folding · Paper Toy

Six-fold, flat-fold, Paper-fold

Paper Folding the Ferozkah Jaali
Paper Folding the Ferozkah Jaali

I found a fold.

If paperfolding graps your attention, prepare to be overwhelmed.  There’s three things to unpack here: the fold, the pattern on the fold, and how they interact.

I had been wondering if I could fold a tetrahedron out of a rectangle.

Tetrahedrons and other shapes
tetrahedrons and other shapes

Turns out, yes. I can make a tetrahedron with a square base or a triangular base out of the same piece of paper using the same folds in different ways.

Looks like a fish
Looks like a fish

Then I started seeing that I could make other shapes out of the same pieces of paper using the same folds differently.

Some shapes are flat, others are dimensional.

I’ve been playing with these all week, and I am still finding different shapes that these folds create.

 

I’ve also been drawing this six-fold pattern from Islamic Geometry called the Ferozkoh Jaali. It occurred to me that it would go perfectly with the folds I was making.

detail of Ferozkah Jaali
detail of Ferozkah Jaali

 

This is just a small portion of the pattern. I’ve been coloringing copies of these in all week, trying to get to know the shapes.

Here’s the fold that I’m using:

 

Mountain and Valley folds
Mountain and Valley folds

It’s four mountain folds (diagonals) and two valley folds (horizontal and vertical) that are created around equilateral triangles. Oh, and there’s a slice in the middle. One horizontal slice.

Now here’s the first wonderful thing about using this image with my folds:

No matter how you use the creases (which are around the equilateral triangles) , the pattern lines up. In the photo above, a corner is peeking through that slice in the paper, and, look, the pattern lines up.

Equilateral triangle(s)
Equilateral triangle(s)

I printed the design on the fronts and backs of my papers, and look, when the paper wraps around itself, the pattern lines up.

Now there is one more thing to mention. Hold on to your seats. This is wonderful. But, first, here’s the foundation of the image I created, first by hand, then on the computer, because I needed the precision of the computer image.

Six-fold-geometry
Six-fold-geometry

Okay, so as I’ve been folding and refolding and refolding again, and finding different shapes all the time, the last final amazing thing that I noticed (and this makes so much sense) ….

Some heart shapes?
Some heart shapes?

…is that every shape I make with these folds can is echoed somewhere in the lines of the  geometric drawing that is printed on the paper.

This makes me so happy, well, I can’t even describe it.

Another heart shape
Another heart shape

Well, there you have it. Hope you love it as much as I do.

covered with NOT geometry
covered with NOT geometry

Oh, and just in case you’re wondering, I think this fold looks good with just about anything on it.

Am seriously considering making a bunch of these and offering them for sale, probably through Etsy. Stay tuned….

 

 

 

folding · geometry and paper · geometry and paperfolding

A Square of Your Own

Squares, scaled
Squares, scaled

Making a plain old square exactly the size that you want can be such a chore. Too much can go wrong.

Squares can become so many different things (think ALL origami) that it’s worth knowing how to make a square just the size you want without getting stressed out.

Here’s how to do it, via the photo essay, followed by the video tutorial.

plain piece of copy paper
plain piece of copy paper

Here’s a regular piece of copy paper. The first steps look like I’m heading towards folding a square whose side is already predetermined by the short edge of the paper, but no. Be patient. This is going to be a surprise. Trust me.

First Step
First Step

Fold that short edge to meet the long edge, making sure that your fold ends (or starts) exactly at that corner. What you are doing here is bisecting that corner angle. If you cut away that rectangular flap on the edge, and unfold the triangle you’d have a square, but this isn’t what we are doing now. (Doing that was a different post, )

Marking the size of the new square
Marking the size of the new square

Unfold the fold then make a mark to indicate size of the square you want to make.

continuing...
continuing…

What I’m going to write next sounds horribly complicated but it’s easy to see and do.

Curl over that corner that has the fold going through it, lining up that corner point with the fold line it is leaning towards, and also lining up the upper edge of the paper with that mark you made. Press the fold.

Here’s a closer to look:

Closer Look
Closer Look

Now draw a line that traces around that folded down corner.

Time to cut out the square
Time to cut out the square

There’s your square!

Confused? Here’s a video to watch:

Got it?

Origami Pockets
Origami Pockets

Now make cool stuff.

