developing teaching materials(2)

Have you considered how many kinds of repetitions there are for patterns on a plane?
There are seventeen!

(A) Without rotations.
p1 Without reflections and glide reflections.
pm With a reflection. Any axis of a glide reflection is necessarily also an axis of a reflection.
pg Without reflections. With a glide reflection.
cm With a reflection. There exists an axis of a glide reflection that is not an axis of any reflection.

(B) With 180 degrees of rotation. Without rotations of 90 degrees or 60 degrees.
p2 Without reflections and glide reflections.
pmm With a reflection. Any axis of a glide reflection is necessarily also an axis of a reflection.
pgg Without reflections. With a glide reflection.
cmm With a reflection. There exists an axis of a glide reflection that is not an axis of any reflection and has a parallel axis of a reflection to it.
pmg With a reflection. There is an axis of a glide reflection without parallel axis of a reflection to it.

(C) With 90 degrees of rotation.
p4 Without reflections and glide reflections.
p4m There is a center of 90 degrees rotation lying on an axis of a reflection.
p4g With a reflection. There is a center of 90 degrees rotation not lying on any axis of a reflection.

(D) With 120 degrees of rotation. Without 60 degrees of rotation.
p3 Without reflections.
p31m With a reflection. There is a center of 120 degrees of rotation not lying on any axis of a reflection.
p3m1 With a reflection. Any center of 120 degrees of rotation lies on an axis of a reflection.

(E) With 60 degrees of rotation.
p6 Without reflections.
p6m With a reflection.

Tohsuke Urabe Mathematics Laboratry
Seventeen Kinds of Wallpaper Patterns
Group Theory of Wallpaper Patterns


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