Hexaflexagons

THE HISTORY

• It was a just another crisp fall day in 1939....
• or was it?
• Actually, it was. It was a normal day for Arthur Stone, a graduate from England studing at Princeton University in New Jersey.

THE HISTORY (cont.)

• Arthur was just getting used to life in the Americas.
• Americans, he came to see, were quite strange.
• While he was putting his American paper in his English binder, he came to realise something... his paper was sticking out from the sides!

THE HISTORY (cont.)

• Arthur was annoyed. He tried to trim the extra paper off while he half-listened to his boring math professor.
• Arthur now had a lot of extra pieces of paper lying on his desk. He decided, for his own enjoyment, to fold them into little shapes.

THE HISTORY (cont.)

• He folded the papers into mini pyramids, little squares, anything really.
• As he was folding the papers, he happened to fold them into a symmetrical hexagon shape.
• How entertaining.

THE HISTORY (still cont.)

• As he fiddled with the hexagon, he noticed he could fold in the sides of the hexagon to make a flap.
• He unfolded the flap. To his amusement, the hexagon seemed to flip inside out!
• He flipped it again... and again.... and it worked!

THE HISTORY (still cont.)

• Arthur's perked at his new discovery, and he immediately made more.
• He knew just what to name it... the hexaflexagon!
• The other students looked at the new exchange student weirdly as he huddled over his desk, completely enamored with a strip of paper.

THE HISTORY (still cont.)

• The other students decided that the British were strange.
• ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
• At lunch, Arthur was still 'flexing' the hexagon he created.
• A few of his friends were interested in Arthur's new obsession.

THE HISTORY (still cont.)

• Among these friends were....
• Richard Feynman
• Bryant Tuckerman
• John W. Tukey
• (not turkey.... he didn't like it when he was called turkey.)

THE HISTORY (im sorry its still cont.)

• As Arthur showed his friends the way of 'flexigation', (it's actually a term guys) they decided to form a flexagon comittee.
• They called themselves hexiflexigators. (I'm not even joking)

Hexaflexagon Categories

• Trihexaflexagons (3 faces, easiest to make)
• Hexahexaflexagons (6 sides, pretty difficult to make; easy to mess up)
• Heptahexaflexagon (7 sides, not for the faint of heart)
• Dodecahexaflexagon (12 sides. Don't even try me m8)
• first one to tell stand up say all of those without stumbling on their words gets anything at snack cart on me!

Tuckerman's Traverse

• Bryant Tuckerman was particularly interested in the mathematics behind hexaflexigation.
• He thinking one day about how to reach all the sides of his hexahexaflexagon the easiest way possible.
• And then it came to him!

Tuckerman's Traverse

• He created a quick sketch of his drawing.
• It was quite sloppy.
• His friend, Richard Feynman was nearby, and saw Bryant stuggling to make a diagram. He decided to help him out.

Tuckerman's Traverse

• And Feynman drew a great map of what Bryant was thinking. With this diagram, it could help anyone reach the goal of finding every side on a hexahexaflexagon with ease.
• Bryant named this method the Tuckerman's Traverse, appropriately so.

Feynman Diagrams

• Feynman also named the diagram he made... the Feynman Diagram, which later became the Feynman Diagrams.
• But later on in life, Feynman became great at these drawings.
• It is decided that without hexaflexagons, Feynman might have never created his diagrams, which his quantam physics diagrams helped create the first atomic bomb.

Tuckerman's Traverse

• We'll leave the Feynman Diagrams for another time. Let's discuss the Tuckerman Traverse!
• First of all, what the heck is a 'traverse'?

Tuckerman Traverse

• According to freedictionary.com, a traverse is "anything that can move to and fro, cross and recross."
• This diagram shows the most efficient way to get to all of the sides, by flexing them
• Yes, someone actually took their time to figure this out...

Tuckerman's Traverse

• The Tuckerman's Traverse shows that when you flex the flexagon, it will somehow cycle around and around again and again. It seemed to Bryant Tuckerman that they just went in a loop, but he actually found out that they were in different states each flex.
• 1 cycles to 2, 2 cyles to 3, and goes back to one. But from 1, 2, and 3, you can cycle to 4, 5 and 6!

Tuckerman's Traverse

Hexahexaflexagon Game!

Catalan Numbers

• Catalan Numbers... yes, there's actually math in this
• Catalan numbers are numbers that are recursively defined. A recursively-defined number is, by dictionary.com, something that is characterized by recurrence or repetition. Basically, this number is showing how many times the hexaflexagon can be in different states.

Catalan Numbers

• The number that defines the hexahexaflexagon is 42. There are 42 ways that the flexagon can be in different states. That's why it made it difficult to make out what shapes were what when we played the game.

Catalan Numbers

• The Catalan numbers can be calculated by using the formula (2n 2)/n+1. The sequence goes as follows: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796… and it goes on and on and on. But we can see ‘42’ on that list. This is how many times the hexaflexagon can be flexed with the faces being in different places and states, all because the handy-dandy Catalan formula figured this out for us.

Catalan Number Formula

Catalan Numbers

• These can be difficult to grasp. It is a level of math that is quite over my head, so I'm sure it's over yours :D
• Basically, the formula shows how to calculate the recursively defined number in a sequence. This is how we can understand the hexahexaflexagon!

Thanks for listening!

Now you can make your own!

Websites :D

• http://britton.disted.camosun.bc.ca/hexahexa/hexahexa.html https://www.easycalculation.com/maths-dictionary/catalan_numbers.html https://www.stat.berkeley.edu/~brill/Papers/life.pdf http://flexagon.net/index.php/hexaflexagons/mpas-8-sides http://encyclopedia.thefreedictionary.com/catalan+number https://prezi.com/jfymwd9wkt9_/the-mathematical-mystery-of-flexagons/ http://www.maa.org/sites/default/files/pdf/pubs/focus/Gardner_Hexaflexagons... http://mathworld.wolfram.com/Hexaflexagon.html