Flexagons are an example of the strange results when one starts playing around with the simplest of things. 1939 Princeton graduate student Arthur Stone could never have guessed the future frenzy when his foldings of trimmed notebook paper finally formed his first flexagon. These fascinating fumblings sparked the formation of a 'flexigation' committee. With this first foursome, flexagon evolution accelerated.

All of the flexagon fame fanning founders distinguished themselves in later life. Arthur H. Stone became a Professor Emeritus of mathematics at the University of Rochester, Bryant Tuckerman became a research mathematician with IBM, John W. Tukey became a statistics professor at Princeton and Richard P. Feynman became a Nobel Laureate.

How Many Unknown Flexagon Episodes Have Occured?
The flexagon's first flurries of far-flexed fame were fronted by Martin Gardner and Scientific American. I'm certain though that other people have experimented with flexagons, and in many different ways. For example, take a look at page 121 in W. Lietzmann's 1955 book, Visual Topology. The folded Möbius Band diagram (Fig. 145) and accomanied discussion sure looks like serious math-related flexagon theory to me!

Lietzmann was at the University of Göttingen when this book was first published in German. It was later translated into English by M. Bruckheimer at the Northampton College of Advanced Technology and published in 1965. Did Lietzmann know about flexagons at this time or was he having fun with them but just didn't tell anyone else? Is it possible that M. Bruckheimer left out references to fun applications and also became fascinated? How many hundreds of other brushes with flexagons have been hidden?

Tastes Of Today And Tomorrow
Creative, fun, and complex studies of the trihexaflexagon are always surfacing. For example, the 1997 study of the trihexaflexagon by Hilton, Pedersen, and Walser provides instructions on making one presenting 'Smiley' faces. At times, a pirate like image is exposed (a Pirate-hexaflexagon?). By incorporating this novel approach, a mathematical analysis of the trihexaflexagon is simplified.

It used to be difficult if not even impossible to learn about almost full-spectrum interest/application phenomena. Now, the Internet instantly allows everyone access to cutting edge (or in this case, cutting 'flex') flexagon phenomena. Infinite potential exists for present and future unsuspected applications.

I have included an extensive list of references as my site is primarily dedicated to my new trihexaflexagon version, the version comparison, and my 'Magic' Flexagon. However, I have also included some of my many other highly diverse fumblings and failings see site map. This is in hope that they may catalyze other findings that I am unable to see from my perspectives.

I am very far from being capable of flexagon mathematics like the following, however, I do include references that will lead you in that direction. This equation was how Oakley and Wisner put one of their conclusions about flexagon theory and it appeared in "The American Mathematical Monthly" (March 1957). I copied the equation from "The Mysterious Flexagons (1966) by Madeline Jones.

Links

Flexagons: Good set of colored instructions for both making and flexing a traditional trihexaflexagon. Includes information on tetrahexaflexagons plus introduces higher order flexagons. Nice set of links and references.
Kathryn Huxtable's Flexagon Page: First off, she has quite the interesting interactive Jave based flexer demonstration for a hexahexaflexagon and then one of her artistic creations; How to make a hexahexaflexagon; How to flex a flexagon; And, how to find all hexagons (with a diagram for the dodecahexaflexagon).
Lee Stemkoski (Mathematrix) Good colored instructions on folding the tri-, tetra-, penta-, and hexahexaflexagon plus print-friendly templates.
Adam Walsh: Another flexagon trihexagon and tetrahexagon site. Has a neat generic triangle grid that can be put to use in constructing flexagons.
Alexandro Kapauan: Bit of flexagon history and instructions for making a hexahexaflexagon.
Kjartan Poskitt: How to make triflexagons and hexaflexagons.
Magnus Enarsson: A bit of history of the flexagon plus instructions on making and flexing a hexahexaflexagon.
Crafty Coaster and Kaleidoscope: Commercial applications for trihexaflexagons and hexahexaflexagons. Also a neat animation demonstrating how to flex a flexagon on the Crafty Coaster Site.
David Mitchell's Origami Heaven: Extremely wide spectrum of Origami creativity complexity potential. Also includes a bit of information on silver and bronze flexagons, plus, the hybrid silver tetraflexagons and likely hybrid bronze tetraflexagon.
Check out Pete's Hexaflexagon Software!

Links For Those That Are Serious In 'Flexing' The Envelope

David King: Lot of flexagon stuff here including tangling with some theory. Instructions for making tri-, tetra-, penta-, and hexahexaflexagon. He also provides templates for tri-, tetra-, penta-, and hexahexaflexagon. He definitely has a cool animation of a flexagon flaunting its flexes.
Flexagons: Harold V. McIntosh provides a copy of the huge Flexagons paper by Anthony S. Conrad and Daniel K. Hartline. This includes almost everything known about flexagons when it was published in 1962.
The Theory of the Flexagon: Harold V. McIntosh also provides a copy of the paper, The Theory of the Flexagon, by Anthony S. Conrad.
Infinity12: Hexaflexagons (Trihexaflexagons, Hexahexaflexagons)
Jill Russell - The Magic of Flexagons: This site should further whet anyone's appetite to the infinite fascinations and phenomana to be found with flexagons. She also has both a triangle and square grid for downloading.

Literature

New Flexagon Book! Flexagons Inside Out
This is the long awaited book excellently bridging the gap between presently available introductory flexagon material and highly sophisticated complex applications and explorations.
Gardner, M. (1959). "Hexaflexagons." Ch. 1 in The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster. (I believe this is where I got the idea for my science project one year. I managed to flex my way to the state level really not knowing what I was doing, but, I was having a lot of fun playing with the trihexaflexagon.)
Gardner, M. (1965). Mathematical Puzzles and Diversions. Penguin Books, or Chicago Press.
Gardner, M. (1961). Ch. 2 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 24-31.
Johnson, D. A. (1974). Mathmagic with Flexagons. Activity Resources Company. Nice introduction to flexagons.
Hilton, P., Pedersen, J., Walser, H. (1997). "The Faces of the Tri-Hexaflexagon." Mathematics magazine, Vol. 70, No. 4 (Oct., 1997), 243-251. Thoroughly fun examination of the Trihexaflexagon including instructions on making one. They creatively employ a 'Smiley' face that occasionally appears as a pirate (a Pirate-hexaflexagon?).
Lietzmann, W. (1955) Anshauliche Topologie and (1965) Visual Topology translated by M. Bruckheimer. New York: American Elsevier Publishing Company, Inc.
Madachy, J. S. (1979). Madachy's Mathematical Recreations. New York: Dover, pp. 62-84.
Oakley, C. O. and Wisner, R. J. (1958). "Flexagons." Amer. Math. Monthly 64, 143-154. Quite the exhaustive mathematical theory traversing the Tuckerman Traverse and trends in the flexagon family and infinitely beyond me!

Links to Major Sections:
General Trailblazing Flexagons Site Map
Trailblazing Flexagons Visual Arts Site Map
Vern's Visual Victuals - Personal Digital and Fine Art Creations
'Weak'ly Wit - Vern's 'Pun'derosa
Vern's Views - Opinions and Musings
Invisible Chronic Illnesses - Information and Analyses
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By Vernon Gutenkunst
(B.S. in Chemistry, Army Musician, B.A. in Interdepartmental Sociology)