What's the largest cube of some material that would not collapse under its own gravity?

Factual Questions
21

It’s the length of a side will be the width of a cube and the diamter will be sqrt(3) times this length.

Polystyrene is very light and when I looked up the value for it’s compressive strength it was very high which is why the figure is so large.

Obviously my approximation is very much an approximation. All I did was to consider basically when the sphere inscribed in the cube will fail due to being crushed by it’s own ‘corners’. There’sobviously a lot of holes than can be picked in that approximation.

22

Polystyrene is pretty strong & pretty light. Polystyrene foam (aka styrofoam or typical coffee cups, egg cartons, etc.*) is a lot lighter. And also has less compressive strength. I wonder which is better for this?
*As DuPont never tires of telling us, stuff like egg cartons & coffee cups is not made of Styrofoam®™; it’s actually a different plastic and process.

23

I think Styrofoam®™ is just polystyrene foam made in a specific process. It’s extruded instead of expanded from little beads, but basically the same stuff. Extruded polystyrene has greater tensile and compressive strength than expanded, so that’s what you’d make the giant cube from if you were going to use polystyrene.

Wait, Superfluous are you trying to put make some kind of 2001 type giant monolith to affect the evolution of some alien species? That would be a violation of the Prime Directive, and I don’t think we should be helping you do that. Isn’t there an SDMB rule about that?

24

A proper analysis would have to have density varying with position, since the core is going to get compressed by the outer layers. But this would require knowledge of the material’s response over a very broad range of pressures, which isn’t known for most materials, and even once known would require some complicated calculus.

If you were to take everything into account, I suspect that no foamy material would work, since it wouldn’t be too hard to compress out the foaminess in the core, leaving the outer portions in a stronger gravitational field. Or at least, it wouldn’t work for the entire thing: You might be able to build a composite structure with a stronger material for the core, then weaker but lighter materials as you move out, culminating with six foam mountains.