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.

geometry and paperfolding · hexagon · How-to

Hexagon via Paper Folding

Regular Hexagon
Regular Hexagon

One of my all-time favorite paper moments was when I learned how to fold a regular hexagon. Many times, with  protractor or compass in hand,  I had tried to draw hexagons, but  they never worked out just right. This post, which features a tutorial page, is something that I have been wanting to do for a long time, but I needed to stumble upon just the right random instant of blog time. Recently, while musing about life, the universe and everything, well it seemed like the right time to finally put these drawings and steps down on paper. After all, the six goes neatly into 42.

Paper Folding and the Regular Hexagon by Paula Krieg
Paper Folding and the Regular Hexagon

 

Now, if you go ahead and make a hexagon for yourself, which of course you will because who could possibly resist trying this, you might notice a few splendid things. Then again, you might not notice them, so I will point them out.  First, you will notice that a preliminary step towards hexagonism is that you create an equilateral triangle , which is just the first of the many perks of this activity. The second, most extraordinary flash will be when you realize that the intersection of the three folded lines within your triangle is actually the center of the triangle. The reason that this is so remarkable is that this intersection point in no way looks like it’s at the center of the triangle. It just looks wrong as a center, and you might not believe it. But when you bring the tips of the triangle in to meet the intersection, well, let’s just say you will believe. 

Silently folding a hexagon drawn by Paula Krieg
Silently folding a hexagon

Just for fun, I decided to include this set of directions, too, because, really, it’s a much more attractive page than the one with all the writing on it. And there are plenty of people who will try it out without reading a thing, so here you have it.

Now after you’ve noticed that you’ve made a big equilateral triangle, there are few more shapes to uncover.  First of all, there are all sorts of little equilateral triangles inside of the hexagon. And if you fold the hexagon in half, well, you will have made an isosceles trapezoid. Now, think back, when is the last time you actually held an isosceles trapezoid in your hands? Next, fold back a third of the trapezoid, and there you have a rhombus. And if you can’t remember what any of these shapes are it’s probably because you never learned to spell them. Really, what it is it with this terminology? Wouldn’t these all these shapes be more memorable if we called them lollipops or kiwis?

A special nod here to Christopher Danielson. math teacherblogger who recently had way too much fun using hexagons in his classroom, so I wanted to add something to the virtual hexagon mix. And I want to acknowledge Steve Morris,  who kept me thinking way too long about the edges and shape of the universe: I no long think that the universe is shaped like a hexagon. That was just silly. Now I think it’s the shape that’s made when hexagons and pentagons are fitted together, -but don’t be looking for a post about that. I think I need to get back to making books. Rectangular books.

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