THE SECRET WORLD OF FLEXAGONS: Fascinating Folded Paper Puzzles by Scott Sherman, Yossi Elran and Ann Schwartz--Love the beauty

 "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

WHAT ARE FLEXAGONS?  

              

Flexagons are fascinating, origami-like puzzles folded from strips of paper that you flex in various ways to reveal previously hidden portions of the flexagon. Flexigon shapes are found in machine designs, jewelry.  I love the unexpected beauty of a flexagon-like leaf pattern or butterfly wing. 

The hexaflexagon is a folded paper strip of colored triangles that has long delighted people with how it “magically” changes its appearance when “flexed”. The Secret World of Flexagons: Fascinating Folded Paper Puzzles (CRC PRESS-Routledge,  5/7 launch) goes beyond the hexaflexagon, the standard version of this folded puzzle,  expanding the barely explored field of flexagons as it brings new options and fresh insights. 

Flexagons sit at the intersection of paper folding, puzzles, and mathematics. This comprehensive book, explores the pleasure of ussing your hands to flip through a kaleidoscope of arrangements. There are  interesting ways to decorate flexagons. You might check-out what's hiding inside a flexagon and  discover new flexes and types of flexagons. There's also the challenge of figuring out how to “solve” a flexagon, as you might a Rubik’s cube. There's also the mathematics that untangle how the twisted strips of paper actually work. You might even discover a universal way to describe and predict their behavior. Flexagons are connected to group theory, computer science, and topology. 

Flexon history began with the random adventure of Arthur Stone, who traveled from England to the U.S. to attend Princeton University. Due to the difference in standard paper sizes between the two countries, he needed to trim off one edge of each sheet to fit them in his binder. After experimenting with different ways of folding those strips of paper, he stumbled upon a particularly interesting arrangement, which he called a flexagon, short for flexible polygon. He and his friends formed the “Flexagon Committee” to explore these mysterious objects. The other members of the committee were Bryant Tuckerman, Richard Feynman, and John Tukey, each of whom went on to have distinguished careers in mathematics or physics. It is believed that Feynman used some of the techniques he developed for flexagons when he later explored concepts in quantum physics!

Popular science and math writer Martin Gardner first introduced flexagons to a broader audience in a 1956 Scientific American article, where he described Arthur Stone’s hexaflexagons. The flexagons proved so popular that that particular issue became the bestselling issue of Scientific American for decades. Many readers sent letters describing all the entertaining ways they decorated their flexagons, including making greeting cards out of them. Scientific American soon gave Gardner his own monthly column to discuss other topics in recreational mathematics.  One of his later columns described tetraflexagons.

Intriguing questions continue. Can you make a hexaflexagon with 3 or 6 faces, what about 12 or 5 or 100?  If you can make a flexagon by folding up a strip of regular triangles, what about right triangles or scalene triangles? If you can make a flexagon out of triangles or squares, what about pentagons or hexagons  or octagons? If you can reveal new faces by flexing it one way, are there other ways to flex it that might behave differently? Can you describe how the internal structure of a flexagon changes as you flex it in order to make predictions?

The answers are yes! And they've helped uncover a huge variety of flexagons. For your explorations all you  need are scissors and clear tape. Ideally, you have access to a computer and can print out paper patterns to make flexagons. Be prepared to cut, fold, and tape to make a lot of different flexagons.

Above all, say the authors, flexagons are fun to play with. Happy flexing!

ABOUT THE AUTHORS

Scott Sherman is a leading software designer who has worked at Tableau and Microsoft, and applies his software skills to exploring flexagon dynamics,

Yossi Elran is a recreational mathematician at the Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science, Israel.

Ann Schwartz is a flexagon enthusiast who has created many new flexagons and led numerous flexagon workshops. 

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