Earlier this year I had been exploring the dodecaflexagon frieze patterns, and have just recently had a chance to put together the maps for the 3, 4, and 5 sided dodecaflexagons.  I have posted diagrams for 7 of these frieze patterns on the first dropdown item of the dodecaflexagon menu item. 

Scott Sherman (see: http://loki3.com/flex/dodeca.html ) recently made an important observation in the Yahoo flexagon-lovers newsgroup  that patterns for flexagons made with polygons of differing length sides can have more variation in flexagon frieze patterns than for flexagons created with the same polygon with all sides equal.  So, the dodecaflexagons not only have more varied face patterns than the traditional hexaflexagons, but, they also have more varied frieze patterns as well.  All three of the 4 sided frieze patterns are rather nice symmetric geometric patterns.  The five sided friezes are comparatively more complex geometric patterns.

For one of the 5 sided dodecaflexagon frieze patterns I have a down loadable template on www.flexagon.net.  A second pattern is composed of a rather straight strip of triangle groupings.  This one will fold up into the 4-sided hexagon frieze pattern after six initial folds and then into the spiked pattern of the 3 sided dodecaflexagon after 6 more folds.  The third 5-sided frieze is composed of 5 triangle units.  Each of these units and it's reflection repeat alternatively  3 times to become the frieze for the pentadodeca flexagon.  The frieze is a 3D twisted spiral which after 6 folds can become the wonderful 6 pointed star pattern for the 4 sided dodecaflexagon. The symmetrical beauty of this flexagon is one to appreciate from the twisted spiral frieze (see photo on www.flexagon.net) to the intermediate patterns as it is folded, to the menagerie of flexed face patterns in the complete model.  Flexagon frieze patterns can be just as interesting as the flexagons themselves.

When I get a chance I plan to post the patterns for the 6 sided hexadodecaflexagons.

Flexagons are perfect for creating dynamic magic squares (magic stars etc.)  There are a great number of ideas that can be explored.  Square flexagons are a nice place to start as they already have the geometry of a magic square.  Starting with 4x4 or 8x8  magic squares where the sub 2x2 and 4x4 squares are also magic, square flexagons can be created that retain magic sums even after flexing.  I created a template for one of these you can download from the Magic Flexagons menu tab.  I choose 4 famous magic squares to display on each of the 4 sides of a cyclic square flexagon. It is easy to make and of course has a "Magic" sense to it.  This idea can be expanded into any of the infinite varieties of square flexagons.

Magic Flexagons can also be created from a traditional three-sided hexaflexagon with excellent dynamic properties.  For example, check out the Magic Trihexaflexagon template on the Magic Flexagon tab.  For this template, the three numbers in the corners of all triangles add up to 42 on all flexagon faces.   The sum of the six numbers around the inside center point on each flexagon face will always add up to twice 42 or 84.  The outside numbers around every face will always add up to 168.  When trying to create 6 magic sums for each of the 6 triangles on the faces of the flexagon while at the same time trying to make magic sums for the two rings of 6 and 12 numbers on the faces of the flexagons and preserving these sums even when the faces are jumbled by flexing is challenging.   I found one way to construct the numbers for this flexagon is to create a 3x6 magic rectangle for each face such that the unique conditions of the flexagon are all met.  As it turns out, 3x6 magic rectangles using the numbers 1-18 is not possible.  Magic rectangles can only be created from rows and columns that are both even or odd (except 2x2).  That is why my flexagon faces have numbers greater than 18 in it. I believe it is possible to created 3x6 magic rectangles with prime numbers, but my initial look at it showed there would be a lot of 3 and maybe 4 digit numbers in the solution.  I am sure there are numerous other solutions using interesting numbers sequences for this flexagon.

Now to look at a Magic Flexagon with even more interesting number combinations, I created a Super Magic Star Trihexaflexagon.  I considered six pointed magic stars (six is the magic number for hexaflexagons!) where there are 4 numbers arranged from each star point to another star point. There are 6 of these four number groups in each super magic star.  Each of these number sums add to 26 and in this case every super magic star face has only the numbers from 1-12.  I chose magic stars where not only the arms add up to the magic number, but also the points add up to the magic number of 26.  There are only six - 6 pointed Super Magic Stars where the points of the star also add up to the magic number (26).  The 3 sided hexaflexagon has exactly 6 face variations.  I decided this was a perfect match and created a Magic star flexagon where each face displays one of the six - 6 pointed Super Magic Stars.  I color coded all the variations so each magic star is easily seen on all six of the flexagon faces.  Download a template for this flexagon also from the Magic Flexagon tab.  The unique concept for this flexagon was in creating a design that would represent Super Magic Stars on the flexagon faces.  The result makes for a fascinating flexagon and also stimulates dozens of new ideas for other Magic Flexagons!  I can imagine a huge number of Magic Flexagons of many different types can be created from the infinite flexagon bestiary.  I would love hearing from anyone who makes these models.

