Is a coral-based planet possible?
Inspired by this answer and the mention of a hyperbolic plane, I've been imagining a world based on that geometry. The closest real-world example of such a geometry I can come up with is hard coral. I've been imagining a planet which is a coral, something which essentially looks like this (on a planet-sized scale), only with deeper folds:
However, I know there are going to be some problems with a world of this design. (Things like structural integrity, inner pressure, and gravity come to mind.)
I don't know what those problems might be, and because of that, I don't know if a coral-based world is utterly impossible, or is entirely possible, or is possible only with certain changes to the design. That's what I want to know.
Very Important Note: This question is primarily concerned with physics, not biology. For this question, assume that space-faring non-aquatic creatures visually identical to coral (since actual coral requires water - assume that requirement isn't a factor) are perfectly viable. I'm more concerned with things like size, mass, gravity, atmosphere, etc.
The Design: Ideally, I would like an Earth-sized spherical coral, with a surface akin to that of a hyperbolic plane (eg, massive folds rather than a flat(ish) surface). If this is possible, or possible but only with changes, let me know. It is very important that the hyperbolic-like surface stays that way. The rest I am flexible on. This includes:
- Size/mass. I don't care about the size of planet. Bigger is better, but if making the planet smaller will make it possible, I'm fine with it.
- Depth of folds. A true hyperbolic surface is made up of folds and trenches. I would like those trenches to reach down as far as possible, to the core if they can, but as long as they are at least as deep as mountains are high (~29K feet), I'm good.
- Planet. I'm after a single coral the size of a planet, not a coral which grows and covers a planet. However, if this is impossible, I'm willing to go the later approach (planet-covering coral), as long as the coral ridges are high enough.
- Material. I don't need a literal coral. The world can be constructed out of anything natural (vs man-made). Actual coral would be ideal though, if possible.
- Occupants. Humans are ideal. I at least want this world habitable by humanoid creatures, meaning bipedal, and more or less the size of a human. Don't worry about biology. I just want to make sure the coral-world can support such beings in terms of gravity, atmospheric pressure, temperature, etc. As long as these factors are close to Earth, I'm good.
Answers: Answers to this question should say whether a world with a hyperbolic surface (meaning coral-like) is possible in the realm of physics. If such a world is entirely impossible under any circumstances, the answer should explain why in general.
If such a world is possible, or is possible only with major changes, the answer should note that, and if applicable, note which changes would be required, and why. Again, in general. I will ask other questions if I need to drill down on any specifics. A brief mention of any important differences between Earth and the coral-world, if applicable, would be appreciated (for example, there will be no rain, constant earthquakes, etc.).
Answers should NOT get into the biology of the coral-planet, or the biology of the inhabitants of the planet. Answers should deal only with the physics of the world, and by extension the physics of inhabitants living on such a world (aka, make sure they aren't getting crushed by gravity or floating away into space).
Question: Assuming that space-faring corals are viable, would a planet-sized one be viable, and if so, would human-like (with adjustments as necessary) life on that world also be viable?
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