![]() ![]() This group contains reflections, 180 degree rotations, and 90 degree rotations. There are not reflections or glide reflections and the lattice is square. The centers of the half turns are midway between the centers of the order 4 rotations. The 90 degree rotation is an order 4 rotation, and the half turn is a order 2 rotation. This group contains a 90 degree rotation, a 180 degree rotation, and translations. The lattice for this specific group is rhombic. This group has perpendicular reflection axes and has half turns. This group also contains half turns (rotations). This group contains reflections and glide reflections. The lattice for this group is rectangular. There are perpendicular axes for the glide reflections, and the fixed points of the 180 degree rotations do not lie on these axes. ![]() This group contains translations, glide reflections, and 180 degree rotations but it does not contain regular reflections. The fixed points of the half turns do not lie on the axes of reflection. It does not contain translations or glide reflections. This symmetry group contains reflections and 180 degree rotations. ![]() The lattice for this symmetry group is rectangular. There are no glide reflections and the only rotations are half-turns whose fixed points lie at intersections of axes of reflection. This group contains reflections and rotations whose axes are perpendicular. This group applies for symmetrically staggered rows of identical objects, which have a symmetry axis perpendicular to the rows. There is at least one glide reflection whose axis is not a reflection axis it is halfway between two adjacent parallel reflection axes. There are no rotations in this group and the lattice is rhombic. The reflections and glide reflections have parallel axes. This group contains reflections, glide reflections, and translations. There are no rotations or regular reflections and the lattice is rectangular. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. This symmetry group contains glide reflections and translations. There are no rotations or glide reflections and the lattice is rectangular. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. This group consists of reflections and translations. The lattice for this symmetry group is parallelogrammatic. Also, the two translation axes may be inclined at any angle to each other. ![]() It does not contain reflections or glide reflections. This group consists of translations and 180 degree rotations within the plane. It does not contain reflections, rotations, or glide reflections and its lattice is parallelogrammatic. This group is the simplest of all the symmetry groups, it consists only of translations throughout the plane. Now, we can continue on to show each of the seventeen wallpaper groups that my group and I have created for our project. Another important definition that is used to describe the movement in the plane is a lattice. This term describes all the centers of rotation and the axes of reflection for the objects in the plane. Glide Reflection – a transformation that is a combination of a reflection and a translation.Īll of these symmetries are important for understanding how these wallpaper groups have been created and how they move around the plane. Translation – a transformation that moved every point in an object the same amount of distance. Reflection – a transformation in which a mirror image is created around an axis of reflection. Rotation – a transformation in which an object is rotated about a specific point, typically rotated in degrees. First, I wanted to define the different types of symmetries used to tile the plane. Each individual group is a collection of these symmetries that are used in unique ways, making each group different from one another. I completed the 17 wallpaper groups as part of my project for this course and decided to complete a blog post on the work that I completed! The 17 wallpaper groups represent the seventeen different ways to cover a two-dimensional plane if one only uses symmetries. For this post, I am going to be completing the ‘doing math’ requirement. ![]()
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