As flexagons aquire more faces, you will find that there are multiple ways to fold a frieze pattern (template) into flexagons that have different flexing patterns (state diagrams or traversals).  As with most flexagons, there are an infinite number of hexaflexagons and all hexaflexagons with more than 6 faces have frieze patterns that will fold up into multiple varieties of unique flexagons. Take a look at this table for hexaflexagons:

The numbers of friezes with multiple foldings grow rapidly. For the decahexaflexagon, only one freize does not have multiple foldings! One decahexaflexagon frieze will fold up into 12 different flexagons. 

I wanted to continue pursuing the possibilities of friezes with multiple foldings.  I choose the 12 faced Hexaflexagon, the dodecahexaflexagon, with a straight strip of 36 triangles. This flexagon has four possible ways to fold up.  You can see the Tukey construction diagrams for each of these four variations and download a complete 9 page template from the hexaflexagon page. 

On this template, you will find that each of the 72 dodecaflexagon triangles is coded with four unique symbols so that it is fairly easy to see how to fold up each of the four variations by folding together matching symbols in one of the four - 12 symbol sets.  This flexagon is rather fun to fold up each variation in different ways to see all the intermediate patterns.  These are really quite fascinating.  Cut and folded accurately, it flexes amazingly well for having 12 faces. Each of the folding variations have their own unique faces. The theme for  the flexagon was inspired by the 12 faces.  It is a Zodiac theme with 48 unique symbols in 4 Zodiac symbol sets and with each symbol numbered from 1-6 for a total of 288 different symbols.  These are the most complex flexagon plates I have ever created.  The Chinese Zodiac would also make for a nice set of 12 symbols ....  maybe for a future project.  Let me know if anyone succeeds in making one of these!
 

There are four variations to the 7 faced hexaflexagons.  Of course, as with all hexaflexagons, there are mirror versions.  So the 4 heptahexaflexagons have 4 mirror foldings.  You can make these by folding the flexagons in the opposite direction.  For example, instead of folding all the G triangles together, fold them apart.  It is interesting to fold up the mirrors to all the flexagons to see the variations in the face patterns. Try it.

A an example, an interesting experiment is to fold up mirror versions of my B & C heptahexaflexagon variation.  If the weekday folding pattern is folded backwards to produce the mirror folding, the result ends up with no faces with all weekdays, and two faces of several arrangement of triangles that have all the same letters (Bs and Ds) from the letter folding variation. If the letter variation is folded backwards, there are no faces with all the same letters, but two faces with all the same weekdays (Tuesdays and Thursdays). The G face which was the same (the only face that is the same in the two variations) in both regular foldings becomes split in both of the symmetric foldings. The surprise in this is that when folding the symmetric to one variation, two faces will show up from the regular opposite variation.  You will also notice that these faces that show up in alternate mirror foldings of the B & C heptahexaflexagon variation have triangles that are not adjacent in the frieze code (template).  Higher order flexagons have more and more faces created from non-adjacent triangles and also more and more frieze patterns that will fold up into flexagons with unique Tuckerman traverses.

It is interesting to look deeper at the higher order flexagons with duplicate foldings from a single frieze.  There is an octahexaflexagon that has a frieze with 3 duplicate foldings, a nonahexaflexagon with a frieze that has 6 duplicate foldings and one dekahexaflexagon has a frieze that has 12 duplicate foldings.  I have been considering some ideas for creating unique patterns for these frieze codes.  I suspect there are many interesting properties for these flexagons and also some v-flexing possibilities.  As long as the mobius half twists in the different foldings are the same direction, it should be possible to v-flex between them.  I do not know how hard it would be to prove this.  I also do not know how to determine what the least number of v-flexes it would take to flex from one folding to another. 

Two final notes;  If you are following my Tuckerman traverse diagrams, they will work when the flexagon is repetitively flexed from one of the sides.  You will have to experiment to find the correct side.  Also note that all my templates use vector graphics for the lettering and triangles.  So, in Adobe Reader, be sure to set the page for landscape and click on "fit to print area".  If you have access to a color printer that will print 11x14, and you click on "fit to print area" you will get a nice size flexagon.  It is also possible to print them even bigger if you have a color printer that will print even larger pages.

