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!

 

 

 

 

Arts in Education · Book Art · Flexagons · group project · Paper Toy

5th Graders’ Kaleidocycles of the Bill of Rights

Piling Kaleidocycles
Piling Kaleidocycles

They did it! This group of fifth grades did this hand-lettering kaleidocycle class project! I described the details of this project a few posts back so check out that post for more details. Here’s the general gist: After introducing the project, which is a 3D paper construction with rotating faces that will be graced with references to the Bill of Rights, students were given pages of letter fonts to choose from.

Tracing

Using the windows of the library as light boxes, students traced out letters to created one phrase each that described one of the first ten constitutional amendments, aka The Bill of Rights.

At the window

Every single student was highly engaged. Really.

Within two class sessions the students produced something that I could take home and scan into my computer . I won’t lie..scanning and cleaning up their work took time. Above you can compare what they gave me, on the left, to what I ended up with on the right. Some pages required much more work than others. The middle example above was so easy to work with that next time I will encourage students to just give me outlines. The most time-consuming letters to work with were those that were colored in and touching other letters. I moved things around a bit, like in the top example you can see I centered the word “OF.”

I brought home their work, scanned them into my graphics program, cleaned them up and laid them into a kaleidocylce template. Brought them back for students to cut around the perimeter and score.

Scoring the paper
Scoring the paper

Students made score lines so that the paper would fold easily and accurately. Scoring is generally done with bone folders but we used glitter pens to score the lines. They worked great, and kids were excited to be using the gel pens.

Assembling the Kaleidocylce

Then came the folding and gluing. I didn’t take many pictures of this process as I was, like, really really occupied helping move this process along.

One of four faces of the Kaleidocycle
One of four faces of the Kaleidocycle

This project turned out so well. Not everyone had a chance to finish up and decorate, but the wonderful school librarian will be able to help with the few than still need finishing.

Kaleidocycle yellow borders
Kaleidocycle yellow borders

Students enjoyed individualizing their own kaleidocycles.

Another side
Another side

I tried to get them to use completely different color schemes on each face, so that the differences between the four rotation of faces were dramatic. Students didn’t much listen to my suggestions.

Here’s one of their kaleidocycles in action:

I consider this project a great success. I got to talk to the students about design, about hand lettering, and they got to work with some cool geometry. I’d even go so far as to say that they are also much more familiar with the Bill of Rights , as they were constantly asking each other, which one do you have, which one is yours, and talking about their own.  I have to say that at first the students were confused about what I was asking them to do, after which the librarian told me that doing a group project was pretty much out of their experience, so the concept was hard to grasp at first.

One thing that made this possible was that this was a small class, just 12 students. I often work with 60 to 70 students in a grade level: I wouldn’t do this project with a big group. OH, but it was so delightful doing this with a small group.

Tower of Kaleidocycles
Tower of Kaleidocycles

Do I get to pick a favorite project of my teaching season? Yes? This is it.

For more about all this take a look at these posts:

https://bookzoompa.wordpress.com/2017/02/19/kaleidocycles-and-tetrahedrons/

and

https://bookzoompa.wordpress.com/2017/03/19/designing-a-seriously-fun-project/

Arts in Education · Book Art · Flexagons · Uncategorized

Designing a Seriously Fun Project

Model for a Kaleidocycle Project for Fifth Graders

Just realized I spelled “Kaleidocycle” wrong on this model for a project I’m putting together for Fifth Graders! Well, good, we’ll talk about mistakes.

Now, in case you missed my post about kaleidocycles, they are this fun paper structure whose sides rotate to reveal a surprising number of new surfaces. This style of kaleidocycle here has four completely different faces, each of which has 3 distinct areas to fill with text or designs. Sort of like a fortune teller, but more 3D.

I’m not a person who is skilled in doing hand lettering to create fancy looking phrases. Nor do I think that I can teach 5th graders to be hand lettering artists in a few class periods. But I love this art form, and am happy to be able to talk to kids about hand lettering.

We’ll be doing a project that references the first 10 amendments to the American Constitution. darn. Spelled Amendments wrong too. Ok, will make a new model with corrections. But will show both to students.

Anyone who does calligraphy or hand lettering will tell you that making spelling mistakes happens frequently when so much effort is being put into forming the letters. People with dyslexia will agree.

Here’s the project. I will talk to students about creating something like a movie title which references each of the  first ten amendments, aka The Bill of Rights (know them or lose them!!).  Students will then be given some alphabets and will trace out the letters.

It can take a few tries, but I’m anticipating that they will not have much trouble coming up with a version that I can then trace in Adobe Illustrator to create a master document.

Flourishes

They can also add flourishes around their words.

My version for the model that I will be showing off, featuring later amendments, looked like this when the computer work was done:I will print the student images on 28 lb, 8 1/2″ x 11″ copy paper. My plan is that each student will design just one of the 12 surfaces on this kaleidocycle (there happen to be just 12 students in this class). I will print up enough copies for everyone and we’ll spend the last of the three sessions I have with them doing cutting, gluing and decorating.

When this project is done I will take photos and post on my blog, recommending either that you try this out with your students or telling you that have to be crazy to think that this can be done as three-meeting classroom project for fifth grade students.

All will be told be the end of the first week in April. Yikes. Starting this project tomorrow, early.

