Our universe began from singularity, an orderly arrangement and underwent expansion accompanied by increase in entropy (refer Gravity, Entropy, Thermodynamics And Universe ). It cooled down as it expanded into a more heterogeneous state. And as it cooled it lost its state of symmetry. Let us see how.
Take a simple cube of ice kept at absolute zero and slowly increase its temperature. You will observe that ice remains more or less same. But when you will reach 0 degree Celsius, ice will undergo a phase transition. Its perfect and ordered lattice of hexagon will get disintegrated into large clumps of molecules which will form a liquid state called water. It should be obvious that chemically there is no difference in composition of ice and water ( both being H2O ). This phase transition causes increase in symmetry*. This is because hexagonal lattice changes its appearance, while the water molecule clump remains unchanged when they undergo rotations.
Upon further heating of the water, another phase transition takes place at temperature around 100 degree Celsius. Here water molecules looses all the forces holding them together and move freely in space. This phase is called steam. When we rotate the clump of water molecules, the rotation hardly get noticed, but in steam there is no set pattern of molecular arrangement. Therefore upon rotation about any axis, the gas (steam) will remain same.
Most (not all) of the substances gain symmetry as they change from solid to liquid or from liquid to gas (same is true for sublimation). If we reverse the process to convert gas into liquid or liquid into solid, symmetry also reverses.
Our cosmos work in the same way like ice cube. Not only the objects, but the universe itself undergoes phase transition. Currently it is expanding and cooling. But maybe at certain temperature ( like 0 and 100 in case of H2O) universe will undergo change in symmetry**. Who knows maybe space as well as time will freeze as universe expands.
* An object is symmetrical if it looks the same way when rotated along a certain axis as it was before rotation. Like when we rotate a cube along axis perpendicular to one of its plane surface, it will look like a cylinder. But a sphere will look the same as it was even after rotation along any axis. So a sphere is more symmetrical than cube.
** Change in symmetry does not mean that entropy will change. Entropy will always increase.
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