I first came accross the pattern for the Tri-dodeca flexagon in Anthony Conrad's paper, The Theory of Flexagons.  That is the pattern that I used to make my first models.  I realized later that this exact pattern shows up in Ann Schwartz's folding sequence of her 12-gon and it also showed up in Harold McIntosh's paper, Trigonal Flexagons.  No one, until Ann's articles, seemed to know, including my self, that this flexagon has such varied dynamic properties.  In the standard flexing of the dodecaflexagons, the variations of faces can only be matched by the hexahexaflexagon when using the V-Flex.  There are also numerous non-standard flex moves for the dodecaflexagons that add another layer of interest and possibilities. 

My pattern for the tetra-dodecaflexagon also shows up in Ann's folding sequence for her 12-gon.  In fact the tetra-dodecagon has the same dynamic properties of Ann's 12-gon.  My pattern for the penta-dodecaflexagon is a new dodecaflexagon model, with an added layer of dynamic complexity.  The front and back toggle triangles are the best example of a new dynamic feature introduced by the penta-dodecaflexagon.  I designed the penta-dodecaflexagon based on general flexagon theory presented in various papers.

Penta-dodeca flexagons have some new twists.  After having created the tri and tetra dodecagon flexagon templates, I was so impressed with this family of flexagons, that I had to create a 5 basic sided version, the penta-dodecagon flexagon.  This turns out to be even more interesting than the tri and tetra versions!  The Penta-dodecaflexagon has 5 basic faces and 5*12 or 60 triangles.  The number of face variations that you can flex is numerous and much bigger than 5.  Flexing will take you into hexagonal, kite, and many other shapes both two and three dimentional.  If you want an easy way to make one for yourself, my net for the penta-dodecaflexagon can be downloaded from the dodecaflexagon tab.

Several people have written about the existence of flexagons built from 30-60-90 triangles, most notably Harold McIntosh in his paper, “Trigonal Flexagons”, Anthony Conrad in his paper, "The Theory of Flexagons" RIAS Technical Report 60-24, and Les Pook in his book, “Flexagons Inside Out”.  Harold McIntosh actually had the net for the dodecaflexagon diagrammed in his paper, but stated that the best that could be done with it was a flexacup.  Anthony Conrad also had the tri-dodecagon template and stated that it could be made into a flexagon but did not elaborate further.  But then it was Ann Schwartz who independently found a net that produced her “Shape Shifting 12-gon” based on the 30-60-90 triangles.  This discovery has brought much new interest and discussion about this type of flexagon due to the fascinating dynamic properties of this flexagon. Ann's 12-gon is actually equillivant to my tetra flexagon template.  The nice aspect to Ann's design is that you can make it from a straight strip, it just has some triangles that do not show up in flexing.  Les Pook has released a draft paper on these new dodecaflexagons, titled, “Dodecaflexagons ” to the Yahoo “Flexagon Lovers” newsgroup.  He describes them as being a stellated degenerate ring dodecagon flexagon. He has an excellent discussion of Tuckerman diagrams (state diagrams) as they relate to these type of flexagons.  As Les states in his paper, there is a lot more to learn about this family of flexagons.

There is a lot to explore with this dodecaflexagon.  The toggle triangles appear on front and back faces for more options when flexing and they also open to new faces by shifting one set or both of them clockwise and/or counterclockwise.   If you fold up my template as I describe, you will find all five sets of colors/letters will each appear on one or more faces in various configurations when flexing.  A first challenge is to find the face with all Ds where all 12 Ds circle the center with the 30 degree angles in the center.  It is not easy at first to find this face, but I assure you it is there to be found via standard 3-corner pinch flexes and shifting of rogue triangles.

Numerous interesting faces can be obtained by three corner pinch flexes and shifting of the rogue triangles on front back or both.  I suggest that faces created in this way are the proper or standard faces of this flexagon.  If you make good creases with the three corner pinch folds after it is first assembled, the penta-dodeca will flex quite well and not get tangled up.  Many additional faces can be obtained by nonstandard flexing.  This will take you into uncharted but quite facinating waters for sure.   The shifting of triangle numbers (eg A1-A12) on the faces is not as orderly as with the three and four faced 12-gons.  Be sure to watch how the numbers lay out on each of the faces.  There is much more to come on this topic….