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 = U v = 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 = U v = 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 Euler construction cos w = cos w = cos W + cos U cos V sin U sin V , , or  U, V, or W cos U, V, W sin U, V, W /2 3/ / 2 1/ /2 1/ /2 3/ / 2 1/ /2 1/2

3 Euler construction cos w = cos W + cos U cos V sin U sin V cos w w

4 Euler construction cos w = cos W + cos U cos V sin U sin V / /2 60 cos w w

5 Euler construction cos w = cos W + cos U cos V sin U sin V / / / / 2 45 cos w w

6 Euler construction cos w = cos W + cos U cos V sin U sin V / / / / / /2 30 cos w w

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

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 = = 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 = = 43m 432 = 432 = 3 More point groups m

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

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