Addendum: See the results https://bookzoompa.wordpress.com/2017/04/13/5th-graders-kaleidocycles-of-the-bill-of-rights/

Flexagons · geometry and paperfolding · Math and Book Arts

Kaleidocycles and Tetrahedrons

img_20170219_101957.jpg

I looked up the definition of a tetrahedron today, I figured out how to spell kaleidocycle a few hours ago. Just saying.

kaleidocycle-colors-1 Paula Krieg
Pattern for one kind of Kaleidocycle

Sometimes an exploration pursues me. It’s always a gift to be preyed upon by ideas, but if my desk is already full and messy, and I think I can’t bear adding one more layer I pretend to kind of ignore the newcomer. No, this strategy doesn’t work.

https://www.facebook.com/artsmathematics/videos/718044448365422/
https://www.facebook.com/artsmathematics/videos/718044448365422/

I didn’t know that tetrahedrons were following me around. Like I said, just this morning I finally looked up the definition (a solid having four plane triangular faces; a triangular pyramid).

The image above is where this all started. This is not such a startling set of pictures until you know that the image shows the same, unchanged structure viewed from front and back. It’s on the facebook page of someone whose name is written in an alphabet I don’t understand. This is the link to the page on facebook https://www.facebook.com/artsmathematics/videos/718044448365422/ . Take a look if you can. It’s such an amazing bit of transformation, which I have yet to figure out how to do. What’s going on here is that this structure to made up of connected 3D shapes that rotate together to reveal different surfaces. It’s very tricky and fun to see the shapes turn, revealing new surfaces.

The next piece of this story is that a teacher just a bit south of me in Upstate NY posted some directions on how to build a certain  geometric shape, and he asked, via twitter, if anyone would be able to test drive his tutorial. It looked simple enough to me, so I thought I’d try it out the following Saturday morning. I thought it would take about 2o minutes. Ha ha.

Tetrahedron
Tetrahedron

Looking back, I think if this teacher, Mr.Kaercher, had done a tutorial on a simple tetrahedron it might have gone more quickly and I might have finished up knowing what a tetrahedron was. But, no, Mr. K provided directions for a tensegrity tetrahedron, and since I didn’t have much of a clue about the definition of either term, I didn’t really have much of an idea of what I was doing.

Tensegrity Tetrahedron
Tensegrity Tetrahedron

Even so, after a megillah of failures, I got it done and was quite pleased with myself.

In the meantime I was still thinking about those images from that facebook page.

I showed the FB clip to book artist Ed Hutchins. He told me that what I was looking at was a type of kaleidocycle.

Oh, and Ed just happened to have a hot-off-the-presses copy of what is probably the world’s most amazing example of a hexagonal kaleidocycle, designed by Simon Arizpe. (This is a fully funded Kickstarter Project, which you can view to see the book in motion.)

This structure tells a story as it rotates. Since these rotating sides can turn forwards or backwards, the sequence of the story is determined by the direction the viewer rotates the kaleidocycle. The way that I choose to turn it, it begins with a bear peeking out at a stream…

img_20170219_092224.jpg

…the bear opens his mouth, a salmon jumps out…

The Wild by Simone Arizpe
The Wild by Simone Arizpe
The Wild by Simon Arizpe
The Wild by Simon Arizpe

… and then the salmon jumps into the river. There’s one more frame, but I’m not going to be a spoiler and show it to you.

So what does this have to do with tetrahedrons? I’m getting there.

As it turns out, the last couple of times I’ve gone lurking at the Lawler family math page, they’ve been looking at, yes, tetrahedrons.

lawler-family tetrahedron
https://mikesmathpage.wordpress.com/2017/02/17/revisiting-james-tantons-tetrahedron-problem/

This shape that the Lawler’s were considering was beginning to look familiar to me. Part of the reason for this was that, ever since Ed had given me the gift of the term kaleidocycle I had been Googling around then assembling kaleidocycles.

Kaleidocycle, unhinged
Kaleidocycle, unhinged

Here’s one of my first attempts. Notice that I forgot to attach the ends together before I closed things up. This turned out to be a good thing, because, wait! these shapes appear to be repeated echoes of the shape that the Lawler family was exploring.

Just to pile it on, it certainly helped that just yesterday a package came in the mail, all the way from France, from Simon Gregg. In the package was, can you guess?… a tetrahedron.

Tetrahedron from Simon Gregg
Tetrahedron from Simon Gregg

That Saturday a few week ago that I tried, time after time, to create my tensegrity tetrahedron, I had been posting my failures publicly on twitter. I imagine that Simon thought that it might be merciful to send me some bamboo, as the straws that I was using would sometimes collapse. Included with the bamboo rods, Simon also gave me a collapsible tetrahedron, held together by stretchy cord.

With all of these pieces floating around me it,  I finally made the connection that units of kaleidocycles are series of tetrahedrons. 

Kaleidocycle/tetrahedrons
Kaleidocycle/tetrahedrons

So cool.

Now to reward you for making it all the way to the end of this post, here is a pattern for a kaleidocycle that you can make yourself.

kaleidocycle-color-2
click the image to get the printable PDF

Just cut it out, use it alone or attach it to the one near the top of this post, but, in either case, do  make sure you attach ends to make it circular. Here’s a pleasant little video to show you how it’s put together.

I still intend to figure out how to make the kaleidocycle that I saw on FB. When I do sit down and try it out, at least, now, I feel like I’m starting with some helpful understandings.

I have no big attachment to figuring it out for myself, so if you are inspired to decipher it, please let me in on its secrets!

img_20170219_102455.jpg

That’s it for now. Thanks for staying with me through these meanderings.

Addendum 2/20/2017

truncated tensegrity tertrahedron
truncated tensegrity tertrahedron

Used bamboo sticks with bobby pins in the ends to make another one of the Mark Kaercher project.  The bamboo worked out great! If I was to make this again with straws, I think I’d try to first put stirrers, like what Starbucks provides to stir coffee, inside the straws. But love the bamboo!

Bamboo