If you like flexagons, you will certainly like the dodecaflexagons!  These come with some new twists and turns.  I found the articles by Ann Schwartz on the 12-gon quite interesting. 

(see: http://www.eighthsquare.com/12-gon.html, and http://www.eighthsquare.com/12gon2.html.  

Ann's 12-gon was so intriguing, I wanted to explore some of the other members of this infinite family of flexagons, especially the Junior members.  I found that the 3-faced version of the 12-gon has some, but not all the features of Ann's 12-gon.  Even so, I found that It does flex into the propeller shape and a number of other faces and shapes.  In fact it is a rather facinating and complex flexagon with excellent face design possibilities.  I created an easy to cut and glue template for this flexagon with graduated colored triangles and with all triangles lettered and numbered for easy recognition of the face permutations.  Feel free to download this template from the dodecaflexagon menu item and see for yourself what this flexagon is all about. The template results in a nice 5 inch flexagon.

The letters and numbers on the triangles in the template show clearly that some of the faces you might think are identical are really not identical; you will find that the triangles are shifted around.  For example, the face with the three arcs of As in my template has two variations, both with the same arcs.  On some of the faces, when the triangles with the 60 degree angles are in the center the numbers can go clockwise around the center and the outside triangles go counterclockwise in number.  

Success with the tri-dodecaflexagon spurred me on to create a tetra-dodecaflexagon.   A template for this one can also be downloaded from the dodecaflexagon menu page as well.  I created this template with some original fractal faces and also triangle numbers, but not as bold as I did for the 3-faced one.  Photos of some of the faces for both the 3 and 4 faced dodecaflexagons are also on the dodecaflexagon page.   Both the tri-dodecaflexagon and the tetra-dodecaflexagon have the "toggle triangles” described by Ann Schwartz.  These are the three triangular flaps that appear on some faces.  If you move these flaps clockwise or counterclockwise, you will be able to flex into new face patterns.  There are a variety of propeller faces and many other interesting patterns that you can explore.   I am now experimenting with the 5-faced and 6-faced dodecaflexagon variations. When I indicate 3,4, or 5 faces, I mean basic faces.  Each basic face will have 12 triangles, so a 5 faced dodecaflexagon will have a total of 60 triangles with 5 basic faces.  I would love to hear from anyone that builds one of these from my templates.