The Secret World of Flexagons: Fascinating Folded Paper Puzzles

by Scott Sherman, Yossi Elran, Ann Schwartz

Overview

The hexaflexagon is a folded paper strip of colored triangles that has long delighted people with how it “magically” changes its appearance when “flexed”. This hands-on, comprehensive book goes beyond the hexaflexagon, the standard version of this folded puzzle, exponentially expanding the barely explored field of flexagons as it brings new options and fresh insights to light.

● Learn over a dozen different flexes, and make dozens of different flexagons with the aid of step-by-step illustrated directions and templates to copy and print.

● Delve into the internal structure of flexagons and discover a universal way to describe and predict their behavior.

● Learn how to create your own custom flexagons with a special computer program.

● Understand how flexagons are connected to group theory, computer science, and topology.

● Have fun decorating flexagons and make flexagon books, puzzles, pop-ups, mazes, and more.

Written in a clear, easy-to-understand, and conversational style and enhanced with challenges and tips to broaden your flexagon skills and spark creativity, The Secret World of Flexagons: Fascinating Folded Paper Puzzles is a must for flexagon enthusiasts, teachers, students, libraries, mathematicians, and everyone who loves to solve a good puzzle.

Product Details

ISBN-13:	9781032560472
Publisher:	CRC Press
Publication date:	03/19/2025
Series:	AK Peters/CRC Recreational Mathematics Series
Pages:	336
Product dimensions:	8.25(w) x 11.00(h) x (d)

About the Author

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.

Table of Contents

Introduction

Part One: Symmetric Flexes
/1: The Pinch Flex
/2: General Tips
/3: More Faces
/4: Triangle Tetraflexagon and Octaflexagon
/5: Different Triangles
/6: Pinch Flex Variations
/7: The Book Flex
/8: The Box Flex

Part Two: Asymmetric Flexes
9: Flex Diagrams
/10: The V-Flex
/11: The Tuck Flex
/12: The Flip Flex
/13: The Pyramid Shuffle Flex
/14: Breaking Down Flexes
/15: Slot Flexes

Part Three: Learning about Flexagons
16: Flexagon Naming
/17: State Diagrams
/18: Flex Sequences
/19: Pat Notation
/20: Atomic Flex Theory
/21: Defining Flexagon and Flex
/22: Groups and Flexagons
/23: Flexagon Computers
/24: Conrad and Hartline’s Flexagon Theory
/25: Les Pook’s Flexagon Theory
/26: Topology
/27: Templates and Labels

Part Four: Exploring Flexagons
28: Square Silver Octaflexagon
/29: Hexagonal Bronze Dodecaflexagon
/30: Hexagonal Silver Dodecaflexagon
/31: Silver Bracelet
/32: Octagonal Ring 14-flexagon
/33: Flexagon Inspector

Part Five: Fun with Flexagons
34: Decorating Flexagons
/35: Flexagon Books
/36: Flexagon Puzzles and Mazes
/37: Flexagon Pop-ups
/38: Cutting Flexagons

Epilogue
Appendices
/A: Definitions
/B: Flexagon Charts
/C: Flex Compendium
Bibliography

Editorial Reviews

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

From the Publisher