Tag Archives: cube

Partial Cubes, Open Cubes, and Spidrons

Welcome to this week’s Math Munch!

Recently the videos that Paul and I made about the Yoshimoto Cube got shared around a bit on the web. That got me to thinking again about splitting cubes apart, because the Yoshimoto Cube is made up of two pieces that are each half of a cube.

A part of Wall Drawing #601 by Sol LeWitt

A part of Wall Drawing #601
by Sol LeWitt

A friend of mine once shared with me some drawings of cubes by the artist Sol LeWitt. The cubes were drawn as solid objects, but parts of them were cut away and removed. It was fun trying to figure out what fraction of a cube remained.

On the web, I found a beautiful image that Sol made called Wall Drawing #601. In the clipping of it to the left, I see 7/8 of a cube and 3/4 of a cube. Do you? You can view the whole of this piece by Sol on the website of the Greater Des Moines Public Art Foundation.

The Cube Vinco by Vaclav Obsivac.

The Cube Vinco by Vaclav Obsivac.

There are other kinds of objects that break a cube into pieces in this way, like this tricky puzzle by Vaclav Obsivac and this “shaved” Rubik’s cube modification. Maybe you’ll design a cube dissection of your own!

As I further researched Sol LeWitt’s art, I found that he had investigated partial cubes in other ways, too. My favorite of Sol’s tinkerings is the sculpture installation called “Variations of Incomplete Cubes“. You can check out this piece of artwork on the SFMOMA site, as well as in the video below.

In the video, a diagram appears that Sol made of all of the incomplete open cubes. He carefully listed out and arranged these pictures to make sure that he had found them all—a very mathematical task. It reminds me of the list of rectangle subdivisions I wrote about in this post.


Sol’s diagram got me to thinking and making: what other shapes might have interesting “incomplete open” variations? I started working on tetrahedra. I think I might try to find and make them all. How about you?

Two open tetrahedra I made. Can you find some more?

Two open tetrahedra I made. Can you find some more?

Finally, as I browsed Google Images for “half cube”, one image in particular jumped out at me.


What are those?!?!

Dániel's original spidron from 1979

Dániel’s original spidron from 1979

These lovely rose-shaped objects are called spidrons—or more precisely, they appear to be half-cubes built out of fold-up spidrons. What are spidrons? I had never heard of them, but there’s one pictured to the right and they have their own Wikipedia article.

The first person who modeled a spidron was Dániel Erdély, a Hungarian designer and artist. Dániel started to work with spidrons as a part of a homework assignment from Ernő Rubik—that’s right, the man who invented the Rubik’s cube.

A cube with spidron faces.

A cube with spidron faces.

Two halves of an icosahedron.

Two halves of an icosahedron.

A hornflake.

A hornflake.

Here are two how-to videos that can help you to make a 3D spidron—the first step to making lovely shapes like those pictured above. The first video shows how to get set up with a template, and the second is brought to you by Dániel himself! Watching these folded spidrons spiral and spring is amazing. There’s more to see and read about spidrons in this Science News article and on Dániel’s website.

And how about a sphidron? Or a hornflake—perhaps a cousin to the flowsnake? So many cool shapes!

To my delight, I found that Dániel has created a video called Yoshimoto Spidronised—bringing my cube splitting adventure back around full circle. You’ll find it below. Bon appetit!

Reflection Sheet – Partial Cubes, Open Cubes, and Spidrons

Turing, Nets, and More Yoshimoto

Welcome to this week’s Math Munch!

The Turing Tenner

What you see there is a 10 pound note. You know, British money. So who’s that guy on there? Must be a president or king or prime minister or something, right? NO! That’s Alan Turing, one of the most important mathematicians of the 20th century. During WWII, he was a codebreaker for the Allies, intercepting German submarine codes. His analysis of the Enigma Machine was a huge turning point in the war. (video explanation)

In England they put the queen on one side of the money, but the other’s used for significant Brits. Charles Darwin is currently on the 10 pound note, but these things change, and there’s a petition to get Turing on the ten. A Turing Tenner, as they call it. It’s all part of Turing’s 100th birthday celebration.

Google’s homage to Alan Turing

Since Turing did some of the earliest work on computing theory and artificial intelligence, Google paid tribute to the computer legend with a recent doodle. It’s a fantastic little puzzle game based on his work. I’ll let you figure it out, but definitely try this one. Click here to play!

In last week’s munch, Justin introduced us to the Yoshimoto Cube, and we’ve kept on thinking about it.  Here’s a couple simple templates for making one cubelet.  (template 1, template 2)  Make 8 of those and hinge them together with some tape.  I made a short video to show you how to connect them.  But it didn’t end there!

A flat template for a 3D model like that is called a net or a mesh.  Do you know any nets for a cube?  There’s actually lots.  Check out this site, where it’s your job to figure out which nets fold up into a cube and which ones don’t.  It’s a lot of fun.  Here’s another net site showing lots of nets for a pyramid, dodecahedron, and a whole bunch of other solid shapes.  How many do you think there are for a tetrahedron?  Can you design one for an octahedron?

The Monster Mesh

I spent some time this week trying to design a better net for the

The Mega-Monster Mesh
A one-sheet model for the Yoshimoto cube.

Yoshimoto cube, and I think I succeeded!  The tape on my hinges kept breaking, so I wanted to try to make paper hinges.  With my first attempt, which I called The Monster Mesh, I was able to design a net for half of the star.  Down from 8 tape hinges to 2 was a big improvement, but last night I got it perfect!  Using my new version, The Mega-Monster Mesh,  you can make the entire cube without any taped hinges!  The model is pretty complicated, so if you want to give it a shot, feel free to email us at MathMunchTeam@gmail.com with any questions.

Finally, something I’m really really proud of.  Justin and I spent most of Sunday afternoon on the floor of my apartment making a stop motion animation of with Yoshimoto Cube models.  It’s called “Yoshimoto Friends,” and we hope you love it as much as we do.  (We used the free iMotionHD app for iPad and iPhone, in case you want to make your own stop motion animation.)

Bon appetit!


I made another video showing how the mega-monster mesh folds up.  Here it is, acting like a transforming bug!