Translational symmetry in the context of "Momentum"

In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Time crystals were first proposed theoretically by Alfred Shapere and Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.

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. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal"

In the mathematics of tessellations, a non-periodic tiling is a tiling that does not have any translational symmetry. An aperiodic set of prototiles is a set of tile-types that can tile, but only non-periodically. The tilings produced by one of these sets of prototiles may be called aperiodic tilings.

The Penrose tilings are a well-known example of aperiodic tilings.

In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with the translational symmetry of three-dimensional crystallographic lattices. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions (Note that inversion centers and mirror planes are included as equivalent operations to one-fold and two-fold rotoinversions). This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups.

In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations, that is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the (geometric) crystal class of the crystal.

Aperiodic crystals are crystals that lack translational symmetry in at least one dimension, but still exhibit long-range order in 3 dimensions. Where as traditional crystals can be viewed by a unit cell containing some set of matter (atoms, molecules, ions, etc) repeated in 3 dimension, an aperiodic crystal has at least some feature which cannot be described in this manner, but does have some defined rule by which it is arranged such that, knowing a finite portion of the structure, you could predict the rest of the structure. They are classified into three different categories: incommensurate modulated structures, incommensurate composite structures, and quasicrystals.