Block Designs: Signature
composition of rotation & reflection creates 4*2
KEY:
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.