## Hexagon-Flexagon meets the Double Arm Trig

### November 22, 2015

Although I intend the title of this post to refer to what I’ve been messing around with this weekend, I’m not really sure it means anything. What’s been happening in my studio is that I’ve wanted to mix up some interesting lines with some interesting folds.

After a person with the handle of GHS Maths posted a group of images made by rotating the graph of a trig equation I got it into my head to see what one of them would look like on a hexagon-flexagon.

If you don’t know what a hexagon-flexagon is, you haven’t watched enough Vy Hart videos. In 2012 Vy offered her own utterly delightful interpretations of what she thought people should know about this piece of paper wizardry in Hexaflexagon, (6 MILLION views!), Hexaflexagon 2, and the sequel

A Hexagon-Flexagon has three distinct sides, which results in six distinct designs: I’ve written about these here, here, and showcased student work here. I haven’t thought about these in a while, but it seemed to me that the image at the top of this post, and others that I had been working with lately, might be interesting to put on a hexaflexagon.

I had  ideas for all sorts of images but I became so enchanted by what the variations of the image above that in the end this is what I went with.

My computer did not like this idea at all. I spent half my weekend redoing what I lost when my program crashed, half my weekend watching that blue swirly thing going around, and half my weekend coddling my computer so it wouldn’t crash. I know that I’ve listed three halves, so if that bothers you, here’s what I did with  fourth half: I was able to actually make an image that became a hexaflexagon.

This is what a hexagon-flexagon looks like on two sides of copy paper. I always want things to fit onto standard copy paper, so I had made this template:

It’s a bit tricky to follow, but it actually works really well. I love being able to print these up on my little printer.

This is a dynamic structure, that is not easily appreciated in still photos. I am going to either get my son to make a video of me working the structure or will post the more appropriate stills that I can come up with. Tomorrow. Edited into this post. See you then.

Update: I made a quick video! My first one! Based on the image in this PDF which is printed on both sides of the paper.

UPdate #2

For a hexaflexagon template that has a snowflake, a Christmas wreath and a Star of David, visit Chalkdust magazine at http://chalkdustmagazine.com/blog/how-to-make-christmas-special/. You’ll find a link there, and now here too, for Martin Gardner’s famous article on Hexaflexagons.

## Hexagon-Flexagon: Post #4, Student Work

### November 28, 2011

I’m ending this series of posts on hexagon-flexagon with images of work done by students in Michele Gannon’s art classes in the Adirondacks.  I taught the students how to do the paper-folding. Michelle’s considerable artistic influence supported the students’ creative output.  Each of images that I am showcasing here is followed by the same image in different configuration on the same structure. Notice how the patterns change.

This lovely design was created with Prismacolor pencils. Some, but not all, of the Prismacolors show up well on black paper.

This next set shows a really simple design that works really well, as the look of it changes dramatically when the image is flexed.

Ta-Dah!

Since each hexagon-flexagon has three distinct surfaces to design and decorate I  brought in a variety of novel tools to help motivate the students to keep working. In the photos below the students used my Chinese Dragon paper punch. Paper punches are a great motivational tool, and the dragon is the most popular of all that I have.

By the way, the students that made these images were about eleven years old.

I like the way the relationship between the dragons changes when the flexagon flexes.

Another tool that makes a big hit with students is Crayon Gel Markers. These markers take about 15 seconds to fully show up on the paper, thus adding a bit a magic to the process of decorating.

I also use some of the Crayon colored pencils. Only some of them work well on black paper. I used to be able to find Crayona FX (?) pencils that worked on black paper, but I haven’t seen them around for awhile.

I have to say that that only reason that I felt comfortable teaching this structure to these students is that I knew I would be working with small groups. I don’t think that I had more than 10 or 12 students in each class.  The fact is, the folding  for this structure is so specific, that I doubt I could successfully teach it to large classes. But I think it would be a great project to teach to home schoolers, or in an after school art class.

The hexagon below has a completely different look when it’s flexed.

The middle circle totally changes, and that white line becomes a whole different thing, too.

These were so much fun to teach.  It’s a real pleasure to watch the expressions on the students’ faces when they see their own design be transformed when the hexagon flexes. That’ s my favorite part of the process, too.

Related posts:

Intro to hexagon-flexagons

Fractions Flexagaon

How to Make a Hexagon-Flexagon

## Hexagon-Flexagon: Post #3, Instructions

### November 23, 2011

How to Make a Hexagon-Flexagon

After posting  about Hexagaon-Flexagons on November 10 and November 16, I started working on making a template for teaching this tricky paper invention. Even though this  structure is well covered on the net, what I want to add to the mix is something to make the folding easier.

