While browsing the Kim Komando website for ideas I could use in my blogging class, I followed a rabbit trail through Kim’s video archive. I think we will try this in Math Club next semester:
If the embedded video doesn’t work on your computer, you can find the original here: Unbelievable Paper Transformer.
Update
Simon posted a template for the puzzle at his MathsClass blog.
Denise Gaskins' Let's Play Math 
We’ve done this in my family – it’s incredible and works just like it looks. Kids quite young can do it. Plus the geometry is pretty interesting.
-John
For a whole book of these sorts of things, you should check out “Metamorphs:
Transforming mathematical surprises” by Robert Byrnes from Tarquin Books. If I remember right, this book has this particular model, and a bunch of others made with cubes and tetrahedrons.
There is a modular origami construction similar to this – a ring of rotating tetrahedrons, that is mentioned in quite a few books, but that I haven’t found an online crease-pattern for. I mentioned it in this post, and there is a video of it in action here.
I just tried to make it from paper. I should have paid more attention to the title of your post, Denise. Mine is probably too sloppy to work. I was wondering if I got the tape on the right sides. It sure looks like it.
I’d love to be able to do it for the math salon in two weeks, but I’ll need to practice a lot – this is not my forte. Do I need to find something the same weight as 3×5 cards, but bigger?
The whole time I was cutting, I was thinking about how to print a pattern for it, so the cutting would be easier, but I don’t know if I can put cardstock through my printer. Thanks for the book mention, Dan, I’m buying it!
I had hoped the info that went along with the video would include a link to the pattern, but no such luck. I wonder if my library has that “Metamorphs” book…
The written info on the video had errors. He wrote 5cmx4cmx3cm for the strips. He meant 5cm+4cm+3cm = 12 cm long. He didn’t give the width, but he gave the paper total, from which I figured a width of 3cm.
If I can find cardstock, maybe I’ll try it again today. If I get it right, I’ll post a pattern.
We had a disrupted day at school yesterday, and I had found this activity on your site the day before.
I made a template to photocopy, and posted it on my blog.
http://mathsclass.net/comments/end-of-term-activities-6/
Thanks for the find, great activity.