![]() 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. It was the case that adding vertical mirrors, glide reflection, and rotation to a frieze with only translation does make the pattern more appealing. 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 generators are the length of a diagonal of the block, as shown. ![]() 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. The composition of two reflections over parallel lines that are h. Thus, glide-reflections constitute a class of planar isometries which is manifestly. A glide-reflection is not conjugate to a translation since one is orientation-reversing and the other preserves orientation of E2 E 2. The translation is in a direction parallel to the line of reflection. Since a reflection fixes a line and a glide reflection has no fixed points, the two are never conjugate to each other. Description of symmetries in design: There are vertical, horizontal reflection mirrors. You can compose any transformations, but here are some of the most common compositions: A glide reflection is a composition of a reflection and a translation. ![]() I know that: the conjugate of a reflection by a translation (or by any isometry, for that matter) is another reflection, by some explicitly calculation (havent try). 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. Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length.
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