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Beauty, Form and Function: An Exploration of Symmetry

Beauty, Form and Function: An Exploration of Symmetry. Asset No. 21. PART II Plane (2D) and Space (3D) Symmetry. Lecture II-7. Space Symmetry Operators. Objectives. By the end of this lecture, you will be able to: describe screw axes are and how they move objects

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Beauty, Form and Function: An Exploration of Symmetry

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  1. Beauty, Form and Function: An Exploration of Symmetry Asset No. 21 PART II Plane (2D) and Space (3D) Symmetry Lecture II-7 Space Symmetry Operators

  2. Objectives By the end of this lecture, you will be able to: • describe screw axes are and how they move objects • interpret axial glide planes and double glide planes are and how they move objects • create a center of inversion • differentiate roto-reflection from roto-inversion.

  3. Three Dimensional Symmetry - Screw Axis To recap.... In 2 dimensions the symmetry operators are reflection, rotation, translation, glide What happens in 3 dimensions? Let’s look a familiar objects.... Spiral Staircase Clearly the stairs are arranged symmetrically - but how? Begin with what we know and can observe • there is vertical translation • there is rotation • there is a periodic repeat • there is an asymmetric unit 15 stairs in the repeat...360o/15 = 24-fold rotation axis Rotation + vertical translation is a screw axis

  4. Two Fold-Screw Axes c 1 1 / / 2 2 1 1 1 2 2 Proper Rotation Axis Screw Axis 2 21 0 0 0

  5. Three Fold-Screw Axes 2 2 2 / / / 3 3 3 1 1 1 1 / / / 3 3 3 1 1 2 3 2 0 1 3 1 2 3 0 0 0 0 Proper Rotation Axis Screw Axis 32 3 31 c

  6. Four Fold-Screw Axes 3 3 3 / / / 4 4 4 1 1 1 1 1 1 1 1 / / / / / / / / 2 4 4 2 2 4 2 2 0 0 1 1 1 1 4 2 3 2 1 3 2 4 3 4 1 0 0 0 0 0 0 4 42 43 41 c

  7. Six Fold-Screw Axis 2 1 5 / / / 6 3 6 1 1 / / 2 3 6 61 62 63 64 65

  8. Screw Axes Screw Axis:A combination of proper rotations with translation along the rotation axis. If the screw axis is c, the combined operation is a counterclockwise rotation about c followed by translation t along +c (or the reverse). In general, there are screw axes Nn with 1 £ n £ N and N = 2, 3, 4, 6. 2 Þ 21 3 Þ 31 32 4 Þ 41 42 43 6 Þ 61 62 63 64 65

  9. Three Dimensional Symmetry - Axial Glide , , , , , , , z z y y x x • there is vertical translation Vertical Ladder • there is reflection • there is a periodic repeat • there is an asymmetric unit This is a ‘c-glide’ z y x Axial Glide:A glide where the translation vector is one-half a unit cell translation parallel to the reflection plane, and referred to as a, b or c according to the axis along which the translation is carried out.

  10. Three Dimensional Symmetry - Double Glide , , z z y , , , y x x 1 1 1 + + + 2 2 2 Straight Staircase z • there is vertical translation • there is horizonal translation y x • there is reflection • there is a periodic repeat • there is an asymmetric unit + + + + 2 stairs in the repeat... 2 perpendicular translations Two orthogonal translations plus a reflection gives a double glide plane There is also n glides involving translations of ¼ the unit cell.

  11. Three Dimensional Symmetry - Inversion (-x, -y, -z) 3 4 4 3 2 2 5 (x, y, z) An inversion centre ( ) is also known as a centre of symmetry. • there is rotation • there is reflection • there is inversion • there is reflection Rotation plus reflection is roto-reflection (or inversion) Rotation plus inversion is roto-inversion 1 1 5 The rotations can be proper or screw types and n-fold (n = 2, 3,4, 6).

  12. Centre of Inversions (or Centres of Symmetry) + , - , 1 - + Roto-Inversion 2/m Roto-Reflection inversion center Twofold with centre of symmetry 21/m Twofold screw axis with centre of symmetry 4/m Fourfold rotation axis with centre of symmetry ‘4 sub 2’ screw axis with centre of symmetry 42/m Sixfold rotation axis with centre of symmetry 6/m ‘6 sub 3’ screw axis with centre of symmetry 63/m

  13. Summary 2 Dimensional Symmetry Glide (Translation + Reflection) Rotation Reflection Translation 10 point groups 5 Bravais lattices 17 plane groups 3 Dimensional Symmetry Glide Rotation Reflection Translation Screw (Translation + Rotation) Roto-inversion (Rotation + Inversion) Roto-reflection (Rotation + Reflection) Inversion 32 point groups 14 Bravais lattices 230 space groups

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