New -  Check out the dodecaflexagon frieze patterns here !

The "Shape-Shifting 12-gon" has been explored extensively by Ann Schwartz.  See: 

http://www.eighthsquare.com/12-gon.html

As Ann describes well in her article, this is a fascinating dodecaflexagon with a large variety of great flex patterns including non-hexagonal shapes.  Each face of the flexagon can have as many as 12 triangles as you flex it.  The 12-gon 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 3-faced, 4-faced, 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 Penta-dodecaflexagon 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 tri-dodecaflexagon or as Ann Schwartz calls it, the Junior 12-gon, the tetra-dodecaflexagon, and the penta-dodecaflexagon.  They are all composed of hexagonal primary faces each with 12 30-60-90 triangles. Other faces may have varying numbers of triangles and arrangement of triangles. The 3-faced dodecaflexagon has surprising dynamic behavior in the face configurations.  All members of the dodecaflexagon family have a feature seen in Ann Schwartz's 12-gon, 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 tri-dodecaflexagon:

Here are a few of the many flexing patterns of the tetra-dodecaflexagon:

Tri-dodecaflexagon with Fractal Faces The Junior 12-gon   Template for the 3 faced dodecaflexagon with original fractal faces. This12-gon template makes a very nice looking flexagon.  I strongly recommend making this one, you will like it!  This is a big 1.7MB file, so be patient when downloading.

 

 

Penta-dodecaflexagon 

 

Template for the 5 faced dodecaflexagon.  This is an incredible flexagon with great dynamic properties.  I recommend making the Tri-12gon or Tetra-12gon before making this one so you can get a good feeling for the dynamic properties of the dodecaflexagons first.  There are two sets of  toggle triangles that will develop in some flexes on the front and back faces of the flexagon.  You can move or toggle these into different positions for access to new faces.  Some faces appear with a gentle twist on the three corner pinch flex.  See blog entry for more discussion on this one.