Hexaflexagons: Mind-Boggling Paper Folding

January 26, 2016 | Elizabeth Knowles

6 different faces of a hexaflexagon in 12 different orientations
Photo credit: Anna Frodesiak/Wikipedia CC0

Math: It’s almost like magic.

We’ve all heard the complaints about math: it’s too hard, it’s boring, when is anyone going to actually use trigonometry? Martin Gardner was a great proponent of mathematical games — a wonderful way to make math fun. One of the neat mathematical concepts he wrote about is hexaflexagons. If you’ve never heard of these before, prepare to be mind blown!

According to a story, Arthur H. Stone came across this neat paper folding when he moved from England to the United States. His paper was too wide for his binder, so he cut a strip off the side and started experimenting with folding it. He realized that he could make a hexagon and that it could be flipped inside out. The really cool thing was that once flipped, it uncovered more than one “inside” face.

A simple hexaflexagon is made by folding a strip of paper into 10 equilateral triangles, and then into a hexagon by gluing the last triangle to the first. This is called a trihexaflexagon because it reveals three different sides.

how to make a flexahexagon

A more complicated hexaflexagon, a hexahexaflexagon, can be made by starting with 19 equilateral triangles and folding them onto each other to create the strip of 10 triangles that you started with for the trihexaflexagon. Here, when fully folded, six sides can appear.

How to make a hexahexaflexagon

Once your hexaflexagon is created, you can experiment with cutting it apart and seeing the pattern of colored triangles that appears on each side of the strip. Bryant Tuckerman and Richard Feynman, two other mathematicians who at the time were grad students with Stone, were fascinated by the order in which the colors appeared. They came up with what is known as a Tuckerman Traverse — the shortest way to get from one color to the next — that can be seen in a Feynman diagram, a diagram that shows what face you can get to from any other.

A Tuckerman Traverse

Interestingly, in a hexahexaflexagon, each of the colored faces can appear in two different orientations such that, if you were to draw a six-pointed a star at the center of your hexagon, it would sometimes appear around the center and sometimes around the outer edges.You’ll also notice that some faces appear more often than others.

In the video below, mathematician Vi Hart folds her hexahexaflexagon at lightening speed and explores the wonders of this mathematical sensation. She also has a second video that goes into more depth about the Tuckerman Traverse and Feynman diagrams:


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