Wikipedia:Reference desk/Archives/Mathematics/2024 January 17

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January 17

Naming the wallpaper patterns

We name the frieze patterns after the kind of alteration each one has. They are named using 5 letters of the alphabet:

• T means horizontal translation
• R means 180 degree rotation
• H means horizontal line reflection (any pattern with H will also have G)
• V means vertical line reflection
• G means glide reflection

There are 7 possible frieze patterns; T, TR, TV, TG, THG, TRVG, and TRHVG.

Anyone have a similar way to name the wallpaper patterns?? Georgia guy (talk) 19:38, 17 January 2024 (UTC)

There are 17 wallpaper groups and various notations for them are given in the article. Your notation and several other notations are given in for the 7 frieze groups in that article. I'm not sure what you mean by a "similar way", but the notation used in the article is at least based on the types of operations that exist. It uses additional information as well, for example the groups cm and pm differ by the locations of their glide axes, not just whether they exist or not. --RDBury (talk) 00:33, 18 January 2024 (UTC)

From what I'm gathering, I believe the question can be framed as whether one can start with a finite set of elements and transformations, then perform a periodic set of transformations so that the 'image' of all the steps taken satisfies the same kinds of symmetries that the wallpaper groups. In the case of the frieze groups, the steps T, R, H, V, and G are such a set of transformations, with the symmetries of the frieze patterns leading to the frieze groups. GalacticShoe (talk) 04:37, 20 January 2024 (UTC)