Block Designs: Signature

composition of rotation & reflection creates 4*2

1) red segments represent reflection mirrors
2) light green segments represent glide mirrors
that are not reflection mirrors
3) dark blue segments represent translation generators
4) dark green segments represent shortest glide vectors
that are not translation generators
5) yellow points represent cyclic centers
6) light blue points represent dihedral centers
7) fundamental region is shaded purple
8) quilt block is identified above design







Symmetries present: reflection, glide reflection, rotation, translation

Description of how design was made: We made it by first rotating a block 3 times (90 degree rotation angles) and then reflecting those 4 blocks vertically and horizontally to made 16 blocks. Incidentally, the same pattern results if reflection is done first and then rotation.

Description of symmetries in design: There are vertical, horizontal reflection mirrors. These mirrors are also glide mirrors. There are diagonal glide mirrors as that are NOT reflection mirrors and horizontal and vertical glide mirrors (that are not reflection mirrors) midway between the reflection mirrors. 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. The shortest horizontal and vertical glide vectors are twice the width and height of a block; translation generators are twice these vectors but are not shown because of lack of space. The shortest diagonal glide vectors are half the sum of a pair of vertical and horizontal glide vectors.

Description of fundamental region: an isosceles right triangle that has mirrors as a base. Note that the 4-fold center is located at the 90 degree angle, while the 2-fold centers are located at the 45 degree angles.

Description of symmetries in block:The block displays only 2-fold cyclic rotation at its center, which is not present in the larger design.

Relationship between block and fundamental region:A block and the fundamental region have the same area. In fact, modifying the fundamental region above would show that a block is another choice for a fundamental region.


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