![]() But the axes of reflection are perpendicular, and none of the rotation centers lie on the reflection axes. This group also contains reflections and rotations of orders 2 and 4. The lattice is square, and an eighth, a triangle, of a fundamental region for the translation group is a fundamental region for the symmetry group. In fact, all the rotation centers lie on the reflection axes. The axes of reflection are inclined to each other by 45° so that four axes of reflection pass through the centers of the order-4 rotations. This group differs from 10 (p4) in that it also has reflections. The lattice is square, and again, a quarter of a fundamental region for the translation group is a fundamental region for the symmetry group. Midway between the centers of the order-4 rotations. This is the first group with a 90° rotation, that is, a rotation of order 4. The lattice is rhomic, and a quarter of a fundamental region for the translation group is a fundamental region for the symmetry group. The centers of the rotations do not lie on the reflection axes. ![]() This group has perpendicular reflection axes, as does group 6(pmm), but it also has rotations of order 2. Again, the lattice is rectangular, and a quarter-rectangle of a fundamental region for the translation group is a fundamental region for the symmetry group. There are perpendicular axes for the glide reflections, and the centers of the rotations do not lie on these axes. This group contains no reflections, but it has glide-reflections and half-turns. The lattice is rectangular, and a quarter-rectangle of a fundamental region for the translation group is a fundamental region for the symmetry group. The centers of rotations do not lie on the axes of reflection. This group contains both a reflection and a rotation of order 2. The lattice is rectanglular, and a rectangle can be chosen for the fundamental region of the translation group so that a quarter-rectangle of it is a fundamental region for the symmetry group. There are no glide-reflections or rotations. This symmetry group contains perpendicular axes of reflection. A fundamental region for the symmetry group is half the rhombus. The translations may be inclined at any angle to each other, but the axes of the reflections bisect the angle formed by the translations, so the fundamental region for the translation group is a rhombus. This group contains reflections and glide reflections with parallel axes. The lattice is rectanglular, and a rectangular fundamental region for the translation group can be chosen that is split by an axis of a glide reflection so that one of the half rectangles forms a fundamental region for the symmetry group. There are neither rotations nor reflections. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. This is the first group that contains glide reflections. A fundamental region for the translation group is a rectangle, and one can be chosen that is split by an axis of reflection so that one of the half rectangles forms a fundamental region for the symmetry group. There are neither rotations nor glide reflections. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. This is the first group that contains reflections. A fundamental region for the symmetry group is half of a parallelogram that is a fundamental region for the translation group. The two translations axes may be inclined at any angle to each other. As in all symmetry groups there are translations, but there neither reflections nor glide reflections. Rotations, that is, rotations of order 2. This group differs only from the first group in that it contains 180° Is parallelogrammatic, so a fundamental region for the symmetry group is the same as that for the translation group, namely, a parallelogram. Translation axes may be inclined at any angle to each other. There are neither reflections, glide-reflections, nor rotations. International Union of Crystallography in 1952. The IUC notation is the notation for the symmetry group adopted by the * = not all rotation centers on reflection axes + = all rotation centers lie on reflection axes There are enough characteristics listed in the table to distinguish the 17 different groups. Note that clicking on a small image below will take you to a discussion of the associated symmetry group (as will selecting the name of the group in the headings below).Ī short table of characteristics of the symmetry groups A mathematical analysis of these groups shows that there are exactly 17 different plane symmetry groups. The various planar patterns can by classified by the transformation groups that leave them invariant, their symmetry groups. Wallpaper Groups: the 17 plane symmetry groups The 17 plane symmetry groups
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