One of the primary components of Dalton's theory is to "express" an atom as an unbreakable spherical shape filled with whatever. The question is, why didn't he suggest any other shape, like for example, cubes?
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I don't know what Dalton was thinking exactly, but I do know that for a very long time philosophers and scientists have assumed that things are spherical in the absence of any indication that they are not.
The reason for this was originally related to religious beliefs, and the idea that God or (The Gods) would only create perfect objects, and the sphere (being perfect) was the obvious choice for heavenly objects and subatomic particles.
The assumption that things are spherical is still used very often, although it is not usually directly related to religious beliefs anymore. These days, the idea is that you should always use the simplest possible explanation that works. The simplest geometric shape is a sphere - you only need one dimension to describe it, and it has some other interesting properties like minimized surface area-to-volume ratio. Mathematics can also be greatly simplified by using spherical coordinates when things are spherical, which is another big advantage.
Therefore, when we imagine particles that we can't see, a sphere is a natural choice, and I would guess that Dalton was using one or both of these lines of reasoning when he came up with the theory.
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1$\begingroup$ What exactly do you mean with "you only need one dimension to describe it [a sphere]"? Clearly you're not talking about spatial dimensions, so I assumed you might mean measure (i.e. a sphere is defined solely by its radius). However, that's not unique of spheres, since cubes and many other polyhedra can be determined by their side length. $\endgroup$ Commented Aug 11, 2014 at 23:21
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1$\begingroup$ Also, I should note for historical interest that atoms have previously been thought of as cubes, and that according to quantum mechanics atoms are not in general spherical. $\endgroup$ Commented Aug 11, 2014 at 23:27
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4$\begingroup$ @Nicolau - What I meant is: you only need one number to describe a sphere. For a regular polyhedron, you need two numbers - the number of vertices or sides, and the length of each side. $\endgroup$– thomijCommented Aug 11, 2014 at 23:37
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1$\begingroup$ @NicolauSakerNeto Just for the record, the conclusion of that topic was the atoms are spherical.. $\endgroup$– GregCommented Jul 7, 2015 at 9:42
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1$\begingroup$ @Greg Ah yes, a few months after I wrote this comment here, DavePHD corrected me in comments of the very question I linked. Indeed, atoms are spherically symmetric, and I was wrong previously. Good thing that you pointed it out here, as I'd already forgotten about this comment. Thanks for helping keep the record straight. $\endgroup$ Commented Jul 7, 2015 at 11:18