Flexagons · geometry and paperfolding · summer art/math

Hexaflexagons in the Summertime

Molly’s

This past Wednesday was my third three-hour meeting with a group of teenagers I am doing projects with this summer. I’m getting to know these young people a bit more, which is the best part of what I do.

It means a lot to me to be able bring projects to them which they enjoy, that are dynamic, and that might teach them something new. I missed writing about last week’s project…oh well.

I’ve been planning three things per week. My thought is to start with with something really short but really cool. This week I started with this amazing puzzle I saw on a post by Mike Lawler :

What you’ll see if you watch this video is a square piece of paper that has a square cut out of the center which a CD must fit through. It looks impossible. It’s quite mind blowing. Take a look.

The main event the afternoon was making hexaflexagons, which I’ve written about numerous times. Basically, they are a tricky foldable structure that, as they flex, transforms the patterns that are applied on them.

For instance,

Jordan’s

these are two views of the same sides of the a flexagon. The way that the paper folds rotates the sides to create an illusion that you are looking at something entirely different.

 

It was great fun to watch these teens discover the different transformations of their designs.

These cats were a surprise. Mostly people were doing purely geometric designs. I had no idea how these cat motifs would work out. Just loved how they paired up!

The fellow that did this one, with the black square and the blue and red circles within, has surprised me during each class. He leaves me wondering if he’s going to participate at all and then I look over after awhile and see that he’s done something stunning.

 

Here;’s a PDF for 11″ x17″ paper. You need just one of these strips of 10 equilateral triangles to make a hexaflexagon

We made the flexagons using a template I created. What is needed is a paper strip which folds into 10 equilateral triangles, so this template I made can be used to make four separate hexagon-flexagons.

For some reason I kept messing up showing the group how to fold. One of the older teenagers, who’s position is counselor, really understood the folding well, so I took a video of her explaining how it goes:

If you still haven’t seen enough, well, I took just a couple more fun photos of the work of this talented group.

the drawing
the Transformation

So cool!

There was one last project we did during the last half hour together. It was doing some origami, but I had them each cut out a separate rectangular piece of paper. Each rectangle was a different size but proportionally the same. I have a thing about scaling: I want all kids to know how to do it.

Similar rectangles

After the paper was cut, I walked them through the steps of making an origami toy boat, not because I wanted a toy boat, but because I wanted to stack the different sizes and see what happened. Each person had a different size paper

This is what happened.

It stands on it’s own, and looks kind of like a ziggurat , or maybe it looks like a big hat.

I think we decided it looked like a hat.

After having spent most of the afternoon making flexagons with this group I came home and checked my twitter feed. Coincidentally, seems that my friends had been all atwitter about flexagons, starting with this from Vincent: https://twitter.com/panlepan/status/835988773875892224

Link to Dave’s PDF

Within the thread was a link to Dave Richeson’s template and instructions for what he calls a Cube Tri-Hexaflexagon, but it’s what I’ve been calling the hexaflexagon.  I made one of these immediately. It’s a great template.

 

Ok. It’s nearly time to start planning my next project with this group. Looking forward to it!

 

 

 

 

Flexagons · geometry and paperfolding · hexagon · Paper Toy

Hexagon-Flexagon meets the Double Arm Trig

double-arm-trig-yellow

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.

https://twitter.com/GHSMaths/status/667147284044513285
https://twitter.com/GHSMaths/status/667147284044513285

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.

Hexaflexagons in progress
https://twitter.com/GHSMaths/status/667147284044513285

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 Hexaflexagon Safety Guide .

 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. 

double-arm-trig-bw

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.

Three-hex

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.

A hexagon-flexagon on 8.5" x 11" copy paper
A hexagon-flexagon on 8.5″ x 11″ copy paper, printed on both sides

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: hexagonflexagontemplatecolor

 

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.

Flexagons · Paper Toy

Hexagon-Flexagon: Post #4, Student Work

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

Flexagons · hexagon · How-to

Hexagon-Flexagon: Post #3, Instructions

How to Make a Hexagon-Flexagon
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
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
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