Wallpaper Patterns
Features
- N-Rotation Point (a point with C_N symmetry) - indicated by the number N before any *
- Mirror Symmetry (a reflection) - indicated by a *
- M-Dihedral Symmetry Point (a point with D_M symmetry) - indicated by the number M after a *
- Glide Reflecion - indicated by a x
- Wandering (only translations) - indicated by o
Prices
| Feature |
Cost ($) |
| N-rotation |
(N-1)/N
|
| M-dihedral |
(M-1)/(2M) |
| Mirror * |
1 |
| Glide x |
1 |
| Wandering o |
2 |
Magic Theorem
Every wallpaper pattern has total cost $2.
Corollary
All possible symmetry types of wallpaper patterns are as follows (click on the picture for more examples)
Procedure For Finding Symmetry Type:
- Draw all mirror symmetry lines you can find
- Use 1. to identify all differnt types of dihedral points
- Find all different points with rotational symmetry
Stop and record what you have found:
- For each (distinct!) N-rotation point write an N.
- Write a * if you found at least one mirror line.
- For each (distinct!) M-dihedral point (intersection of mirror lines) write an M.
Now you have a label which looks like: abcd.. *qrst.. and should be one of the following:
- cost $2: done.
- 22 : there must be a glide reflection (22x)
- * : there must either be two mirror types (**) or a glide reflection (*x)
- nothing : either this a wandering (o) or there are at least two glide reflections (xx)
Tiles
Every wallpaper patern can be generated by a tile which is then replicated using the symmetry group.
Credits & References
- This website is a low grade duplication of a piece of Steve Edward's Tiling Page
- For a very thorough (and enjoyable!) exposition of this material see "The Symmetries of Things" by Conway, Burgiel, Goodman-Strauss
