Carpenter's Wheel




Analysis of Carpenter's Wheel

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 pink
8) quilt block is shaded yellow




Symmetries present: reflection, glide reflection, rotation, translation

Description of symmetries: There are vertical, horizontal, and diagonal reflection mirrors
(i.e. at 45 degree angles) as shown. These mirrors are also glide mirrors. 4-fold dihedral rotation centers
are located at the intersection of 4 mirror lines; 2-fold dihedral rotation centers are located at the
intersection of 2 mirror lines. The translation generators we chose are twice the distance between the
horizontal and vertical mirrors. The shortest glide vectors are the same length and direction as the shortest translation
vectors (the shortest diagonal glide vectors are marked because they are not either of the two translation generators we chose.)

Description of fundamental region: an isosceles right triangle that has mirrors for sides (smallest one that can have mirrors for sides.) Note that the 4-fold centers are located at the 45 degree angle vertices, while the 2-fold center
is located at the right angle.

Description of symmetries in a block:Blocks have an 8-fold dihedral center, located at the center of the block. Note that this symmetry does not appear in the quilt as a whole.

Relationship between block and fundamental region:A block is approximately 8 fundamental regions; the relationship is only approximate because of the red rectangular strips used to piece together the blocks. That is, the chosen fundamental region includes a portion of these strips, but the block does not. This quilt seems to have the exact same structure as Red Geese.

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