It's Monday, which means once again, it's time to highlight some of the recent community submissions posted to the Math Craft corkboard. I also thought that we'd try and create something known as an "Orderly Tangle" or "Polylink".

This week we had a few submissions based off the projects of the week: creating parabolic arcs from straight lines and creating concentric circles.

Watermelonlemon shared two incredibly detailed pieces:

Cerek Tunca extended the idea of drawing parabolic curves using straight lines by also connecting all of the lines that wouldn't cover over the curve.

Justin Meyers of Scrabble world posted several pictures of a translucent cube with curve stitching designs that was colored like a rubiks cube. There are several more pictures and an explanation of his thought process and how he accomplished it in his corkboard post.

We also had two submissions from Imatfaal Avidya. The first shows a beautiful octahedron based off Richard Sweeney's modular curved paper sculptures. A couple of pictures can be seen in his corkboard post.

The second submission is his recreation of Thomas Hull's Five Intersecting Tetrahedra. I hope that Imatfaal posts a How-To on this. It's a very beautiful, but very challenging model to make.

Five Intersecting Tetrahedra is a class of object that is sometimes called an Orderly Tangle or Polylink. It's made up of 5 polyhedra that are all woven together to make a single object that is rigid. Robert Lang has created several more of these in origami, and George Hart has a page discussing them. It's fun to use his program that you can download on the bottom of the page to try and find your own.

Someday we'll look at recreating a few of the origami models, but for now I thought we'd create the simplest polylink, the Orderly Tangle of 4 triangles.

## Materials and Tools

- Paper (I recommend thick cardstock)
- Template
- Clear tape
- Scissors or X-Acto knife

## Download, Resize, Print and Cut Out the Template

Download the template. Print it out at the size you want. One easy way to make it smaller is to go into your printer settings and set it to print multiple pages on one sheet. I recommend printing it full size the first time you make it. Alternatively, you can make your own template by having an equilateral triangle with an equilateral triangle hole with sides half as long as the original triangle.

Use scissors or X-Acto knife to cut out the template.

Make a cut through each of the triangles on one side. Don't put it at the middle of the side or it will be harder to assemble the model.

## Link Two of the Triangles Together

Link two of the triangles together and move the triangles so that the midpoint of one side is at the vertex of the triangular hole. The opposite sides should look the same. I recommend using a little tape to hold the triangles in place at this point.

## Continue Linking Triangles

Now the puzzle begins. It's not exceptionally difficult, but considering it's only 4 pieces, it is more challenging than it looks.

You must link the triangles so that each triangle has another triangle side's midpoint in each of the vertexes of its triangular hole. This will also make it so that the midpoint of the side opposite this vertex is in the vertex of the triangular hole of the opposing triangle. It might help to think you are creating regular hexagrams or stars of David where each triangle is rotated with respect to the other.

Here are a few pictures showing the process.

Third triangle linked:

Fourth triangle linked (you will probably have to do a lot of bending and twisting to weave this one into its proper place):

Tape the triangles back together:

You can make them larger or smaller. Here's a picture showing them double-sized, regular-sized, half-sized and quarter-sized. Double-sized is floppy. You would probably want to make it out of thicker material. Half-sized works really well I think.

At one quarter the size, I think you could make some attractive earrings if you used some metallic coating for the paper. You might need to make them a little smaller.

Here's a full sized one with metallic origami paper glued to both sides of the card stock. Yet another Christmas ornament. My tree is going to be full this year.

## Show Off Your Work

If you complete any of these or any of the Math Craft projects, please share with us by posting to the corkboard. Maybe you just have something cool that you made or saw on the web—you can share that, too!

If you like these types of projects, let me know in the comments. If you have any other ideas you would like to pursue, let me know in the forum.

Next post, we're going to create some geometric shapes using simple modular origami.

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## 6 Comments

I just noticed the quarter-sized one is the size of a quarter. I swear that was unintentional. Really. I'll let you know when I'm telling bad puns.

Woah youre right- thats hilarious!

the five intersecting tetrahedra can be found in youtube under the name barbabellaatje

Looks like that user has quite a bit of cool origami how to videos. Thanks.

no

The template link is broken. Do you have an updated link, or can this be made with any equilateral triangles?

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