The Dangerous and Exciting World of Flexagons!

Tri-hexaflexagons, Tetra-Tetraflexagons, and Hexa-

dodecaflexagons

Fortune-Tellers

• Have you ever played with a “fortune- teller” or “cootie-catcher” like the one below?

• Today we are going to learn about a math oddity called flexagons, which some people say remind them of these!

How Were Flexagons Discovered?• Arthur Stone, a graduate student at

Princeton in 1939, was in the practice of doodling with the strips of paper that he cut from around the edges of his notebook paper.

• One of the constructions he made caught his attention in particular.

• A committee of graduate students formed to solve the mystery of the “flexible hexagon,” or, as it soon became know, the “flexagon.”

What the Heck is a Flexagon?• At first glance it looks like a folded

hexagon or square, a fortune teller or a piece of origami.

• But look closely and you'll see hidden layers lurking between the front and back.

• When you fold or pinch corners together, the flexagon "flexes," meaning a formerly hidden layer will come to light as the top layer folds underneath.

Mathematicians and Flexagons• Mathematicians refer to flexagons as

"mathematical oddities." • That's because flexagons have very

complex mathematical structures. • Mathematicians enjoy analyzing the

structure and behavior of flexagons.

The Tri-Hexaflexagon

• The Tri-Hexaflexagon is the simplest of the flexagons.

• See an online example.• Now make your own!

The Tri-HexaFlexagon

• How many vertices does your flexagon have?

• How many edges does your flexagon have?

• How many faces does your flexagon have?

The Tri-HexaFlexagon

• How many triangles make up your flexagon?

• How many rhombi make up your flexagon?

Click here to see a movie!

Will's website2.mov

The Tetra-Tetraflexagon

• Another kind of flexagon is the tetra-tetraflexagon

The Tetra-Tetaflexagon

• Let’s build our very own tetra-tetraflexagon

Front Back

The Tetra-Tetraflexagon

• Why do you think there are TWO tetra-’s in the name?

• How many faces does it have?• How many edges does it have?• How many vertices does it have?• Online Example

The Hexa-Hexaflexagon• The hexa-hexaflexagon is a most

famous flexagon!• How many faces and sides do you think

it has?• We won’t be building one, but here is a

simulation of the Hexa-Hexaflexagon.

The Hexa-Dodecaflexagon

• The amazing flexagon has twelve sides and six faces!

• It flexes into a non-hexagonal shape and mixes triangles from its six basic faces to form new different faces.

• Wow!

Re’ flex’ ions

• I hope you’ve enjoyed learning about flexagons!

• You are now a certified

Flexa-maniac!!