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Euler construction

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 =. U = a /2. w. C 2. C 1. V = b /2. v. u. cos W + cos U cos V. sin U sin V. W = g /2. C 3.

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Euler construction

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  1. Euler construction Spherical C1C2C3 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 = U = a /2 w C2 C1 V = b /2 v u cos W + cos U cos V sin U sin V W = g /2 C3

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

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

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

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

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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  24. More point groups 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 3 2 3 2

  25. More point groups 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 3 2 3 2 4 2 m m

  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 = = 3 = mmm mmm m 2 2 2 2 2 2 = = mmm mmm 4 2 2 6 2 2

  27. Summary 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 2 2 2 mmm 4 2 2 mmm 4 2 2 m m m 2 m 6 2 2 mmm 3

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