Necessity is the mother of invention, if sometimes only indirectly, as was the case with structures called flexagons. English graduate student Arthur H. Stone faced a challenge when he arrived at Princeton University in 1939: how to fit large American notepaper into a small British binder. So he began trimming the margins from his notes and folding the leftover strips for fun. His idle origami ultimately led to a remarkable entity named a trihexaflexagon, which was shaped like a simple, flat hexagon but could be manipulated to reveal three different faces. He shared it with his friends, including math graduate student Bryant Tuckerman, physics graduate student Richard P. Feynman and mathematics instructor John W. Tukey. Soon a variety of flexagons—with four, five, six, even 48 faces—were born.

Martin Gardner, Scientific American’s long-time Mathematical Games columnist popularized flexagons in his very first article for the magazine in December 1956. Watch him discuss the structures in the video clip below. In honor of what would have been Gardner’s 100th birthday this month the magazine’s October issue features a celebration of Gardner and his work, which still inspires mathematicians and puzzlers today.

How to Make a Three-Faced Hexaflexagon
A trihexaflexagon is formed from a strip having 10 equilateral triangles on each side. Here is how to make one.

Click here for this template that you can cut out
Click here for a blank template for a trihexaflexagon

Step 1. Fold the cutout shown along the central horizontal line (“Fold 1”) and glue.

Step 2. With the side showing letters A and B facing you, fold the crease between the fourth and fifth triangles (when counting from the left), following the line labeled “Fold 2,”) so that the fourth triangle goes behind the fifth (2c faces 2d).

Step 3. Fold the crease between the third and fourth triangles (when counting from the right) along “Fold 3,” so that the third triangle goes on top of the fourth (2f faces 2e).

Step 4. Tuck the bottom triangle behind the one already there (2a faces 2b, going under 1a).

Step 5. Fold the triangle hanging off the bottom (3a) to the underside and glue faces A and B together. Your trihexaflexagon is complete. It will look like a flat, hexagon with two faces.

How to find the third face in a Trihexaflexagon
Step 1. Fold the resulting hexagon shape along all of its seams to break it in.

Step 2. To reveal the hidden third face, pinch two triangles together with each hand and open from the center out. (See the first video here, at about the 1:40 mark.)

It may take a little practice but once your trihexaflexagon loosens up it should rotate easily to reveal three different faces.

Submit Your Photos
Care to try your hand at something trickier? There are excellent nets for six-faced hexahexaflexagons from Gathering4Gardner here. Anyone for a five-faced pentahexaflexagon? A 10-faced decahexaflexagon? A square flexagon?

Whichever kind you make, we invite you to take photos of your creations and submit them to us below. All images of the flexagons shown must fully owned by you. We plan to produce a slide show of some of the most attractive or interesting specimens.