This page shows how one can construct hexaflexagons with any number of sides or flex patterns desired. I am not going to discuss the theory behind this method, but rather give you the steps to follow without explanation of why. This procedure is based on the 1962 papers of Antony S. Conrad and Daniel K. Hartline, see their papers, "Flexagons", and "the Theory of Flexagons" if you want to understand the underlying mathematical concepts. Additional papers by Harold V. McIntosh can be found at the same link . Of course the following procedure is also based on the same principals that the original Princeton flexagon committee developed.

First, one needs what I will call flexagon construction mapping units, they look like this:

You can use different colored pencils or pens to draw a larger triangle with one color and a smaller triange in the middle of another color. Now, decide how many faces you want on your hexaflexagon. If you want your hexaflexagon to have n sides, then you need n-2 of the above construction units. So for a tri-hexaflexagon, you need one construction unit and for a 9 sided hexaflexagon, you need 7 construction units.

Arrange the construction units in any configuration you like, with two rules.

1. The sides of the larger triangles must match each other.

2. The inside triangles must form a single path that traverses all side points. There cannot be isolated loops in the small triangle network. As an example, one cannot arrange 6 construction units into a large hexagon which results in the small triangles forming 2 separate inside loops.

For this article, I will use a six sided hexaflexagon as an example. So, if we want to create a six sided hexaflexagon, here is one way to arrange the construction units:

Contiunue placing the triangles until you have placed all three repetitions of the table, which will be 36 triangles. In the case of our example the frieze pattern results in a hexagon pattern with one open edge when we complete the three table repetitions. (Remember that it is the first six columns of the table that repeat.)

The next step is to mark each of the triangles, front and back, as to which flexagon side they belong to. By doing this, it will be easy to fold and complete the hexaflexagon. The numbers for the top view of the frieze are given by the first row of the table starting from the initial triangle that we started with in creating the frieze pattern. In the diagram below, that is the yellow #1. Procede to number the tirangles in the same order they were layed out in the R and L mapping. Note the three repetitions of the table pattern in the below diagram by the three different colors for the number sequence. The reverse side of the frieze will be numbered by using the second row of the table we created. Again the numbering follows the placement of the triangles as they were constructed. (the back side will be numbered clockwise and front side counter clockwise). See below how this looks.

This flexagon is now ready to be cut out and folded! When folding, match pairs of numbers. For example, fold the three pairs of 4's together, then the 2's, etc. until the front and back face of the hexaflexagon have the same numbers on their faces. Tape the last two triangles together and flex!

There are a few more advanced topics if one wants to discover and create every variation. It turns out that the flexagon map we built at the start of this article can twist and turn on itself. So the map can be 3D for some flexagons. This is also true for the friezes as well. Many of the friezes for hexaflexagons of large numbers of sides will twist back and around on themselves. An example of this is on the page for the 12 sided Zodiac flexagon model of this website. Sometime I will write up an advanced hexagon creation article to cover these cases and a few other topics.