From Frize Patterns to Wallpaper Patterns
From Prep08Wiki
Some of the Wallpaper Patterns are simply the Frieze Patterns you have already learned about repeated in vertical rows. Let's introduce them.
We will use letters since they are easy to see (and type).
The pattern 11 looks like this: & & & & & & & &
Repeat it in vertical rows and you get the wallpaper pattern P1:
& & & & & & & & & & & & & & & & & & & & & & & &
The pattern 12 looks like this: Z Z Z Z Z Z
Repeat it in vertical rows and you get the wallpaper pattern P2:
Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
The pattern 1m looks like this: C C C C C C
Repeat it in vertical rows and you get the wallpaper pattern ??:
C C C C C C C C C C C C C C C C C C C C C C C C
The pattern 1g looks like this: p b p b p b
Repeat it in vertical rows and you get the wallpaper pattern Pg:
p b p b p b p b p b p b p b p b p b p b p b p b
If the glide reflections occur both horizontally and vertically, you get the wallpaper pattern Pgg:
p b p b p b b p b p b p p b p b p b b p b p b p
The pattern m1 looks like this: U U U U U U
Repeat it in vertical rows and you get the wallpaper pattern ??:
U U U U U U U U U U U U U U U U U U U U U U U U
The pattern mm looks like this: H H H H H H
Repeat it in vertical rows and you get the wallpaper pattern P4(?):
H H H H H H H H H H H H H H H H H H H H H H H H
Finally, the pattern mg looks like this: W M W M W M
Repeat it in vertical rows and you get the wallpaper pattern ??:
W M W M W M W M W M W M W M W M W M W M W M W M
Note: This page reflects an idea I have had for awhile, but never actually written up. You can see there are some places in it where I am not sure of the correct answer myself. But, I thought if I could do this successfully, it might allow students to be able to recognize at least some of the wallpaper patterns, even if all of them are not covered. I would be interested in Feedback on this page, or feel free to edit it yourself if you think the approach has any promise.--Bill 22:25, 25 June 2008 (UTC)
