It's once again Monday, which means it's time to highlight some of the most recent community submissions posted to the Math Craft corkboard. I also thought we'd take a look at building a model that has appeared in numerous posts. It's the simplest of the intersecting plane modular origami sculptures: The WXYZ Intersecting Planes model.

My wife, Elizabeth Poole, made some kirigami snowflakes based off of last week's instructions. She posted up a few examples for everyone. I really like the snow angel:

I made a few more, as well. You can see more examples here. So far, this is my favorite:

My wife also gave me an idea—making snowflakes based off of tessellations. I used some of M.C. Escher's drawings to make some Escher "Snowflakes". Here's a picture of a few of them:

Justin Meyers of Scrabble World posted up a link to a really cool how-to on making beautiful origami Christmas trees. Here's what the finished product is supposed to look like:

Here's a "quick" version I made. It does take a couple of hours to make, but it is pretty fantastic.

Rachel Mansur posted up a plethora of links to some amazing origami artworks. My favorite is probably this post on the Sierpinski tetrahedron.

Or maybe it was this picture of a giant Sonobe icosahedron:

Or this picture of the Linux Penguin from Sonobe units:

Or this post on models folded and photographed by Michal Kosmulski. I really like the Menger Sponge:

And the STUVWXYZ (8 planes) intersecting stars:

In fact, since these intersecting stars keep coming up, I think we should learn how to build the simplest version of them—the WXYZ Intersecting Triangles. This really is the same object as the orderly tangle of 4 triangles, but a very different construction.

Here's what the finished model should look like:

## Materials and Tools

- Square paper (12 sheets)
- Folding diagram

150-cm (6-inch) origami paper works great, but if you need to make your own square paper, you can find that in this post.

## Make a WXYZ Intersecting Planes Unit

Read the folding diagram. Take a square piece of paper.

Flip it over to the uncolored side.

Fold the top over the bottom at the middle.

Fold the left side over to the right side at the middle.

Unfold the last step leaving a crease.

Fold the right side over to the middle crease.

Unfold the last step leaving a crease.

Take the top left corner and fold it over so that the top goes from the top center line to where the corner touches the last crease you made. This forms a 60-degree angle, since it is a right triangle where the longest side is twice as long as the shortest side.

Fold across by bisecting that last 60-degree angle.

Fold the top right corner across so it lays right on the top left corner forming another 60-degree angle on the other side.

I skipped a step. Fold the top right corner back over to the right bisecting the 60-degree angle as you did for the left side. Now unfold both of the angles leaving just the creases that you made.

The next step is to squash fold both of corners back in to make a 60-degree angle. You are just reversing the top creases. Here's a picture showing the top right edge of the paper. We are going to push this edge down into the paper.

Push the edge down using the creases as your fold lines. You will have to make the creases fold the other way.

Here's what it should look like when you've made the squash fold.

Squash fold in the other side.

Fold the top flaps over to the middle.

Fold the other flaps around to the other side. You will end up with an object that looks the same on both sides.

Fold the white parts up along the colored edges.

Repeat with the back flaps. Here's what the completed unit should look like.

## How to Make the WXYZ Intersecting Planes from 12 WXYZ Units.

First you need to have 12 Units. The completed model will look much better if you use 3 units of 4 different colors.

Read the folding diagram. Take one of these units and put the flaps into the slots. The flaps should go both directions so that the slots hold it from 2 sides.

Now take another unit and insert the flaps into one of the other units and use the slots to hold the flaps of the other.

Now you can keep adding units, and as long as you follow these rules, you should be successful:

1. If a unit's flaps are inserted into the slots of another unit, then there should be a unit of the same color on the other side doing exactly the same. This can be seen in the first picture below these rules.

2. If a unit's flaps are inserted into the slots of another unit, at a spot exactly on the other side of the model the reverse situation should be happening with units of the same color.

Here's a picture of the completed model. It can be a little tricky to build, but it looks amazing.

## Show Off Your Work

If you make the intersecting planes or any of the other previous Math Craft projects, please share with us by posting to the corkboard. Perhaps you have some original project or something you've seen on the web that you'd like to share.

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.

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## 18 Comments

Cory, thanks so much for posting instructions for the intersecting triangles... this is definitely at the top of my math craft to do list. I will tackle these before next week and post results :)

Just take your time with both the units and the construction. It gets a little tricky keeping everything together at a couple of points in the construction process. Just relax and think about the shape. You'll figure it out. It really is a pretty fun little puzzle.

after you learn the ropes, assembling is the best part with these module based models... it really is like a puzzle.

Yep. It's pretty fun!

Stop tempting me!

Never!

@ cerek ... fold one!

So many things to do! Between making how-tos and all of this oragami, I am overwhelmed!

I'd love to learn how to fold the menger sponge. maybe Michal Kosmulski would guest post a tutorial... (wishful thinking)

here's a similar menger sponge - http://saston.deviantart.com/art/Origami-Menger-Sponge-112305388

having hard time finding instructions online for this type of model. i keep seeing the business card model.

There is also this one.

It is made out of sonobe units- if you can somehow connect them it would make the Menger Sponge

Wow. I always want to build a good Menger sponge but the problem is that unlike many other fractals going up an iteration costs you so much. 20 times the units for each level up... The Sierpinski tetrahedron only costs you 4 times as much for each level...

Nice job and great demonstration you really teach us how to do it. 1916 Bungalow House

Thanks. If you ever make any of the projects or anything else that's even a little bit "mathy" post it up on the corkboard. I just googled you and checked out your blog. Pretty cool. I love Ebelskivers! Now you've made me hungry!

thanx a bunch!!!!!!!!!!!!!!!!!!!!!!!!!

it's amazing wow.......

Very Cool. My kirigami patterns as of now are very simple. I'm hoping to learn more of this kind of designs for more intricate patterns. Brilliant!

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