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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

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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

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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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