I came across a lovely way of folding stars. It was in a youtube video by someone named Tobias.

As lovely as these stars are, what really caught my attention was the way Tobias showed how to use paper folding to make a pentagon from a square. This square-to-pentagon transformation was in a separate video, and since it will take me about two days to forget everything I saw in the video I drew out the directions.

How to fold a Pentagon from a Square

How to fold a Pentagon from a Square. For the Video of this that Tobias made, go to https://www.youtube.com/watch?v=4kJmJUQVbO0

 

After the novelty (but not the thrill) wore off of making a pentagon from a square I began to look at the angles that I was making and figured that I could make the star with less steps (and perhaps with more accuracy) if I just started out with the net of the shape, so I made this map of the paper star’s fold lines:

Lines for a Folded Paper Star

Lines for a Folded Paper Star

If you make Tobias’s stars, after you get the hang of which lines fold in which direction, I highly recommend printing out lines above, score the lines with an inkless ink pen, and make that same star using just its essential folds.

The back of the paper sta

The back of the paper star

The photo above shows the backside of these stars. Quite a nice backside!

I’m sure that there are all sorts of things to do with pentagons, but something I want to mention is something that is fast and impressive, sort of the pentagon version of snowflake cuts. If you cut off an angled slice at the bottom of the folded up pentagon (step 12 in my tutorial drawing) there are all sorts of star possibilities.

36-54-90 triangles, with cutting lines on their tips

36-54-90 triangles, with cutting lines on their tips

These little beauties turn into:

Stars in Pentagons

Stars in Pentagons

The stars inscribed into these pentagons were made by cutting through all layers on the tips of the folded shapes.

 

And look, below there’s something extra for my friends who teach Geometry, and who might like a holiday themed angle activity. Part of the working out the folding pattern for the star was deciphering certain angles.

Find the Angles with degrees of 90, 45, and ~72, 18, 36, 54, and 108

Find the Angles with degrees of 90, 45, and ~72, 18, 23, 36, 54, 63 and 108

I had a good bit of help with the especially tricky parts of understanding the angle relationships. I’m sharing two twitter threads here, just because it was such a pleasure to get help from my friends.

and

That’s about it for now. Oh, and if you need to directions on how to fold a square from a rectangle, take a look at https://bookzoompa.wordpress.com/2014/12/10/paper-folding-squares-and-equilateral-triangles/


 

To be two cubes

To be two cubes

I wanted to transfer this image to a big piece of paper. Way too big for my printer. It’s just under 24 square inches.

One the way to being two cubes

One the way to being two cubes

I made the pattern with the intention that it would fold into two cubes. BTW, I recently learned that the correct term to use here is net:” A pattern that you can cut and fold to make a model of a solid shape. This is a net of a cube.” (quoteth from the internet)

While I was scheming how to break the net into prints that I could piece back together, it occurred to me to just overlap the artboards in Illustrator. Set them up to be negative one inches apart. Here’s a snip of what the Illustrator workspace looked:

Six overlapping artboards in the Adobe Illustrator workspace

Six overlapping artboards in the Adobe Illustrator workspace

All I did, after setting up the six artboards was to overlay my net onto the artboards. No figuring, no scheming, just laid it right on top. Honestly I didn’t know what would happen. Would the overlapped parts not print? Just didn’t know.

Amazing. Everything printed everywhere. What I mean is that the parts of the image that were on the overlap printed on both papers. This made it really easy to piece together. Of course the best use of this technology is to print Happy Birthday banners. But what I did was piece them together, cover the back of the paper with blue crayon, and, using a ballpoint pen, trace over the lines to transfer to my larger paper.

net of the Cubes, cut out

I didn’t take any more photos of the process, but here’s my fully cut out net.

On the way to cubeness

The blue crayon showed up just enough, but what was really great is that the force of the tracing created score lines, making this easy to fold.

Weighted by a train

Weighted by a train

Here’s the cube. Hard to imagine how that image becomes these two two-inch cubes. So I made a video:

 

Book, peeking out of its box by Paula Beardell Krieg

Book, peeking out of its box

When Miriam Schaer was assembling her teaching collection to send to Telavi University in the Republic of Georgia, I very much wanted to contribute, but nothing I had on hand seemed right. In the nick of time, some thoughts came colliding together. Polygon Fractal book by Paula Beardell Krieg

This structure started out with an exploration of a shape which I wrote about a few weeks ago after watching family math video made by the Lawlers.

Inside Outside Book by Paula Beardell KriegThe book  opens in an accordion-like fashion, but front and back are structurally different.

Polygon Fractal book by Paula Beardell KriegThe colorful pages rotate open to create these double layered corners. The polygon fractals on the pages here are harvested from Dan Anderson’s openprocessing page then toyed with in Photoshop.

To see the fractals in their full radiant radial symmetry one must rotate the book. There are six completely different images to be seen. But it gets more interesting, because there is a whole other side to see.

The folds of those double layered corners completely reverse to form a cube!

You can’t imagine how excited I was when I saw this cube emerge from the folds!

This folded structure totally suggested that, whatever I use on it, that it be about the dual nature of….something….a suitcase (no, too obvious), a politician’s statements (ugh, too boring)…actually wanted to use images that didn’t imply any hierarchy, hiding, agendas, or judgement about contrasting inner and outer manifestations.

It was this thinking, about duality but equality of visuals, which led me to using Dan’s code along with the polygon fractals that it creates. So perfect. Code and images are perfectly linked, simply completely different ways of seeing the same thing. You know, like Blonde Brunette Redhead 

Now, I do have a lingering unresolved issue with this book. I’m not thrilled with the paper that I’ve used. It’s 32lb Finch Fine Color Copy paper. It takes color beautifully, folds well, but I’m thinking that the folds might be more prone to tearing than is comfortable. Not sure what else to use…am open for suggestions. Miriam’s copy has been shipped, but I’m still happy to check out different papers to use.

I can’t help but wonder if people will be able to figure the transformation of these  pages without seeing this post or reading the brief explanation I’ve provided on the back page of the book? Dunno.

Oh, and here’s my favorite variation:

Hanging a tea light from a pencil so I can see the inside and outside at the same time.

Happy.

Kaleidocycles and Tetrahedrons

February 19, 2017

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.

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.

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

 

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