How-to

Spiraling Paper Ornament

I came across this complicated looking but simple paper structure and have been happily playing around with it for days.

I’ve been able to locate only once source for these, which is at http://hattifant.com/triskele-paper-globes-flower-edition/, an exquisite site by German artist Manja Burton. Fact is, her site has enough about these globes that, really, no one else needs to ever write about them again ever, but, oh well, here I go.

Manja calls these Triskele Globes. I have no idea whether these are a traditional paper-folding design, or if she developed it herself. “Triskele” is a symbol which depicts three interlocking spirals. These paper globes appear to be interlocking spirals, but the spiraling is simply a wonderful illusion.

Bonus Update: just as I was about to hit the Publish button for this post, I received a note from Manja, responding to my questions about this structure.  I’ve added her response at the end of the post.

Three rectangles that will become globes
Three rectangles that will become globes

The globes are made with three interlocking strips. 

After interlocking the strips, the arcs are folded in. It really helps to pre-score the curved fold lines. Here’s a short video:

I made three different pdfs for paper strips, which will be at the bottom of this post. Each page will make two ornaments. My templates are a bit smaller than the ones on Manja Burton’s site. I like this smaller size mostly because I think it work so well with standard copy paper.

I’ve embellished with the paper with simple shapes, so that it’s easier to distinguish the strips from each other, otherwise it can be confusing to see what’s going on with the construction. I’ve also provided a pdf below that has no embellishments, so my shapes won’t interfere with your own vision.

Image of template for paper strips to make spiral paper ornament
Image of template for paper strips to make spiral paper ornament. Look for PDF below

You’ll notice that there are triangles on the template. These triangles are hidden in the final product. I put them there to help orient the designs, hopefully making it easier to see how the rings of paper strips are aligned to each other.

If you don’t have access to a copy machine, it’s absolutely possible to construct your own template for these paper strips. Here’s a grid that can be used to understand the proportions of the shapes.

The Paper Strips for the Spiraling ornament, mapped on a grid
The paper strips for the Spiraling ornament, mapped on a grid. The strips are 48 squares tall (plus two more for the tab), 10 square wide, and the circles have an a radius of 8 squares, but only 1/3 of the circle is on the paper strip. There, that should get you started.

I liked the look of the grid so much that I made a PDF of templates that includes the grid.

Grid on paper strips
See PDF for this at the bottom of this post

I’ve had such a great time with these. I’m  surprised every time I see the transformation happen as the strips of paper become a spiraling globe.

Here are the PDF’s of the strips.

Spiral Paper Ornament strips with shapes

Spiral Paper Ornament strips plain

Spiral Paper Ornament graphed strips

and here the PDF of the grid so you can construct your own strips without a copy machine:

Spiral Paper Ornament strip construction  (will be much  easier to use if you have graph paper)

A Bevy of Paper globes
A Bevy of Paper globes

Have fun. Use bling. Be colorful. Experiment with different papers, different designs. Make them into dice, fortune tellers, add quotes. Go wild.

Here’s the bonus update: I wrote a note to Manja which said.

Manja, I am enchanted by the Triskele balls, and am currently writing a blog post about them, which will include links to your site. Can you tell me something about history of these balls? did you invent the form or did you come across it in your travels? Many thanks for your beautiful work.

She wrote back, saying:

It is a couple of years back now that I saw an image of one of these globes and all I knew was that it was made out of three strips of paper. I spent a whole day on figuring out how it works… I never found that picture again online. I did do some more research on it back then and couldn’t find anything. So I asked the Hattifant community to help me find a name. And that was absolute fun and we in the end came up with Triskele Paper Globes.  Today, I have seen more of them and also in the Scandinavian area… there they seem to be called “click balls”. So please I would love to know more, too!  If you find out more do let me know!”
Okay! If anyone knows something about the history of these balls, let us know!

Addendum, December 18.2018

After reading my post about these ornaments a twitter friend examined the curves and came up with some tweeks https://twitter.com/mathforge/status/934826360970924033

Then this friend, who I know only by the name Loop Space wrote a great post for the mathematically curious which is https://loopspace.mathforge.org/CountingOnMyFingers/Triskele/. I especially like how   Loop Space illustrated the explanation.

Geometric Drawings · geometry and paper · geometry and paperfolding · origami

Pentagons, Paper Folding, Stars & Origami

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/


addendum March 2018

Here’s someone making this star. She makes it looks so easy! https://www.instagram.com/p/BfuSgYdnmY5/

geometry and paper · geometry and paperfolding · Paper Toy

A Cool Adobe-Illustrator Artboards Thing

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: