Mark Gritter (markgritter) wrote,
Mark Gritter
markgritter

Symmetries

I'm re-reading "The Symmetries of Things" (Conway, Burgiel, Goodman-Strauss) which starts with a categorization of plane symmetries.

I don't understand, though, why their approach ignores a particular sort of translational symmetry. One of their examples is (ascii-fied):

... AAAAAA ...
... YYYYYY ...
    OOOOOO
    AAAAAA
    YYYYYY 
    OOOOOO
    AAAAAA
    YYYYYY
    OOOOOO 


This has two kinds of "mirror signatures", which they denote **. There is a vertical mirror line between the letters, and a mirror line through the midpoint of the letters. But this doesn't describe the fact that each set of letters repeats vertically as well!

For example, we would expect a nonrepeating infinite vertical stripe to be denoted differently:

... 888888 ...
... UUUUUU ...
    888888
    888888
    UUUUUU 
    UUUUUU
    888888
    888888
    888888
    UUUUUU
    UUUUUU
    UUUUUU 
    .
    .
    .


This has the same mirror symmetries but lacks the translation symmetry. (It's not that they ignore translation--- there is a type composed just of translations--- but don't seem to consider a repeating vertical stripe with two mirror symmetries different from a nonrepeating vertical stripe with two mirror symmetries. Maybe there is a technical reason when they get more formal later on.)

You can see the 17 planar symmetries here:
http://en.wikipedia.org/wiki/List_of_planar_symmetry_groups

That page suggests the the technical difference is between finite and infinite "fundamental domains". But, ** still seems odd--- all the other groups extend naturally across the plane, but ** needs a translation tacked on.
Tags: books, mathematics, symmetry
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