New  Check out the dodecaflexagon frieze patterns here !
The "ShapeShifting 12gon" has been explored extensively by Ann Schwartz. See:
http://www.eighthsquare.com/12gon.html
As Ann describes well in her article, this is a fascinating dodecaflexagon with a large variety of great flex patterns including nonhexagonal shapes. Each face of the flexagon can have as many as 12 triangles as you flex it. The 12gon discovery of Ann Schwartz is one of an infinite family of dodecaflexagons with a lot of similarity to traditional hexaflexagons but more dynamic behavior when flexing. There are 3faced, 4faced, etc. versions of this flexagon, several of which you can see below. I consider a 3 faced dodecaflexagon to have three basic faces or 3*12=36 triangles. There are actually many actual face patterns and shapes. The Pentadodecaflexagon below has 60 triangles, 5 basic faces and numerous face patterns. The flex variations and shapes of the dodecaflexagons allow for some incredible design patterns to be created.
Les Pook has also published information on dodecaflexagons. See:
http://tech.groups.yahoo.com/group/Flexagon_Lovers/message/310
Les gives some great details of how dodecaflexagons fit into the known theory about flexagons. Les has a nice description of the six fold flex in his recent paper.
Below are great templates for several members of the dodecaflexagons, the tridodecaflexagon or as Ann Schwartz calls it, the Junior 12gon, the tetradodecaflexagon, and the pentadodecaflexagon. They are all composed of hexagonal primary faces each with 12 306090 triangles. Other faces may have varying numbers of triangles and arrangement of triangles. The 3faced dodecaflexagon has surprising dynamic behavior in the face configurations. All members of the dodecaflexagon family have a feature seen in Ann Schwartz's 12gon, the toggle triangles. They are three triangular flaps on some faces that can be folded over to allow new face patterns to be seen and new paths for flexing. See the articles by Ann Schwartz for more discussion on this and many other features of the dodecaflexagons.
A sample of the flexing patterns for the tridodecaflexagon:
Tridodecaflexagon
Template for the 3 faced dodecaflexagon. See below for this template with original fractal faces. 

Here are a few of the many flexing patterns of the tetradodecaflexagon:
Tridodecaflexagon with Fractal Faces
