Symmetry Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern Operation: some act that reproduces the motif to create the pattern Element: an operation located at a particular point in space 2-D Symmetry = 360o/2 rotation to reproduce a motif in a symmetrical pattern A Symmetrical Pattern 6 6 Symmetry Elements 1. Rotation a. Two-fold rotation 2-D Symmetry Symmetry Elements Operation 1. Rotation a. Two-fold rotation = the symbol for a two-fold rotation 6 Element 6 = 360o/2 rotation to reproduce a motif in a symmetrical pattern Motif 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation = the symbol for a two-fold rotation 6 second operation step 6 = 360o/2 rotation to reproduce a motif in a symmetrical pattern first operation step 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 180o rotation makes it coincident Second 180o brings the object back to its original position What’s the motif here?? 2-D Symmetry Symmetry Elements 1. Rotation b. Three-fold rotation = 360o/3 rotation to reproduce a motif in a symmetrical pattern 2-D Symmetry Symmetry Elements 1. Rotation b. Three-fold rotation step 1 = 360o/3 rotation to reproduce a motif in a symmetrical pattern step 3 step 2 2-D Symmetry Symmetry Elements 1. Rotation 6 6 6 6 6 6 6 2-fold 6 1-fold 3-fold Objects with symmetry: a identity Z t 4-fold 9 6-fold d 5-fold and > 6-fold rotations will not work in combination with translations in crystals (as we shall see later). Thus we will exclude them now. 4-fold, 2-fold, and 3-fold rotations in a cube Click on image to run animation 2-D Symmetry Symmetry Elements 2. Inversion (i) inversion is identical to 2-fold rotation in 2-D, but is unique in 3-D (try it with your hands) 6 6 inversion through a center to reproduce a motif in a symmetrical pattern = symbol for an inversion center 2-D Symmetry Symmetry Elements 3. Reflection (m) Reflection across a “mirror plane” reproduces a motif = symbol for a mirror plane 2-D Symmetry We now have 6 unique 2-D symmetry operations: 1 2 3 4 6 m Rotations are congruent operations reproductions are identical Inversion and reflection are enantiomorphic operations reproductions are “opposite-handed” 2-D Symmetry Combinations of symmetry elements are also possible To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements In the interest of clarity and ease of illustration, we continue to consider only 2-D examples 2-D Symmetry Try combining a 2-fold rotation axis with a mirror 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect (could do either step first) 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) Is that all?? 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) No! A second mirror is required 2-D Symmetry Try combining a 2-fold rotation axis with a mirror The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different (not equivalent by symmetry) 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 1 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 2 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 3 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 4mm Why not 4mmmm? 2-D Symmetry 3-fold rotation axis with a mirror creates point group 3m Why not 3mmm? 2-D Symmetry 6-fold rotation axis with a mirror creates point group 6mm 2-D Symmetry All other combinations are either: Incompatible (2 + 2 cannot be done in 2-D) Redundant with others already tried m + m 2mm because creates 2-fold This is the same as 2 + m 2mm 2-D Symmetry The original 6 elements plus the 4 combinations creates 10 possible 2-D Point Groups: 1 2 3 4 6 m 2mm 3m 4mm 6mm Any 2-D pattern of objects surrounding a point must conform to one of these groups 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) Step 2: invert This is the same as i, so not a new operation 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) The result: 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) This is the same as m, so not a new operation 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 1 Step 1: rotate 360o/3 Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 2: invert through center 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 1 Completion of the first sequence 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/3 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 3 1 Complete second step to create face 3 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Third step creates face 4 (3 (1) 4) 3 1 4 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4 (2) 5) 1 5 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5 (3) 6) Sixth step returns to face 1 5 1 6 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 3 This is unique 5 1 4 6 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) This is also a unique operation 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) A more fundamental representative of the pattern 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework: 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 2 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 2 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 5 2 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 5 2 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 e. 6-fold rotoinversion ( 6 ) 3 5 2 6 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane Top View (combinations of elements follows) 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) A simpler pattern Top View 3-D Symmetry We now have 10 unique 3-D symmetry operations: 1 2 3 4 6 i m 3 4 6 Combinations of these elements are also possible A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements 3-D Symmetry 3-D symmetry element combinations a. Rotation axis parallel to a mirror Same as 2-D 2 || m = 2mm 3 || m = 3m, also 4mm, 6mm b. Rotation axis mirror 2 m = 2/m 3 m = 3/m, also 4/m, 6/m c. Most other rotations + m are impossible 2-fold axis at odd angle to mirror? Some cases at 45o or 30o are possible, as we shall see 3-D Symmetry 3-D symmetry element combinations d. Combinations of rotations 2 + 2 at 90o 222 (third 2 required from combination) 4 + 2 at 90o 422 ( “ “ “ ) 6 + 2 at 90o 622 ( “ “ “ ) 3-D Symmetry As in 2-D, the number of possible combinations is limited only by incompatibility and redundancy There are only 22 possible unique 3-D combinations, when combined with the 10 original 3-D elements yields the 32 3-D Point Groups 3-D Symmetry But it soon gets hard to visualize (or at least portray 3-D on paper) Fig. 5.18 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 3-D Symmetry The 32 3-D Point Groups Every 3-D pattern must conform to one of them. This includes every crystal, and every point within a crystal Increasing Rotational Symmetry Rotation axis only 1 2 3 4 6 1 (= i ) 2 (= m) 3 4 6 (= 3/m) Combination of rotation axes 222 32 422 622 One rotation axis mirror 2/m 3/m (= 6) 4/m 6/m One rotation axis || mirror 2mm 3m 4mm 6mm 3 2/m 4 2/m 6 2/m 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m 23 432 4/m 3 2/m 2/m 3 43m Rotoinversion axis only Rotoinversion with rotation and mirror Three rotation axes and mirrors Additional Isometric patterns Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System (more later when we consider translations) Crystal System No Center Center 1 1 Monoclinic 2, 2 (= m) 2/m Orthorhombic 222, 2mm 2/m 2/m 2/m Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m Hexagonal 3, 32, 3m 3, 3 2/m 6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m 23, 432, 43m 2/m 3, 4/m 3 2/m Triclinic Isometric Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 3-D Symmetry The 32 3-D Point Groups After Bloss, Crystallography and Crystal Chemistry. © MSA