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A homogeneous and isotropic liquid or a perfectly periodic crystal lattice which one has higher symmetry. I was thinking that both have same order of symmetry as if I am at a lattice point and translated to another point I can not distinguish between the two points. In a similar way for liquid I can not distinguish where I am. But I have a doubt because lattice is discrete but liquid is continuous. I do not know if I am thinking in the right way.

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The discrete symmetry group of a lattice is a proper subgroup of the continuous symmetry group of homogeneous, isotropic space, and therefore represents “less” symmetry.

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The symmetry of a liquid is an average symmetry. That is, the liquid is homogeneous and isotropic only when you average out on a scale much larger than the size of the atoms or molecules in the liquid.

If you take the large scale average then the liquid has a higher symmetry. However at the atomic/molecular scale the liquid is less symmetric since it has no long range order. So whether you regard a liquid as more or less symmetric than a crystal depends on the application.

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The apparently odd answer is that a homogeneous isotropic liquid has a higher symmetry than a crystal.

To be more precise, one should say that the group of spatial symmetry transformations of the one-particle density contains more elements for a liquid than for a crystal.

As far as the counting (cardinality) of the set of symmetry operations there is no surprise. However, the statement may sound weird because the highly symmetric uniform one-particle density of a liquid is obtained by averaging over a huge set of configurations, a vast majority of them being highly disordered (no symmetry), while in the case of a crystal, the dominant fraction of configurations is very close to symmetric configurations. It is the average over all configurations which allows to end up whit a liquid more symmetric than a solid.

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