# 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 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.

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?

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

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. Two halves of an icosahedron.

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

### 87 responses »

1. Oh, wow! These are amazing – I think I have another weekend project. Thanks, Justin.

• I’m glad the post inspired you, Jen! And thanks for sharing the link to LeWitt’s work in Hartford

2. I think it was very cool . my mind is still wondering how they twist and turned the spidrons. It was interesting how at the end the gold spidron went to the silver one, then I realize that I can conclude that the silver one had to be bigger than the gold one to fit inside the silver spidron. The silver one also had to have patterns to fit other side of it and outside of the gold one. thanks for the great videos

• Hi Ginger,

I’m glad you thought it was cool. I thought so, too.
The best way to get a sense for how the spidrons twist and turn is to make one for yourself. I just finished my first one yesterday, and I feel like I have a much better sense for how they work now than I did when I had only watched the video.

Also, the amazing thing is that the gold and silver Spidronised Yoshimoto pieces fit inside of each other! That’s part of what’s so amazing about the Yoshimoto Cube. You can read more about this fascinating object here: http://mathmunch.org/2012/06/18/music-box-fatfonts-and-the-yoshimoto-cube/

You’re welcome for the videos, and good luck making spidrons!

Justin

3. This video is a very touching as in interesting please make more cube videos they can fascinate you!

4. Sol LeWitt’s Incomplete Open Cubes was so cool to me because I loved how he made them with not only geometry, but his ideas. Also, it is so cool on how the human eye works, on how at one angle, it can look dis-organized and messy, but at another angle it can look very organized.

5. The video With the Partial cubes Was confusing, but then i understood that all of those are part of a cube, and i never knew there were so many ways, I wonder how many ways there is for a pyramid?!

6. That is cool how those two pieces of papers are the same shape but if you mess with one you can put the secound one inside it. It looks almost impossible to do.

• Hi Natalie,
Totally. If you liked that video, you might enjoy checking out our previous posts about the Yoshimoto Cube (http://mathmunch.org/?s=yoshimoto). They’ll show you how you can make a Yoshimoto Cube of your own if you want to!
Justin

7. This video was really cool. The spidrons changing shapes really made me think. Hopefully I can find out how to make one. Thanks for the great videos!

• Hi Diego,
You bet! And do try making a spidron of your own. I just finished making one, and it wasn’t as hard as I thought it would be. Just check out this video: http://www.youtube.com/watch?v=rdwts5jSjmo
Good luck!
Justin

8. I just watched the Sol LeWitt’s Incomplete Open Cubes video. It’s so interesting to me because only when you look from up close, can you actually see the individual open cubes.
When you back up and look at it as a whole, it looks like a jumbled mess of a maze, almost impossible to see individual structures. This is really interesting that open cubes really seem that complex. Amazing!

• Hey Ryan,
You’ve made a great point about perspective—how looking at a math object at different levels of scale can show you its simplicity and its complexity. Something like this sometimes happens with fractals:
http://mathmunch.org/tag/fractals/
I’m glad you enjoyed the video and the post. Happy reading!
Justin

9. That is super cool, I love it! How did you do that with only two boxes? Or did you made it in the computer?

10. This video was incredible. The video did not make sense at first, then you amazed me because I thought you were just gonna make the two pieces of paper together. But instead, you put a piece of paper into a cube. Thanks for making this video!

• Hi Seth,
Glad you like it. Happy munching!
Justin

11. This is my first time reading an article from this cool website! It was really cool to read about things that I’ve never thought I’d learn in my life. Partial cubes seem difficult to make and hard to understand, from my eyes. Spidrons, or half-cubes, is something that I would never look up on Google, only because I’ve never heard of them until today. The Variation of Incomplete Open Cubes chart was hard to follow, but again was something that I’ve never learned or heard of before. This was cooland I can’t wait to check out another article and play some games and watch videos!!

• Hi Annalise!
I love your enthusiasm, and I’m happy that you enjoyed your first experience with Math Munch. One of our big goals is to share math with young people (and really, all people) that they probably wouldn’t run across otherwise. There’s a lot of fantastic math out there that can be hard to find if you don’t know where to look. But now you know that you can come to Math Munch to find some of those cool math things!
I hope you have fun exploring the site and coming back again and again!
Cheers,
Justin

12. This video is really cool I like how the two
Cubes change shapes and at the end they merge and turn into one.

• Hi Syanna,
I’m glad you liked it. Can you think of other shape that this might work with?
Justin

13. I thought it was amazing and super cool. It was also very confusing how the spidrons reflected so much on the paper itself when they were bent differnet ways because it looked super fake, like it was impossible! One thing that mostly got my asttention, was when the golden spidron went into the silver one at the end and how it fit so perfectly inside. Also, how the silver spidron didn’t follow in sinc with the golden one for only two moves and it made such a difference on the shape of the spidron.

