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These are two types of spatial periodicity, as opposed to temporal periodicity. Periodicity basically refers to the repetition or recurrence of a phenomenon or value at regular intervals, be they time intervals or space intervals. The interval of repetition is termed the period.

In translational periodicity, there is a straight line direction of periodicity, and an associated finite spatial distance (the period). The scalar fields, and the components of vector fields referred to a global rectangular Cartesian coordinate system, on any surface S in the domain, are identical to the respective scalar fields and rectangular components of vector fields on every surface obtained by translating S along the direction of periodicity by an integral multiple of the period.

In rotational periodicity, there is a (straight line) axis of periodicity, and an associated angular period. If the domain encompasses the entire 2 Pi radians of azimuth about the axis, then the period must be an integral fraction of 2 Pi radians. The scalar fields, and the components of vector fields referred to a global cylindrical Cartesian coordinate system with the axis of periodicity as the cylindrical axis, on any surface S in the domain, are identical to the respective scalar fields and cylindrical components of vector fields on every surface (in the domain) obtained by rotating S about the axis of periodicity by an integral multiple of the angular period.

Trivial example. The fully developed steady flow in a straight pipe of circular cross-section exhibits both translational periodicity (with all real periods) along the axis of the pipe, and rotational periodicity (with all real angular periods) about the axis of the pipe.

Simple example. The steady flow of a stream uniform at upstream infinity, past an infinite two-dimensional cascade of identical airfoils displays translational periodicity in a direction tangential to the tips of all the airfoils, with a period equal to the pitch of the cascade along that direction.

Simple example. The flow through a fan (with periodic blades) blowing air between infinite reservoirs displays rotational symmetry about the axis of the fan, with a period equal to 2 Pi divided by the number of blades on the fan.

Trivial example. The letter "S", if written appropriately, can display rotational symmetry about an axis centered in the "S" and normal to the plane of the paper, with an angular period of Pi radians.

Simple example. The string "english english english" on your computer screen displays translational periodicity (in a finite domain) with axis parallel to the baseline of the string and period the length of the string "english ".

Periodicity of the geometry of bodies and of material properties is a necessary (but not sufficient) condition for periodicity of other continuum fields.