1. 1 Euler construction Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere cos w = cos u cos v + sin u sin v cos W u = 180 - U v = 180 - V w = 180 -W cos w = Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere cos w = cos u cos v + sin u sin v cos W u = 180 - U v = 180 - V w = 180 -W cos w = cos W + cos U cos V sin U sin V C1C1 C1C1 C2C2 C2C2 C3C3 C3C3 V = b /2 W = g /2 v v w w u u U = a /2

2. 2 Euler construction cos w = 1 2 3 4 6 cos w = 1 2 3 4 6 cos W + cos U cos V sin U sin V , , or  U, V, or W cos U, V, W sin U, V, W 360 180 -1 0 180 90 0 1 120 60 1/2 3/2 90 45 1/ 2 1/ 2 60 30 3/2 1/2 360 180 -1 0 180 90 0 1 120 60 1/2 3/2 90 45 1/ 2 1/ 2 60 30 3/2 1/2

3. 3 Euler construction cos w = cos W + cos U cos V sin U sin V 222 0 90 cos w w

4. 4 Euler construction cos w = cos W + cos U cos V sin U sin V 222 0 90 223 1/2 60 222 0 90 223 1/2 60 cos w w

5. 5 Euler construction cos w = cos W + cos U cos V sin U sin V 222 0 90 223 1/2 60 224 1/ 2 45 222 0 90 223 1/2 60 224 1/ 2 45 cos w w

6. 6 Euler construction cos w = cos W + cos U cos V sin U sin V 222 0 90 223 1/2 60 224 1/ 2 45 226 3/2 30 222 0 90 223 1/2 60 224 1/ 2 45 226 3/2 30 cos w w

7. 7 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6

8. 8 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I

9. 9 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m

10. 10 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m

11. 11 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m

12. 12 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m

13. 13 Point groups so far Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m 13 monaxial groups …….. 19 to go Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6 What about combining proper & improper along same axis? P I I 1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m 13 monaxial groups …….. 19 to go

14. 14 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm

15. 15 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm

16. 16 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 222 = 2mm

17. 17 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm

18. 18 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm

19. 19 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 223 = 3mm

20. 20 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 4mm Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 4mm

21. 21 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 42m Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 42m

22. 22 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 42m Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 224 = 42m

23. 23 Point groups so far Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 13 to go Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 13 to go

24. 24 Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 332 = 3 = 2/m 3 = 332 Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 332 = 3 = 2/m 3 = 332 More point groups 3 2

25. 25 Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 332 = 3 = 2/m 3 = 332 432 = 43m 432 = 432 = 3 Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go How?Combine P & I - remember P I I & I I P 226 = 6mm 226 = 62m 332 = 3 = 2/m 3 = 332 432 = 43m 432 = 432 = 3 More point groups 3 2 4 2 m

26. 26 More point groups 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m 2 2 2 2 2 2 3 2 2 3 2 2 2 4 2 2 4 2 2 6 2 2 6 2 2 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m 2 2 2 2 2 2 3 2 2 3 2 2 2 4 2 2 4 2 2 6 2 2 6 2 2 = = 2 2 2 mmm = = 2 2 2 mmm m m 3 3 = = = = 4 2 2 mmm = = 6 2 2 mmm

27. 27 Triclinic 1 1 Monoclinic 2 m 2/m Orthorhmbic 222 2mm Tetragonal 4 4 4/m 422 4mm 42m Cubic 23 432 43m 3 Trigonal 3 3 32 3m 3 Hexagonal 6 6 6/m 622 6mm 62m Triclinic 1 1 Monoclinic 2 m 2/m Orthorhmbic 222 2mm Tetragonal 4 4 4/m 422 4mm 42m Cubic 23 432 43m 3 Trigonal 3 3 32 3m 3 Hexagonal 6 6 6/m 622 6mm 62m Summary mmm 2 2 2 4 2 2 mmm m m 2 2 3 3 4 2 m m m 2 2 6 2 2 mmm