• Great observations, Raphaella! Know that this video was made of computer models of spidron Yoshimoto Cubes, which may be why it looks “fake” to you.
If you haven’t already, you might also enjoy checking out the videos about the Yoshimoto Cube that Paul and I made.
Happy Munching,
Justin

14. Above the video it says there are two videos. One for the templates, but I only see one video where is the second because i would like to make one. It’s cool to learn how these and the Rubics cube have a connection, although they are very much alike! The artist of the video must have lots of practice because this is a very well made video, and did the creator of the Rubics cube make this video. I was amazed with the video, but a bit confused too. Please continue posting videos showing illusions and paper folding in awesome ways.

• Hi Ryan,
There are links on the words “first” and “second” in the paragraph about the how-to videos. They should take you to the two videos.
The inventor of the Rubik’s cube did not make any of these videos, but Daniel Erdely did make the second how-to video!
We’ll definitely continue to post great mathy videos. We love finding them and sharing them with you. Good luck with your templates!
Justin

15. I thought that was awesome!! Also how did you get that to do that is it paper or something else. Also is the video stop motion how is it moving if we cant see anything moving it. That was a very Unique video its cool that they always fit together!

• Hi Nico,
The Spidronised Yoshimoto Cube video was done with computer animation. The one that Paul and I made was done with stop-motion.
Glad you enjoyed it!
Justin

16. The video of the partial cubes was great in the beginning of the video you can see a pyramid made of them and also giant cube. When they were talking about how the partial cubes looked like a something tangled I looked at it and it did. When I looked at it closer I could see that the partial cubes were actually made of right angles. The best thing about this video to me was what kinds of things you can make from partial cubes.

• Hi Christian,
Thanks for sharing your reflection on the Sol LeWitt video. You make some great observations.
Happy Munching!
Justin

17. Hi Everybody!
I’ve just got this link. First of all I have to give credits to my collegaues who helped me to create all these artworks and video: Rinus Roelofs, Walt van Ballegooijen, Lajos Szilassi, Paul Gailiunas, Amina Buhler-Allen, Marc Pelletier and other members of the Spidron Team which has been working on this issue for about 6-8 years. Another remark is that the first two picture are not parts of a cube. They are something esle, what is very difficult to describe. By the way you can see ‘real’ Spidron cuts in half cubes in the animation, where the parts are really halfcubes.

Thank you anyway to mention our work here.

Daniel Erdély
the Spidronist

• Hi Daniel!
Thanks so much for dropping by!
I had thought that the first picture was something like these half-cubes, but with the hexagonal face spidronised:

I’d love to hear more about what the “something else” in those two picture actually are. I’ll follow up with you by email.
I hope you enjoyed checking out Math Munch. Thanks for creating such cool math!
Yours,
Justin

18. I didn’t know that there were so many ways to make an incomplete cube. I kind of see why there are so many ways because there are so many edges so there could be a lot. Also why did you want to know how many ways there were, why did you want to do this project?

• Hi again, Nico!
The incomplete cube project wasn’t mine—it was Sol LeWitt’s. If you want to know more about his motivations for his project, you’ll have to do some research. I would like to explore the related question about pyramids. Maybe I’ll try that this weekend!
Best,
Justin

19. I really enjoyed the video and was fascinated by the way the spidron works and how at the end when the gold one went into the silver one and how it fit so perfectly.I am really excited to watch more of these videos and maybe I will be able to make one myself.

20. Wow i really enjoyed this video and i found itreally interesting how at the end they both turn into one this is one of the best cideos i’ve seen!!

21. Awesome! how did he do that?! Its amazing when the gold cube closed and jumped inside that silver cube! i love learning about these kinds of things. I will definitely look Sol LeWitt up!!!! I always wondered about the science of folding aper. I mean seriously, it looks like that many folds would rip and wear out the paper. Well what kind of paper is that anyway?

22. This video was really amazing i was astonished how the spidron worked i was amazed at the end of the video by the gold cube jumping into the silver cube without a dent.

23. I didn’t quite understand the unfinished cubes topic. However, it was astonishing to think about all the different unfinished objects. The spidrons were breathtaking. The gold one was able to fit in the silver one perfectly. Thank you for teaching me about spidrons and unfinished objects and how amazing they can be.

