Why do crystals have discrete translational symmetry rather than continuous translational symmetry?

Crystals possess discrete translational symmetry rather than continuous translational symmetry because they are composed of an orderly, repeating arrangement of atoms in space.12

In empty space, one experiences continuous translational symmetry because every position is identical, and moving any distance in any direction results in the same view. In contrast, a crystal lattice breaks this continuous symmetry because atoms are fixed at specific, finite intervals. When you move through a crystal, you must travel a specific distance in a specific direction to reach an identical atomic environment; this property is known as discrete translational periodicity.2

Essentially, the formation of a crystal is a process of symmetry breaking. As matter cools, it transitions from a disordered liquid state—which possesses continuous translational and rotational symmetries—into a more ordered, lower-energy state. In this crystalline state, the atoms occupy regular, repeating positions within a unit cell, which restricts the symmetry of the system to discrete, periodic displacements rather than the continuous freedom found in liquids or empty space.321