I stumbled upon Hexaflexagons from Vi Hart's YouTube videos.

She seems to prefer making them by hand. But I like to make them using these templates I designed.

I found that Escher tessellations make some cool hexaflexagons. However, they don't always work. Especially in some of the hexa-hexaflexagon configurations. Analyzing why some worked and some didn't led me down the path of figuring out the state-transition diagram for hexa-hexaflexagons.

For this to work, you need to use a pattern like this with indexed faces. You can find them at the bottom of this page.

The state-transition diagram for this hexa-hexaflexagon will look like this:

Here is how to read this diagram. Each colored rectangle node represents one face, and its backside. I numbered and colored each face for clarity. And each face can have up to three configurations, A, B, or C. Which configuration you are in is designated by the letters in the center of the hexagon.

For example 1A/3A in the middle of the diagram: . Has this on front: , and this on the back: . Or vice, versa.

Folding one corner "down" is what follows that edge. For example, if we are looking at face 1A, then fold corner "B" down and open up the hexaflexagon, we will then be looking at face 2B.

That is basically all there is to it. The only thing else to point out is that some transitions are impossible.

Inspecting this diagram, we can find out some interesting things. For example, there are 9 unique double sided states. This is 18 faces. But only 15 faces are unique.

The inner circles go clockwise (A → B → C → ...). The outer circles go counter-clockwise (C → B → A → ...). Each of these double-concentric circles is the state transition diagram for a tri-hexaflexagon.

Some faces show up more than once, in different locations. Namely, 1C, 2A, and 3B each show up twice. And some faces don't show up at all. Namely, there is no 4C, 5B, or 6A. Here is the list of all faces:

1: A B C C
2: A A B C
3: A B B C
4: A B
5: A C
6: B C

Tri-Hexaflexagons

These templates use a standard Letter or A4 size sheet of paper.

PDF Format - SVG Format - A Hawaii themed pattern.

PDF Format - SVG Format - Solid colors and numbered faces for figuring out your own patterns. The black and white pattern makes a neat trick: Write a secrete message on the white face then fold it into the inside to keep it safe. Only people who know how to unfold a hexaflexagon will be able to find it.

PDF Format - SVG Format - Simple color diamonds, cool patterns, and a squirrel themed pattern.

PDF Format - SVG Format - A Roomba 980 themed pattern.

PDF Format - SVG Format - Another iRobot themed pattern with different robots.

PDF Format - SVG Format - This is probably my favorite pattern. On the left, we have the faces of the three discoverers of hexaflexagons: Richard Feynman, John Tukey, and Bryant Tuckerman. In the middle we have some Escher tessellations. And on the right we have some other cool patterns that look different in each configuration.

PDF Format - SVG Format - Some Celtic Knot type patterns, a blank, and smiley faces. The smiley faces are fun because the turn happy and sad. And the one with the eye-patch changes which eye it is on.

PDF Format - SVG Format - Some animal themed patterns.

Hexa-Hexaflexagons

These templates require an 11x17 inch sheet of paper.

Folding a hexa-hexaflexagon can be challenging. So here are some folding instructions. They follow the same color pattern that I use on all the solid color patterns here.

PDF Format - SVG Format - A blank, solid colors, and some tessellations. I'm not sure the tessellations work in all configurations.

PDF Format - SVG Format - A blank, colored and indexed faces, and some tessellations. I'm not sure the tessellations work in all configurations.

PDF Format - SVG Format - Solid colors, indexed faces, and a work in progress. I couldn't decide what I wanted on the other faces.