Paper Sphericon, a shape that wobbles

 

Sphericons

Just in time for the holidays, a shape that you probably never heard of. Maybe that’s just as well. If you want something that’s kind of awesome & easy to make, you are in the wrong place (go here instead). This shape is awesome and endearing, but takes more finessing than is comfortable for the average bear. I do not recommend making these with children. Or adults. Proceed at your own risk.

I was introduced to this shape by Vincent Pantaloni, who has a knack for distraction. Near the bottom of this post I will link some of our sphericon-related twitter threads.

Sphericons, nets and shapes

What’s most endearing about this shape is that it wobbles as it rolls. You have to give it a flick right near one of its edges and it will roll like a drunken sphere. Its net (the flattened out version of the shape) reminds me of a duckbill platypus. I bet they wobble too.

If you have a template you can probably figure out how to make one of these without any help from me. Here are the templates, in black and white, in color, and in two different sizes. I’m pretty sure these will work with A3 and A4. (Let me know if I’m wrong about this!)

Spericon color 8.5 x 11 A4

Spherhicon B&W 8.5 x 11 A4

Large Sphericon B&W 11 x 17 A3

Large Sphericon Color 11 x 17 A3

After you fail a few times you can watch my video to see me struggle through making one. I do have a few good tips to offer.

Score and fold the curves and the edges

I recommend printing these on heavier papers than standard copy papers. They don’t do their rolly-wobble really well when made from lighter papers. Really, who would want a non-rolly sphericon? I use 67lb cover paper.

Once you cut out the shape, score the curves and the straight lines within the net.

Adhesives matter

To get everything to stick together I recommend using some kind of double-sided adhesive.

I’ve tried regular tape and white glue and glue sticks: I do not recommend using regular tape because it messes up the rolling edge. I do not recommend using white glue or glue sticks because I have to hold everything together while the glue dries and this takes too long.

I had some big glue dots around. I like the way they worked, especially as I could stretch them over a larger area that one would expect. When I ran out of glue dots I discovered adhesive LINES. Very cool.

Stretching Adhesive LInes

See the the adhesive line stretching in the photo above?

I put the adhesive on all the glue surfaces before I actually try to adhere anything to anything else.

The first sticking

I find that it’s best to stick the long tab to the one straight edge before doing anything else. You will likely disagree with me and try to glue one set of the teeth first, then the other set, then do the tab. Then you will realize that this was a mistake. Oh well. Told you so.

So Adorable

I think these look so adorable at this stage that I had to post both of these photos. Now you just have to somehow get those teethy things to stick to the inside edge of that arc (which, according to Vincent, is about about 127.3 degrees of a circle. About)

Getting these to stick together perfectly is just not possible. But good enough is actually good enough. You have to do a 3D print for perfection.

Here’s a video of me struggling through making this. It’s worthwhile to watch but there are some reaaaalllllly boring stretches. The whitish on the bottom of the video never goes away. Sorry.

 

Here are are some interesting twitter threads to look at. Click on them then scroll up.

Here’s the video that inspired my son to print some sphericons: https://t.co/6JydqEh1Ul

It includes links to 3D printing patterns as well as details about his experimentations with sphericons based on different shapes.

Cc: @PaulaKrieg

— Sally Biskupic (@SallyBiskupic) November 11, 2018

 

and

Ta-dah! @GotMathHelp pic.twitter.com/pM29EdBsru

— Paula Beardell Krieg (@PaulaKrieg) November 11, 2018

Round-up of Holiday Season Projects

A Bevy of Paper globes Directions at https://bookzoompa.wordpress.com/2017/11/24/spiraling-paper-ornament/

Making things out of paper seems to be something we do during the holiday season.

Here’s some projects that I’ve written about with links to the full posts that explain them, that seem appropriate for the season.

This first one, the Spiraling Ornaments was surprisingly well-liked. All year, when I visited different places, I’d see ones that people I know had made.

Video tutorial (not mine) for Kaleidocycle https://www.youtube.com/watch?time_continue=4&v=tdv1GwY025M

 

 

Here’s a template for a kaleidocycle. Not particularly holiday-ish, but fun and colorful, folds into something like the image below. More about this at https://bookzoompa.wordpress.com/2017/02/19/kaleidocycles-and-tetrahedrons/

Kaleidocycle/tetrahedrons

 

Next, directions for a six-sided snowflake. My big tip is to use paper napkins, as they already are the right shape: no extra prep needed! Also, paper napkins cut quite easily. They are perfect for snowflakes.

https://bookzoompa.wordpress.com/2010/12/14/tis-the-season-to-make-paper-snowflakes/

 

If you want to understand how the cuts of your snowflakes affect the final design, see below:

 

https://bookzoompa.wordpress.com/2014/12/02/paper-snowflake-cutting-tips/

 

 

Festive Jumping Jacks are quite fun. I’ve made these with kids just a few times, as all the knot tying makes this an intense project for anything more than a small group, but so worth the effort!

To work out how to make these you might have to look at a few posts, which are all listed at https://bookzoompa.wordpress.com/2012/02/04/wat-meer-trekpoppen-more-jumping-jacks/

The stars below are tricky to make, until you get the hang of them. I still have the ones I made on display from last year.

https://bookzoompa.wordpress.com/2017/11/09/pentagons-paper-folding-stars-origami/

The original post contains a good bit of discussion about the geometry embedded in these shapes.

Finally, making little books with stories or messages is always worth doing.

Origami books made from a multiple folded papers, to create a Star Book and a Cascading Book, aka Origami Caterpillar Book

Here’s a post that can get you started on some simple books to make with kids https://bookzoompa.wordpress.com/2017/10/10/four-books-students-can-figure-out-how-to-make-on-their-own/

For more an overwhelming amount of other book ideas, check out what I’ve tagged as making books with children

There you have it. Enough to do to keep you out of trouble at least until January.

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

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 fits so well on 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. 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 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.

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

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.