|MadSci Network: Earth Sciences|
Hey, Robert, that is a neat question!
I am not surprised, though, that you got several different answers, all quite reasonable ones, from your colleagues.
The problem is, you see, that the question is not really well-defined. We have to do a lot of philosophical analysis before we can even start on the science. It all hinges around the exact meaning of 'stable', and whether, and how the notion of 'stability' could be seen as an attribute of a shape rather than as an attribute of an object.
Even for objects, scientists use the notion of stability in a few different ways. I am primarily a chemist, and the objects chemists are usually interested in are substances or molecules. I am going to give an example that has absolutely nothing to do with shapes, but will illustrate some of the ambiguities and difficulties that chemists have with the notion of stability.
There is a substance called formaldehyde that has a fairly simple molecule: a central carbon atom bonded to one oxygen and two hydrogen atoms. In one sense this is a perfectly stable gaseous substance. Its molecule shows no inclination to fall apart and turn into something else, even if you jostle it quite hard with some other unreactive molecules, for example nitrogen gas at 200°C. But in a second sense it is not stable. You cannot keep a sample of gaseous formaldehyde for long, because one of the substances that a formaldehyde molecule will react with is formaldehyde! Formaldehyde molecules react with other formaldehyde molecules to make either a white solid polymer from an indefinite number of formaldehyde molecules bonded together in a daisy-chain, or a white crystalline substance called sym-trioxane from three formaldehyde molecules joined in a ring. A third sense of stable is 'environmentally stable'.
Formaldehyde is released into the atmosphere in tiny quantities from plastics and chipboard and similar materials. But it is not stable in the atmosphere. Oxygen and related substances react with it to turn it into carbon monoxide and carbon dioxide and water. Finally, there is the notion of 'thermodynamic stability'. For any set of atoms there is a single, lowest energy chemcial arrangement, that is ultimately stable. You could not possibly turn it into anything else without providing extra energy. In the case of formaldehyde, the substance is not 'thermodynamically stable'. The low energy (stable) form of those atoms is as a mixture of water, H2O, and graphite, a solid form of carbon.
Here is a much simpler and more familiar example. If you dig a hole in the soil and put a post in it, you can get it to stand up and be stable. You can make it more stable by compacting the soil around it and maybe propping it with a few rocks. You can make it more stable still by concreting it in. A yet more stable arrangement would be to bury the entire pole, but that probably defeats the purpose of what you are trying to do with it!
So there is a problem with 'stable', but what does that have to do with shapes? Well we have to start out by agreeing on what type of stability we mean. Then it will turn out that for a particular type of object in a particular context, there probably is a 'most stable shape' in many instances. But that will not always be the same shape, depending on the type of object and the context. Thus the most stable shape for a planet is a slightly flattened sphere (check this answer from our archives). But a flattened sphere is not a stable nor suitable shape for a girder or a foundation if you are making bridges or buildings!
What about the three shapes that you suggested?
For any object made of uniform material that is held together by central forces, the most stable (lowest energy) shape is a sphere. If the object is spinning the sphere will be slightly flattened at the poles. That accounts for the shape of planets and stars, and, to some extent, of atoms.
A sphere is also the shape that encloses the largest volume for a given surface area. That accounts for the shape of liquid drops and soap bubbles.
The hexagonal network of a beehive is the shape that gives the minimum amount of wall per unit area for a building where every point has to be closer than a particular fixed distance from a wall. Strong lightweight buildings or materials can be based around a honeycomb network of cells.
The tetrahedron is the shape that you get if an object is holding four other objects close to it, but those four are trying to get as far away from one another as they can. It occurs a lot in chemistry, especially in the shapes of molecules with carbon atoms.
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