The Tetra-Octa-Flexagon

Watch the video and follow the instructions to build the tetra-octa-flexagon:

• Download the file below (template number 17 in the flexagon booklet) which has the templates for the back and front of the paper strip that will be folded into a tetra-octa-flexagon.
• Print the two templates using a colour printer.
• If necessary, glue together the back and the front, so that the paper strip is printed on both sides. Make sure you do this the right way, and that your back and front are correctly positioned.
• Crease the paper along the lines, back and forth, so that the paper is flexible and easy to work.
• Fold the paper strip into a square. To assemble, fold each yellow triangle on top of its adjacent twin. Then fold the pairs of green triangles in the same fashion. A square will form with red on one side and blue on the other. Position the white triangles face to face and fix with adhesive tape – and that’s it!

The tetra-octa-flexagon, has eight triangles on its face, hence, octa. It is shaped like a square and has four faces. This is a straight strip flexagon, similar to the tri-hexaflexagon family. Once the flexagon is assembled you get a square, with one side blue, and the other side red.

Notice that the inner side colours are yellow and green and the outer side colours are red and blue. Flexing this flexagon is really puzzling. If you pinch flex red face up, blue face down you go through an intermediate stage which can be opened up into a red and green rectangle, ending with a square that is blue face up and green face down. Continue to pinch flex from this state. You get straight back to red up blue down. So this is the cycle: Red-Blue square faces – intermediate state rectangle – Blue-Green square faces – Red-Blue square faces.

Pinch flexing exposes only three of the four faces: red, blue, and green. These are the dominant faces. The yellow face is the hidden face and it can only be exposed if you can do a special reverse pass-through flex.

To do the reverse pass-through flex, make a pinch flex from the red face but just before you open up to the intermediate rectangle stage, switch all mountain folds to valleys and vice versa. This can be done by a kind of “peel and close” movement for all of the four sides of the flexagon as shown in the video – and there it is – the yellow face!

The reverse pass-through needs time and practice so if you don’t get it at first, try again and don’t give up! You’ll get it in the end.