What does it mean to be less symmetrical than the equations governing the system? I found this line on the wiki for time crystals :

The existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system.

And I didn't understand a word of it. Can someone explain in simple terms what they are trying to say?
 A: An example would a Lennard Jones crystal. In two dimensions, at low enough temperature, it forms a hexagonal phase

Yet in the Hamiltonian there is nothing that hints at 6-fold symmetry at all. The potential is given by
$$U_{total}=\sum_{i<j}u(|\mathbf r_i-\mathbf r_j|)$$
where
$$u(r)=4\epsilon\left[\left(\frac{\sigma}r\right)^{12}-\left(\frac{\sigma}r\right)^6\right].$$
The potential is rotationally symmetric which you can see because it only depends on distance. The symmetry group of the Hamiltonian is $O(1)$ which is as big as $\mathbb R$. The ground state is only symmetric under rotations of $60^\circ$. Its symmetry group is $\mathbb Z_6$ which consists of merely six elements. In case you have never heard of these groups the takeaway is that the number of symmetries of the ground state is much smaller than that of the Hamiltonian.
Another famous example is the Mexican hat potential. This potential rotationally symmetric about the orign. You can imagine a ball starting out at the top/center. When the ball is at the top its state is rotationally symmetric: if you rotate everything the position stays at the same place. But if you give the ball a little nudge it will roll down hill and when it does the state is no longer rotationally symmetric. It has picked a preferred direction and it has "broken the symmetry".

source of first image: https://www.youtube.com/watch?v=J76wEASh1sw&ab_channel=NilsBerglund
