# Geometry Q: Collapsable Polyhedra (Work related!)

Let me say that I’m quite pleased to pose this question, since it’s truly work-related, which gives me an iron-clad excuse to check in with the SDMB throughout the day (or week) to see if it’s been answered.

I’m mulling over the possibility of designing an airtight, collapsable aluminum structure as a replacement for a rubber bladder (there’s various reasons why this might, or might not, work better). We’ve looked at bellows, but in the scale I’m talking about, the cost is prohibative.

So then I thought, “well gee, there must be some sort of polyhedron that, if hinged at all its edges, will collapse in on itself.” So I tried constructing something out of paper, and couldn’t find any that seemed to work. So, my question is:

1. Is it possible to construct a polyhedron, of any sort, that will collapse? (i.e., for a given construction of surface polygons, the internal volume of the polyhedron is variable.)

2. Is there a method for constructing such a polyhedron? I’m particularly interested in constructing on that has an “inflated” shape that most closely approximates a long circular cylinder, and has a minimum number of sides.

If you have any suggestions on reference material that would be helpful, I’m interested in that, too.

Well, a cube will collapse, which is why carpenters spend so much time and effort bracing their framing.

I’m not trying to construct a “stick” polyhedron, which is what you’re thinking of in your cube example; rather, I’m trying to construct a polyhedron out of sheets.

If you construct a cube out of two-by-fours along the edges, as you allude to, the structure will collapse because the square polygons making up the sides are allowed to change shape. If you add plywood sheathing to your two-by-four construct, the structure will not collapse. I’d like to construct some shape that will collapse, even if covered in plywood sheathing.

OK. Take one face of your proposed polyhedron and look at vertex V. Since the face is fixed in area and any change in the angle at V will change that area, the angle is fixed. Since this holds for all V, the polyhedron is impossible.

That seems right to me, but I’m willing to be shown differently. As I’m sure you know, many topological constructions are non-intuitive.

I’m sorry to do this, as I know it’ll curtail your SDMB visits (o, it won’t? nevermind), but Cauchy’s theorem is what you’re looking for. A convex polyhedron with rigid faces and hinged edges is rigid. That theorem is over a hundred years old.

Follow the link to the flexible polyhedra. It was only in the past twenty-five years that a non-self-intersecting flexible polyhedron was found. Apparently, Connelly et al. (1997) proved that a flexible polyhedron must keep its volume constant.

The faces are sheathed in plywood–they don’t change shape, and the angles of the faces don’t change. Still, you can have flexible polyhedrons. One of my math professors was working on the problem in the mid-seventies. We brought the guy who first found a flexible polyhedron to campus, and he gave some talks.

RM Mentock: Well, I did mention non-intuitive results. Thanks for the link. Gotta build me one of those flexible things.

Crap! I was afraid that the answer would be something along those lines. I find it intriguing, though, that flexible poyhedra actually can be built; although, since I’m looking for a volume change, it looks like Connelly put the final nail in the coffin of that idea.

However, a couple bright spots:

1. I think it’s neat that a semingly esoteric geometric proof has a practical use (at least for me). Feel free to refer the next whiner who asks “But what goooooooood is all this maaaaaaath?” to this thread.

2. I’m thinking I can use this to win a few nerd bar bets…

And, like jcgmoi, I’m off to build one of my own.