Relationship between block and fundamental region: Four fundamental regions make a block. Note that the 4-fold cyclic center is located at right angle while the 2-fold dihedral centers are located at the 45 degree angles (and are equivalent by rotation.) Description of symmetries in block: The block displays only 4-fold cyclic rotation at its center, which is present in the larger design. The shortest glide vectors are half the translation generators (and sums of these create the shortest horizontal and vertical glide vectors.) Description of fundamental region: an isosceles right triangle with mirror on its hypotenuse. Translation simply means moving, every point of the shape must move the same distance, and in. Therefore, Glide reflection is also known as trans-flection. First, a translation is performed on the figure, and then it is reflected over a line. Can you think of a way of making a glide-reflection by combining three reflections. Definition: A glide reflection in math is a combination of transformations in 2-dimensional geometry. The second map is composition of reflection z z. Because two reflection axes which meet at an angle. The first crucial step is known as the crystallographic restriction : A rotation symmetry of a wallpaper pattern must be a rotation of order 2, 3, 4, or 6. Translation generators are the length of a diagonal of the block, as shown. On squared paper, design your own frieze with a single axis of reflection. By definition, a glide reflection is composition of a translation by some vector v v and reflection in a line parallel to v v (composition order is immaterial they commute) The first map is composition of the reflection z z¯ z z ¯ and the translation z z + a z z + a. As with frieze groups, the classification of wallpaper symmetry groups is done by a process of elimination. 4-fold cyclic rotation centers are located at the intersection of 2 glide mirror lines 2-fold dihedral rotation centers are located at the intersection of 2 mirror lines. Mid-way in between these mirrors are glide mirrors that are NOT reflection mirrors. Represent transformations as compositions of simpler transformations. Description of symmetries in design: There are vertical, horizontal reflection mirrors. Geometry Glide Reflections and Compositions Goals Identify glide reflections in the plane. Block Designs: Flywheel reflection creates 4*2 KEY:ġ) red segments represent reflection mirrorsĢ) light green segments represent glide mirrors that are not reflection mirrors3) dark blue segments represent translation generatorsĤ) dark green segments represent shortest glide vectors that are not translation generatorsĥ) yellow points represent cyclic centersĦ) light blue points represent dihedral centersĨ) quilt block is identified above design Symmetries present: reflection, glide reflection, rotation, translation Description of how design was made: We made this pattern by reflecting the original block horizontally, then reflecting 2 blocks vertically, then 4 blocks horizontally, etc.
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