ArtCraftingMath

How to Flatten a Cube

Y'know, other than the obvious way.

A bit of background on this:

In 2008, Mythbusters made a lead balloon. To construct the balloon, Adam taped together a stack of square lead foil sheets, assembled in such a way that, when inflated, the sheets would unfold into a cube.

And it worked! The cube’s sides unfolded, and the top of the cube rose to its full height, executing a quarter-twist in the process (starting at about the 1:30 mark). Very nice and elegant.

But it started to bug me. There must be some way, I thought, to fold the sides of a cube so that the entire thing can collapse or expand without the top rotating.

After several tries, I came up with a design, constructed one out of paper, and tried it out. Satisfied that it worked, I promptly forgot about it.

Then, a few days ago, it started to bug me again. No idea why. I dug through my files, looking for the design, and couldn’t find it. And I had only a vague recollection of what it looked like. It took me a couple of days of playing with scraps of paper before I hit on the original design.

Here’s the cube in various stages of collapse. Yes, it’s a bit wonky. Probably would work better if I’d used a more rigid material, but paper’s good enough for a proof of concept.

cube, mostly intact
(mostly) cube-shaped
cube, partially collapsed
partially collapsed

 

animation of paper cube being crushed

I can’t think of a practical purpose for this thing. But, hey, sometimes you just feel strangely compelled to make something just for the sake of making it.

Here’s how it works:

  1. The center of each side pushes in, forming the two large triangles (eight total).
  2. The center of each edge pushes in, forming a diamond (four total), joined to the large triangles by smaller obtuse triangles.
  3. As the cube collapses, each diamond folds in half, and the large triangles fold towards the center of the cube.
  4. When the cube is completely flattened, the tips of all eight large triangles meet in the center of the cube… erm, square.

Now, here’s the thing: I have never found an example of a cube being folded in this particular way. I don’t know if that’s because no one’s ever done it before, or if my Google skills are lacking.

On the other hand, I half-expect that, thirty seconds after this post goes live, someone will comment with a link to this exact configuration being used in something completely commonplace. “Oh, that’d be a standard Østberg bellows mechanism, most famously used in Houtman F37 “Flying Mongoose” steam engines from 1930 to 1937. I’ve got three of them in my garage.”  Because of course.

If you want to make one yourself, here’s an image of three faces of the cube (two sides and either the top or bottom). You’ll need to print two copies to make a cube. The blank faces are top and bottom; the ones with crease lines are the sides.

layout of three sides of cube
Layout of three sides

 

Detail of one side
Detail of one side

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Steve DeGroof

Steve is an expat Canadian who now lives in North Carolina. He has worked, at one time or other, as: a TV repairman, a security guard at a children's hospital, and a janitor in a strip club. His current day job is as a computer programmer for a bank, which doesn't involve nearly as much being electrocuted and cleaning up vomit. He has a patent for a "Folding Stereoscopic Computer Display", which sounds a lot more impressive than it really is. He has created various "artworks", including: a baby woolly mammoth with a jetpack (which doesn't actually fly), a Giger counter (not a typo), a clockwork orange (a bowler-hat-wearing, wind-up piece of fruit that plays "Singing in the Rain"), a clock in the shape of Rick Astley that chimes "Never Gonna Give You Up" on the hour (for which he is sincerely sorry). His first book, "Dandelion Seeds", was written largely by accident (it's... complicated).

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