Addendum: here’s a post that shows what I showed, plus another great way of making a square. https://mikesmathpage.wordpress.com/2018/01/23/a-fun-folding-exercise-for-kids-from-paula-beardell-krieg/

Arts in Education · folding · Uncategorized

Seeing Differently, Teaching Differently

I don’t understand  why certain of my projects with kids get more attention than others. My original post about Sight Word Pockets Book for Kindergarteners  from 2011 still gets viewed every day. My teaching season doesn’t go by without requests for this project. By now I have taught literally thousands of young students how to make origami pockets, but it’s never easy. I’m always looking for a better way to explain this folding method.

A titled square is still a square
A tilted square is still a square

I’ve come into this teaching season thinking differently about folding paper.

For so many years I have been telling students what to do. This year I have prioritized trying to draw their attention towards what to see.

They all recognize a square, but when I tilt it, students say its a diamond, a rhombus, or a kite. Last week I suggested to students that this shape was still a square, then I was relieved when a  classroom teacher chimed in and emphasized that tilt or no tilt, the shape was still a square. The main thing, though, is that this simple rotation and the conversation riveted students’ attention to the shape.

Folding a square point to point, from the top down
Folding a square point to point, from the top down

I’ve been asking students what happens to the square if I fold it corner to corner. They all seem to be able to predict that folding a square point-to-point can make a triangle. This makes them happy. They seem to like triangles. Then I tell them that with just one fold more I can make many more triangles. What magic can this be? I have their full attention. They are watching to see if this can possibly be true.

Folding just one layer of paper, I fold the tip of the triangle down to the base. The children are delighted. We count the triangles. Are there four triangles? Are there five?

The next step is the tricky step but now students are attached to the shapes that the folds make and have a heightened awareness of triangles. When I talk about tucking the edge of one of the triangles under the flap, they see what I mean. Here, look at this video. It absolutely blows me aware that this five-year-old just learned this folding sequence about a half-hour before I filmed her doing it. Notice how sure her hands are as they move through the steps.

This change in my teaching, prioritizing seeing & predicting over telling & doing feels really good.  It’s happening because I’ve begun to be able to answer a question I’ve been carrying around in my mind for years: I’ve really looked hard at origami , trying to figure out what about it compels some people say that origami is somehow like math.

Now I’m coming to understand that it isn’t origami that I needed to see differently, it’s been my understanding of math that I needed to adjust before I could make the connection. Now that I am seeing math less as addition and subtraction, and more as relationships and transformations, the boundary between origami and math vaporizes.

More and more I am trying  to be  attuned to childrens’ seemingly intuitive connection to ideas that are aligned to a broader understanding math, and I am able to tap into this with great results. I help them see what is already familiar to them, and what happens next is that they better understanding what’s going on, and can figure out what to do next. Yes, even five-year-olds can do this.

 

Art and Math · folding · geometry and paperfolding · Math and Paper Folding

Constructing a Shape then Admiring It

Paper folding in the morning
Paper folding in the morning

Yesterday I watched a video that showed the Lawler family looking at shapes.

https://mikesmathpage.wordpress.com/2017/01/13/exploring-3-intersecting-cylinders-with-3d-printing/
https://mikesmathpage.wordpress.com/2017/01/13/exploring-3-intersecting-cylinders-with-3d-printing/

One of Mike’s sons said he liked the top shape in the image above. You can’t see from the photo, but it’s a full sphere. The image above is only half of the sphere, the other half looks no different than the half that is showing. I’ve played around with constructing foldable versions of shapes that look something like the one above, and I thought I’d be able to make a foldable version of what was on Mike’s screen.

Template for folding and cutting
Template for folding and cutting

I’m not showing all the steps that led to this map of folding and cutting because what I’m most interested in showing here are the wonderful visuals I got to experience along the way of creating the final structure.

Folding and Cuttin
Folding and Cutting

Silly as it may seem, one of my first realizations was that the indoor, nighttime lighting in my workspace was just all wrong for photographing what I was about to fold. Morning light would be best. So I went to bed.

Assembling in the morning light
Assembling in the morning light

Of course I forgot to recharge the battery of my phone camera before I went to bed, so I didn’t get the earliest light.

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Hmm, I don’t really want to say much more about this process. I’m just going to post pictures now. (Haven’t had my coffee yet.)

wp-1484487718619.jpg

wp-1484487640608.jpg

img_20170115_082242.jpg

img_20170115_082605.jpg

zzzz

Ok. Time for coffee. Am heading to Rochester today to bring my daughter back to college. Will be thinking about all shapes that this structure made. (Which reminds me of a question someone once asked me, “What, do you just sit around thinking about folding paper?” Well, yeah.)