24. I just got finished watching the video about the incomplete open cubes and it was all so crazy and confusing to try and distinguish the incomplete open cubes one by one! I did learn that Sol Lewitt used a lot of geometry in order to create the open cubes. I also learned that creating a 3D open cube compared to drawing and actual 3D model, is the different ways the mind and the eye interact with each other as you observe them seperately.

25. The video about the incomplete cubes was confusing at first but once I thought about it more it was like stepping into another world of imagination. Also the video gave me a little more sense of how math changes we people look at things and in that result makes us think on a whole new level.

26. I was very interested in how the gold spidon went inside the silver spidon. At first I did not know what was going on then I saw them turning in a how to certant way that just clicked. I don’t know to explain it but it just clicked.

27. This video was awesome, it was very interesting how the shapes can move around and at the end the gold one went inside the silver one.These shapes have a lot of design and look hard to create.

28. These shapes look amazing, I especially like the one that looked like a rose, or the spidron. I want to make one of these but i don’t know where the instructional video is. From a mathematical stand point these are very intriguing, it is hard to see how those shapes put together make cube. It is neat that they were invented by a guy who was working with the inventor the Rubik’s Cube.

• Hi Matt,
Thanks for your comment. I agree that those rose shapes are awesome. I’ll ask their creator, Daniel, if he has any recommendations for how to make them. There are some instructional videos about other spidron creations in the post; check them out!
Justin

29. I just watched the Yoshimoto Spidronised video–it was amazing! I realized that the gold and silver cubes were somewhat like crystals. The shapes formed were just different variations of how the cubes were cut/split inside.

The only part that didn’t really make sense was at the end when the gold cube went inside the silver cube and was entirely enclosed by it. Throughout the video the cubes looked to be the same size and shape exactly, but maybe it was perspective because the gold cube was further away. But if it was based on perspective, I would think the gold cube would be bigger due to it being further away and looking the same size. At the end, it seemed that the silver cube was larger. Was this the case?

• Hi Kylie,
I’m glad you enjoyed the video. I appreciate your observations, and you question is one that’s cropped up a lot in the comments. The gold and silver pieces are in fact identical–same size, same shape. But because of hinges, each can take on several forms. When one is put into its hollow cube form, and the other into its star form, the second one exactly fits inside of the first one.
It’s pretty wild, but it’s true! If you really want to convince yourself, you can make a Yoshimoto cube of your own! Just head to this post:
http://mathmunch.org/2012/06/18/music-box-fatfonts-and-the-yoshimoto-cube/
Bon appetit!
Justin

30. I like the way they showed all the way because I had thought that their had been like six so when they showed all the ways I thought that was pretty cool and I also like the way they merge.

31. Its cool how LeWitt’s lines all the incomplete cubes up, so that when you look at it a certain way, its a patter.

• Hi Noah,
How many complete cubes do you think there would be if you used all of the parts from the incomplete cubes? An interesting question…
Justin

32. l thought that the video was very cool. The spidons changing shapes was interesting to me. At first, I thought they were the same size. But at the end, I saw the gold one went into the silver one perfectly. So I realized the silver one was bigger.

• Hi Nina,
I’m glad you enjoyed the video.The silver and gold spidronised Yoshimoto pieces are actually identical–same size, same shape. It’s just that, because they’re hinged, one can be made solid and the other hollow. And the solid one is just the right shape to fit exactly inside the hollow one!
If that helps, great! Something that’s sure to help, though, is to make one of your own. Check out the templates here:
http://mathmunch.org/2012/06/18/music-box-fatfonts-and-the-yoshimoto-cube/
Bon appetit!
Justin

33. Hi math much team i just have to say thank you for posting all these cool things about me. This was also another one of my favorite videos besides the video about how to turn a sphere inside out. I thought this video was awesome and very complex, I thought it was awesome beacusce first i thought that the two of them were the same size, but they actually weren’t which i found out towards the end of the video. I found out that the silver one was bigger then the gold one. Also, in the beginning of the video i thought that they were big sparkly diamonds, but i later figured that out. Any ways thank you math munch for getting me to like math.

