Hexaflexagons

Last week in my College Geometry class, we had to do a presentation on Origami. For the presentation we did the usual: history,origins, etc... However, when it came time for the group participation, we decided to use something that we saw in our Math Seminar class. I recalled a video that Dr. Milou showed us on hexaflexagons and thought that this would be a perfect example of origami to have the class make. So, my group members and I watched Vi Hart's video on YouTube, then made step-by-step instructions on how to make them.

To make the hexaflexagons, I was able to find templates online that were in color, and came pre-numbered. If the hexaflexagon was constructed properly, each "face" would display the same number, but with 2 alternating colors. When it cam time for the participation part, we just showed Vi Hart's video, but didn't tell our classmates until after the video that they would be making hexaflexagons too. Needless to say, those that have seen, or heard of, hexaflexagons were excited (I actually heard a few "yay's" when we told them). For the classmates that hadn't heard of hexaflexagons, by the time they were done constructing them, they were excited as well. Once they figured out how to flip and fold it to get to the different "faces", most of them couldn't put it down! The video to see what exactly a hexaflexagon is can be seen here: https://www.youtube.com/watch?v=VIVIegSt81k

Here are the steps to make a hexaflexagon using the template found on auntannie.com

.Score and fold back and forth on all the solid lines. Do this for each of the 19 triangles of the hexa-hexaflexagon.
Place the strip of 19 triangles with the side with 6s, 5s, and 4s face up and the glue tab, *, to the left. Fold the strip back on itself matching two 6s, then two 5s, then two 4s, on like this for the entire strip (nine folds). Now you have a strip of 10 triangles.

Place the strip with the side without glue tabs facing up and the 3 end to the left. Fold back on the line between the third and fourth triangles (2 and 1)—four 2s are together now. Fold back on the line between the next 2 and 1. Now you should be able to see five 2s. Pull the 2 forward that is behind the 3. There should be six 2s on top and a 1 to the side. Fold the 1 back.

The two glue tabs marked with '*' are now face-to-face. Glue them together. Wait for the glue to dry before flexing your flexagon.

Step 6: Flex

Fold the flexagon to bring together three alternate corners of the flexagon. The top of the flexagon will pop open if there is a new face to reveal. If it doesn't pop open, flatten the flexagon and try again with the other three corners.
As you flex the flexagon in this project, see if you can find all the possible combinations of faces. The hexa-hexaflexagon has six faces with 3 different arrangements each, for a total of 18 possible faces. Only 15 of the faces versions can be seen with the straight strip pattern for a hexa-hexaflexagon. Can you find all 15 possibilities?