The isometry group generated by just a glide reflection is an infinite cyclic group. This isometry maps the x-axis to itself any other line which is parallel to the x-axis gets reflected in the x-axis, so this system of parallel lines is left invariant. It can also be given a Schoenflies notation as S 2∞, Coxeter notation as, and orbifold notation as ∞×. A glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. In group theory, the glide plane is classified as a type of opposite isometry of the Euclidean plane.Ī single glide is represented as frieze group p11g. The intermediate step between reflection and translation can look different from the starting configuration, so objects with glide symmetry are in general, not symmetrical under reflection alone. In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. Note that in the example, although there are translations and glide reflections that are symmetries, no rotations or reflections are symmetries.Since this footprint trail has glide reflection symmetry, applying the operation of glide reflection will map each left footprint into a right footprint and each right footprint to a left footprint, leading to a final configuration which is indistinguishable from the original. Moreover, the whole pattern will fall onto itself if you apply the glide reflection, so this is a symmetry of the pattern.

If you reflect across the magenta axis, then translate along it (either up or down), one green arrow will fall onto another green arrow on the other side of the axis. The green arrows on either side of it suggest the glide reflection. In the example, the axis of the glide reflection is drawn in magenta. It's composed of a reflection across an axis and a translation along the axis. The fourth kind of isometry, the glide reflection, is not nearly as easy to see as the other three. In all our patterns, if there is one reflection, then the translations will guarantee that there are infinitely many reflections. It's called a 'reflection' because similar things happen with a reflection in a mirror. The double-ended red arrows are supposed to suggest the reflection. In the example you see a diagonal axis in white. And doing that rotation twice gives a rotation of 180° that preserves the pattern.Ī reflection fixes one line in the plane, called the axis of reflection, and exchanges points on one side of the axis with points on the other side of the axis at the same distance from the axis. In the example you see that a rotation of 90° preserves the pattern. A 180° rotation is also called a half turn. So a 60°-rotation has order 6, a 90°-rotation has order 4, a 120°-rotation has order 3, and an 180°-rotation has order 2. The order of a rotation is the number of times it has to be performed to bring the plane back to where it started. In fact, the angle of rotation can only be 180°, 120°, 90°, or 60°.

In general a rotation could be by any angle, but for patterns like we have, the angle has to divide 360°, and a little more analysis finds further restrictions. There are three others: rotations, reflections, and glide reflections.Ī rotation fixes one point in the plane and turns the rest of it some angle around that point. Translations are just one kind of planar isometry. Indeed, there are infinitely many different translation symmetries for this and any other pattern that we consider. There are more, of course, you could go up two units, or down one unit, or down and to the left. One points directly up, one up and to the right. In the example you see two arrows suggesting two of the translation symmetries. The only patterns we'll consider are those with translation symmetries in at least two different directions. If you have a given repeating pattern, you can slide it along a certain direction a certain distance and it will fall back upon itself with all the patterns exactly matching. (Self-similar fractals have symmetries on different scales, and so other transformations must be considered to understand them.) There are four kinds of planar isometries: translations, rotations, reflections, and glide reflections. The only transformations that we'll consider are those that preserve distance, called isometries. We can identify a symmetry as a transformation of the plane that moves the pattern so that it falls back on itself. We're interested in the symmetries of a planar pattern.

$glide reflection$

Symmetries of patterns are transformations Wallpaper Groups: transformations Transformations of the plane

Glide reflection