This article presents origami designs realizing the 17 wallpaper patterns, which are distinguished by the isomorphism types of the groups of plane isometries that preserve them. The origami models are adaptations of each other, physically realized through a selective reorientation of pleats throughout the pattern. All 17 are adapted from two basic patterns, one for the five patterns that involve 3-fold rotational symmetry, and another for the 12 patterns whose rotation orders are limited to 1, 2 or 4. It is proposed that origami provides an excellent environment for artistic and mathematical exploration of wallpaper patterns and the relationships among their symmetry groups. This is due to the fact that origami models may include local elements that are physically adjustable, and such adjustments may be selected to alter the symmetry group of the overall pattern.