When I’ve taught this structure the part that people have the hardest time with is creating precise folds. I made the instruction sheet above because it provides a way to create score lines so that the folding is easier.

Ruler and paperclip to use to score lines onto paper

Scoring is PRESSING lines onto paper, so to help facilitate folding.  If a score line is firmly pressed into paper, it will fold easily on the score line. A pen that has no ink in it, or a paper clip, both make good scoring tools, as they will make a thin line. Bookbinders use bonefolders for scoring, but for this template I prefer a paper clip because it makes a thinner score line.

Scoring

Here’s a picture of my daughter’s hand scoring  my template. The template is on a firm but not hard surface: the surface ideally should have a bit of ‘give” so that it can be pressed into. (A stack of newspapers or a catalogue works well.)  Place the ruler on the indicated line, holding it securing in place, then run the edge of the paper clip along the edge of the ruler. Press firmly, but not too hard, as you don’t want to rip through the paper. If you look closely at the image above you can see the the score lines that have already been pressed into the paper.

Cick on this image for to print or to enlarge

That’s about all I have to say for now. Hopefully the instruction sheet above clear enough to follow. If you want to print out template without the instructions, this is the a link to my non-annotated hexagon-flexagon .

And for some more inspiration, here’s a hexagon-flexagon decorated by Michele Gannon:

By Michele Gannon

Now, here it is again, flexing….

The Flexing of a Flexagon

….to next reveal the design that was formerly tucked inside:

This next picture is that same design, after the image has been flexed. Be sure to notice how the dragonfly images change direction.

addendum: Here are a couple of links to another person’s take on the hexagon-flexagon: http://plbrown.blogspot.com/2011/03/amazing-trihexaflexagon.html and http://plbrown.blogspot.com/2011/12/art-math-magic-its-trihexaflexagon.html

Happy flexing!

Addendum 3/12/2017: just came across another great flexagon resource! http://www.puzzles.com/hexaflexagon/activities.html

## Hexagon-Flexagons: Post 2, Fractions

### November 16, 2011

Except for the last picture, all of the images in this post exist on one hexagon-flexagon. In my last post I showed hexagon-flexagons that were about making designs with pencil, paper, and gouache. When I was making the design for this post I was thinking more about math.

Hesxagon-Flexagon divided into three rhombuses

Okay, now here I go,showcasing my hexagon-flexagon as way to illustrate fractions, rather than to be a kaleidoscopic toy.
The hexagon-flexagon above has been illustrated to be understood as thirds,showing that three thirds create a whole.
Now, this concept might be better understood if the parts were actually labeled.

Flexing the structure to reveal another design

I worried that labeling the parts could cause a problem, because, after flexing the structure, the design changes. I did a mock up to see what would happen, and this is what I got:

Fractioned parts, labeled

Not bad. I like this way of working with the hexagon-flexagon.

Hexagon-Flexagon, two trapezoid showing dividing the structure in half

One thing is really clear to me, and that is that I would love to work with a graphic designer who would add words and numbers that looks snazzier than my handwriting. Oh, and another thing that is clear is that labeling the hexagon as two halves doesn’t work well after it’s been flexed. (I’m not providing a picture here of how the broken up halves looks).

Six equilateral triangles dividing the hexagon-flexagon into sixths

This side of the hexagon-flexagon shows that six parts equal a whole. Ideally, I would label each of the blue triangles with their own “1/6′ fraction, then write equations all around the outside edges (such as 1/6 + 1/6 = 2/6 =1/3)

Flexed version of the previous image

I think that this is going to be a long-term project, perfecting these images with better text graphics. I like how the geometric patterns work out here. It’s just the labeling that I can’t get right.

So, everyone has heard students question why they have to learn math, particularly algebra and trig, as they don’t foresee ever using it. I have at least one good answer to that age-old question. The reason to master math is that it keeps a person’s options open. I recently spoke to the someone who was helping her 30ish-year single parent daughter with Algebra because it is required for a nursing degree. Today I spoke a woman who has gone back to school to get a degree is Public Health. She is struggling with her required economics course. My son needs a to complete two semesters of Calculus towards his Biology degree, which will qualify him to go on to Chiropractic studies. So, there you have it: learning math a keeps options open for the future.

Now here’s another way of decorating a hexagon-flexagon, and hey, it’s even seasonally correct, as it can easily pass for a six-sided snowflake.

Hexagon-Flexagon by Michele Gannon

Math and art together: always a great idea.