• Hi Megan,
It’s our pleasure!
The gold and silver are actually identical objects. Check the comments above for an explanation.
I’m glad you like math better now than you did before. Do come back by to Math Munch!
Justin

34. i meant about math

35. Wow! How did you do that? I tried looking at a couple videos on youtube after I saw this one. My dad and I plan on making it. It seems complex to make but its worth a shot to try and make it. Any way I can’t believe you were able to fit one of the spidron’s into the other spidron even though they were the same size. The spidron I would like to make is the one that is shaped like a rose because rose’s are by favorite flowers and I have always been intrested by their beauty.Anyway (sorry for getting of topic) I really enjoyed this video so thank you for posting this video have a nice day!!!!!!

36. wow math art is so cool, its like a more extreme origami. i think i have a weekend project to do now. thanks for the great ideas. Ryan Williams.

37. Wow the video about open cubes was really cool! I did not know that there was that many forms of open cubes i also really liked how from one angle the cube looked really messy and jumbeld up, but from another angle it looked nice and neat and how it was suppose to.

• Hi Anthony,
I’m glad you enjoyed the video. Great observations!
Justin

38. i like how the video was 3D and that the video was kinda easy to follow than a hard confusing puzzle the cubes were more understandable. i love the videos you guys make. =)

• Hi Gabe,
Most of these videos we just share—we don’t make them. But I’m glad you enjoy them so much. :)
Bon appetit!
Justin

39. In Yoshimoto Spidronised it was amazing how they changed. I saw that the silver was bigger than the gold and so they fit in just right. I have something like that that shoes pictures of landmarks. You flip it and each time you see a different one. Great video.

40. I really liked this video. It was so cool how two simple cubes could make such amazing designs. Do you guys have any more videos like this?

• Hi Monse!
You can browse the videos we’ve posted to Math Munch on our YouTube channel:

I might recommend these video as places to start if you like simple shapes creating complex patterns:

and

Enjoy!
Justin

41. I don’t get how an unfinished cube can be look at as a finished cube. it just looked like a bunch of 3D lines to me. how do you see it. I thought it was interesting though.

42. After watching the Sol LeWitt video of incomplete cubes, I really had to think. It seems like such a simple concept, incomplete cubes, but when you really think about it, there’s a lot more behind the concept than you might originally consider. I guess I’d never realized all of the various combinations that are possible to create. Plus, those are just the combinations that you can make that will work structurally; you can make even more when you’re just doing it on paper. From afar, it seems like just a tangled mess, but when you move in closer, you can see all of the details that Sol put into it. This video was very thought-provoking and I’m glad that I watched it. It really gives you something to think about.

43. I thought this was a very interesting video. at first I thought it they were the same size but at the end I saw that the gold one fit right into the silver.

44. IT WAS AN INTERSTING VIDEO IT WAS COOL WHEN THE INCOMPLETE CUBE COMBINED WITH THE OTHER CUBE TO MAKE A WHOLE CUBE

45. The video was very interesting but can you do this in real life not just on a computer. Lastly some of the foldes seemed to be impossible to do.

46. I just watched the LeWitt video, and i must say.. Wow! it was very interesting, i watched it once and to make sure i understood i had to watch again. its interesting to me because you have to look at his work a different way to see what he saw.

47. LeWitt’s art is really mind boggling. At first when you look at it it looks like a mess, but taking a closer look at it can make you decipher the different orientations of shapes and your mind breaks everything down piece by piece and I think thats really amazing.

48. Fascinating post. I’ve been puzzling over Sol Lewitt’s open cubes recently and it brought me to Math Munch for the first time — what a great site! Thank you!

If Sol Lewitt’s diagram is a list of every incomplete cube which has a segment in each of the three dimensions — flattened into a 2D line drawing — then would it be possible to work out a “variations of incomplete open hypercubes”, with every incomplete hypercube which has a segment in each of the four dimensions — flattened into a series of 3D sculptures?

I can sit down and try to recreate the incomplete cubes diagram just imagining an open cube in my mind’s eye, then rotating it around to see if I’ve already drawn it or not. I can’t quite do the same with a hypercube, though — just too much to hold in my head at once! So I guess there are two places where I’m getting stuck trying to solve the hypercube list:

First, is there a good way of representing hypercubes in 3D space, akin to Sol Lewitt’s line drawings? Second is there a way I can use math to rigorously determine which are the unique incomplete hypercubes, since mental imagery won’t cut it anymore?

49. I don’t get how the cubes in the video really works but it was really cool how they folded.

• Michelle,
Try watching it again, or leave it be for now and come back to it later. It’s surprising how new ideas can suddenly click even when they haven’t before.
Cheers,
Justin

50. Spidrons are just too cool. Can geometry get any better? From spidrons to polygon model files (from MMD), geometry is what makes modernized modernized!