1. Wallpaper Symmetry

2. A wallpaper pattern has translational symmetry in two independent directions.

4. “Look! A pattern!”

5. Translational symmetry wallpaper

6. Mirror Symmetry This mirror reflection is a symmetry of the pattern Mirror axis

7. Rotational Symmetry

8. 6-center3-center 2-center Can symmetries be selected independently?

9. Rotational Symmetry 3-center 2-center There are mathematical limitations on decorative art! By group property, if you have some symmetries, you must have all others that follow by composition.

10. Classify wallpaper patterns by symmetries

11. Too many to list! Mirrors 4-CentersGlide reflections

12. Group concept: Compose two symmetries, get a symmetry

13. Build up the group: All symmetries generated by a few

14. Simplify: Track symmetries in a “fundamental cell”

15. Group concept: a few symmetries can generate all

16. p4g can be generated by one mirror, one rotation

17. Two patterns have the same “type” if their symmetry groups are the same

18. Same or different?

19. Each has 3-centers and mirrors

20. Every subgroup that fixes a point has mirrors and rotations Some subgroup that fixes a point has no mirrors p3m1p31m

21. “symmetric yet organic”

22. Wallpaper surprise: Exactly 17 types

23. Wallpaper surprise: Exactly 17 types Can’t have 6-fold rotation without 3- and 2-fold as well

24. Wallpaper surprise: Exactly 17 isomorphism classes Proved by Federov 1891 Proved by Federov 1891 Polya’s 1924 version (with figures) known to M.C. Escher Polya’s 1924 version (with figures) known to M.C. Escher See “The Polya-Escher Connection” by Doris Schattschneider in Mathematics Magazine See “The Polya-Escher Connection” by Doris Schattschneider in Mathematics Magazine

25. M.C. Escher’s Angel-Devil (No. 45)

27. Chalk Slam at Carleton

28. See them everywhere! pmg pattern from The Shining

29. Lightening Round

32. Tomorrow: color-reversing symmetry