering for Gardner, which she has attended since 2006, and has given flexagon presentations and workshops at the National Museum of Mathematics (MOMATH) in New York City and that museum’s MOVES conferences. In 2015 she was the guest speaker at the Recreational Math, Puzzles and Games Conference at the Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science in Rehovot, Israel; and in 2017 gave the only flexagon presentation at the Seventh International Meeting on Origami in Science, Mathematics and Education (7OSME) in Oxford. With Scott Sherman and Dr. Yossi Elran, Ann co-led the Neil Shore Flexagon Workshop at the Davidson Institute in 2019. She coauthored "The Hexa-Dodeca-Flexagon," a chapter in Homage to a Pied Puzzler (2009), and the chapter "Should We Call Them Flexa-Bands?" that was published in The Mathematics of Various Entertaining Subjects - Volume 3.

• IntroductionPart One: Symmetric Flexes1: The Pinch Flex2: General Tips3: More Faces4: Triangle Tetraflexagon and Octaflexagon5: Different Triangles6: Pinch Flex Variations7: The Book Flex8: The Box FlexPart Two: Asymmetric Flexes9: Flex Diagrams10: The V-Flex11: The Tuck Flex12: The Flip Flex13: The Pyramid Shuffle Flex14: Breaking Down Flexes15: Slot FlexesPart Three: Learning about Flexagons16: Flexagon Naming17: State Diagrams18: Flex Sequences19: Pat Notation20: Atomic Flex Theory21: Defining Flexagon and Flex22: Groups and Flexagons23: Flexagon Computers24: Conrad and Hartline’s Flexagon Theory25: Les Pook’s Flexagon Theory26: Topology27: Templates and LabelsPart Four: Exploring Flexagons28: Square Silver Octaflexagon29: Hexagonal Bronze Dodecaflexagon30: Hexagonal Silver Dodecaflexagon31: Silver Bracelet32: Octagonal Ring 14-flexagon33: Flexagon InspectorPart Five: Fun with Flexagons34: Decorating Flexagons35: Flexagon Books36: Flexagon Puzzles and Mazes37: Flexagon Pop-ups38: Cutting FlexagonsEpilogueAppendicesA: DefinitionsB: Flexagon ChartsC: Flex CompendiumBibliography

“COMPREHENSIVE AND ACCESSIBLE...A MUST HAVE for anyone interested in flexagons. The authors combine scholarly analysis with infectious enthusiasm. Instructions are included for dozens of examples that are easy to make and fun to explore.”—Paul Jackson, award-winning origami artist, paper engineer, teacher, and author“HIGHLY RECOMMENDED! A lucid and comprehensive treatment of every aspect of flexagons. Beginners will find a gentle way into a fascinating subject, and experienced flexers will find themselves learning something new on nearly every page of this beautifully illustrated book.”—Jason Rosenhouse, professor of mathematics, James Madison University, author of "Games for Your Mind: The History and Future of Logic Puzzles”“Wow! The possibilities just keep going and going. I had no idea there was so much variety in flexagons! Quite an impressive book. This book goes into amazing depth in the incredible variety of flexagons. There’s something for everyone: if you just want to make cool gadgets, there’s clear instructions (and downloadable templates) for a wide variety of shapes, but for those who want to go deep, the mathematics of flexagons are also presented, rich and rewarding. Be forewarned, though: they’re addictive.”—Robert J. Lang, physicist, virtuoso origami artist, and master of origami mathematics, theory, and real-world applications“A great book for the right audience […] If you find the concepts here get you excited, you absolutely need this book.”—Brian Clegg, PopularScience“This book is one hundred percent hands-on, with templates for every flexagon for readers to print out and fold. Starting with the simplest flexagons and flexes, it then progresses to more complex new ones. Step-by-step instructions for folding and flexing are always provided. Color illustrations and color photos make everything as clear as possible. The idea is to make learning about these wonderful puzzles fun. Different parts of the book can be applied to grade school, high school, and college classes. Perhaps even post-graduate mathematics programs. Flexagons show how math can be fun, and the authors hope elementary teachers will use them to teach their students just that.”—Mathematics Matters, Newsletter for Nova Scotia Mathematics Teachers"Very few books feature the flexagon family in such detail, even though these objects have fascinated mathematicians since their original discovery by Arthur Stone in 1939 and the general public since they were popularized by Martin Gardner in a 1956 Scientific American column. This fun and detailed book is sure to delight all who appreciate mathematical curiosities with deep mathematical underpinnings."—Notices of the American Mathematical Society"I like this book, which is quite comprehensive: The Secret World of Flexagons by Sherman, Elran, and Schwartz (2025)"—Keith Enevoldsen, thinkzone.wlonk.com“[T]his book is written at a level unsurpassed in the field.”—Nick Robinson, https://nickorigami.com/