# abstract algebra – Distinguishing two frieze patterns with symmetry group cyclic

Among the seven frieze patterns, there are two, whose symmetry group is infinite cyclic.

For example, one may see first two in picture below (Ref: Gallian’s book on algebra, Chapter on Frieze Groups and Crystallographic Groups) or in the link here.

The two frieze patterns with cyclic symmetry group are obtained as follows:

(1) Walk only by left foot in a straight line; then the pattern of foot-print is a frieze pattern.

(2) Walk as usual in a straight line; then the pattern of left-right foot prints is a frieze patterns.

Question: What is correct geometric way to distinguish these patterns? These patters are generated by repeated application of generator of symmetry group to a single foot-print (in positive and negative directions).
For first one, it is just translation. For the second pattern, we repeatedly apply glide reflection.

Since, these patterns are one dimensional, so I was unable to compare translation and glide-reflection because they are different as motions in “2-dimensional plane” (orientation preserving-reversing).