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 · math · Paper Toy

Hexagon-Flexagons: Post 2, Fractions

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.