# Carpenter's Wheel

# *442

# Analysis of Carpenter's Wheel

#### 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 pink

8) quilt block is shaded yellow

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#### 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.