The Mathematical Mystery of Flexagons It all started in math class... But wait...

What is a anyways? Flexagon A "flexagon" is a "flexible hexagon" In 1939, at Princeton University, Arthur H. Stone was the new kid from England. Annoyingly enough,

his American paper was

TOO WIDE

for his British binder So while his math teacher lectured about parabolas

and his classmates obsessed over his accent,

Stone simply cut off the ends of his notebook paper And he was left with these strips of paper

just BEGGING to be FOLDED! Stone found many...um...creative..ways to fold his paper scraps,

until he finally discovered... FLEXAGON! The Which would bring about an addictive & puzzling

form of mathematical recreation

for years to come! * Flexagon Committee Arthur H. Stone Bryant Tuckerman Martin Gardner John W. Tukey Richard P. Feynman Do the Tuckerman Traverse! The Revival of the Flexagons Or a hexagon that you can flex,

if you will. As you can see, the mystery of flexagons comes in when you realize that a flexagon

is a simple folded strip of paper that can be flexed or folded to hide and reveal different faces besides the two original faces that were on the

front and back. You may be wondering, are there any variations of the original flexagon?

I'm glad you asked... There are many different types of Flexagons: Typically, a flexagon is made by folding the strip of paper into triangles.

In which case, you might end up with a... If you decide to fold your strip of paper into quadrilaterals instead, you might end up with a... Tetraflexagon Pentaflexagon Hexaflexagon Heptaflexagon Octaflexagon Enneaflexagon Decaflexagon Dodecaflexagon Square Tetraflexagon Kite Flexagon Rhombus Flexagon Trapezoid Flexagon * Evidence of flexagons in pre-war Vienna's elementary schools supports that Arthur Stone was not the first to discover the flexagon. Still, he was the first to bring its mysteries into the modern world. Or, if you're a real over-achiever, a As the story goes, it started in math class & continued in the cafeteria... Or the rather strange-looking Hello chap, check out what I made during math class!

I call it... THE FLEXAGON!

I colored the sides and everything! Dude, that's mind blowing! It reveals new sides out of nowhere! But what's the quickest way to get to all of the colors (sides)? Flexagon

Safety Guide *Please read before embarking

on your own flexagational journey! By George,

I haven't quite figured that out yet. Hey! If you guys don't

make a diagram, I will! Yo man, I have an idea!... During lunch, Arthur decided to show his friend, Bryant Tuckerman, the wonders of his mystically mathematical discovery. Another friend, Richard Feynman, budged into the action.

Thus, forming the Flexagon Committee! But before we review the committee, let's take a look at what Tuckerman's idea was. is camara shy Bibliography The Tuckerman Traverse is the fastest path one can take to reach all six sides of a hexaflexagon (even though it may sound like a dance craze), named after Bryant Tuckerman. 1 3 3 1 3 2 2 4 5 6 Richard Feynman, perhaps being more of a visual learner, created a model of Tuckerman's method. The model was titled a Feynman Diagram Wondering how it works? Well, for example, let's say you started on side 1. According to the diagram you could "flex" to side 6, 2, or 3. Therefore, if, for example, you flexed to side 6, you could flex to side 3 followed by side 1, or simply flex backwards to side 1. Basically, a Feynman Diagram is a

map to the sides of a flexagon. 1 3 1 3 2 2 4 5 6 Now, that we've already established some of its accomplishments, let's take a look at the Flexagon Committee. consists of people dedicated to the exploration, discovery, and expanse of the Flexagon. The You may also be wondering why there are two of some numbers on Feynman's diagram.

John W. Tukey, yet another friend of the flexagoners, discovered that there is more than one side for some numbers on a flexagon.

It sounds strange, but this is yet another exmple of the mysteries of the flexagon! You may remember Tukey as the guy who was snickering at the kid across the lunchroom who spewed milk from his nose. Wait, who's this guy? Unfortunately, after the original committee disbanded, the flexagon was forgotten for many years. Until, Martin Gardner earned his rightful honorary position on the committee for sparking... In order to meet Gardner, we must fast forward from the 1930s-40s to about a decade later. "Flexagon." AP CALCULUS RSS. Alphy Emily, n.d. Web. 21 Dec. 2012. <http://apcalculusbc.com/flexagon-2/>.

"Flexagon." Flexagon. N.p., n.d. Web. 21 Dec. 2012. <http://www.dcgeorge.com/gpage12.html>.

Hart, Vi. "Hexaflexagon Safety Guide." YouTube. YouTube, 15 Oct. 2012. Web. 21 Dec. 2012. <http://www.youtube.com/watch?v=AmN0YyaTD60>.

Hart, Vi. "Hexaflexagons 2." YouTube. YouTube, 08 Oct. 2012. Web. 21 Dec. 2012. <http://www.youtube.com/watch?v=paQ10POrZh8>.

Hart, Vi. "Hexaflexagons." YouTube. YouTube, 01 Oct. 2012. Web. 21 Dec. 2012. <http://www.youtube.com/watch?v=VIVIegSt81k>.

Lamb, Evelyn. "Flexagon but Not Forgotten: Celebrating Martin Gardnerâs Birthday | Observations, Scientific American Blog Network." Flexagon but Not Forgotten: Celebrating Martin Gardnerâs Birthday. Scientific American, 19 Oct. 2012. Web. 21 Dec. 2012.

"Martin Gardner and Flexagons." Video -. Japan on Top, n.d. Web. 21 Dec. 2012. <http://japanontop.com/view.php?video=zo2XG9_pXvQ>.

Sherman, Scott. "Flexagons." Flexagons. N.p., 2007. Web. 21 Dec. 2012. <http://loki3.com/flex/index.html>. His first article told the magical story of the flexagon. For 25 years, Gardner continued to write about the wonders of mathematical recreation. Also, he went on to write multiple books, short stories, poems, and articles on topics ranging from mathematics to religion.

His works have remained popular among mathematicians and the average Joe alike. Many were intrigued by Gardner's column.

The popularity of the article gave a rapid revival to the popularity of the flexagon! Starting in the 1950's, Gardner was writing for the magazine, "Scientific American".

His column was known as "Mathematical Games". Soon countless people were enjoying the mathematical mysteries of the flexagon, and the flexagon finally gained the fame it so rightfully deserved. YES!!

I finally found the purple side!!! That's right, Jimmy! That's a hexaflexagon! Duuuude, we've GOT TO get some

of these flexagons everyone's talking about! Excuse me!

Has anyone seen my dodecaflexagon? Epologue Thanks to the work of many dedicated flexagoners, flexagons have been able to withstand the test of time. From pre-war Vienna into the 21st century.

Today, you can find your nearest flexagon in Caroline Young's math binder, having been inspired by her research. Math is much more than using numbers in the correct order to find a specific answer. At its core, mathematics is about the innovation, creativity, and excitement of the discovery of our world in ways unheard of. With a flexagon and all the possibilities in the world at hand, who wouldn't be awaiting the day her paper is too wide or too short so that she might just make the next great discovery.

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# The Mathematical Mystery of Flexagons

Caroline Young
Honors Pre-Calculus
period 3

by

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