1. Symmetry Rotation Translation Reflection

3. A line on which a figure can be folded so that both sides match.

4. To be a line of symmetry, the shape must have two halves that match exactly. When you trace a heart onto a piece of folded paper, and then cut it out, the two half hearts make a whole heart. The two halves are symmetrical.

5. Which of these flags have a line of symmetry? United States of America Canada MarylandEngland

6. Do they have symmetry?

7. Do you see a pattern?

8. You can look to see if your name has symmetrical letters in it too! A B C D E FG H I J K L M N O P Q R S T U V W X Y Z Infinite number

9. The action of turning a figure around a point or a vertex.

10. Click the triangle to see rotation Turning a figure around a point or a vertex

11. The action of sliding a figure in any direction.

12. The act of sliding a figure in any direction. Click the Octagon to see Translation.

13. The result of a figure flipped over a line.

14. Click on this trapezoid to see reflection. The result of a figure flipped over a line.

15. Which shapes show reflection, translation, or rotation?

17. Fundamental organizing principle in nature and art Preserves distances, angles, sizes and shapes

18. Sort the letters of the alphabet into groups according to their symmetries Divide letters into two categories: symmetrical not symmetrical

19. Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z Not Symmetrical: F, G, J, L, P, Q, R

20. Rotation Translation Reflection Glide Reflection

21. To rotate an object means to turn it around Every rotation must have a center and an angle

22. Move it without rotating or reflecting it Every translation has a direction and a distance

23. Produce an object’s mirror image A reflection must have a mirror line

24. Involves more than one step Combination of a reflection and a translation along the direction of the mirror line

25. Choose a letter (other than R) with no symmetries On a piece of paper perform the following tasks on the chosen letter: rotation translation reflection glide reflection

27. What happens if you do the same transformation twice? How many combinations of two transformations are there? What happens if you combine more than two transformations?

28. Plants and animals exhibit many forms of symmetry

29. Dutch graphic artist No formal training in math or science Used intricate repeating patterns in his artwork

45. Symmetry Element VS Symmetry Operation Symmetry Element is a point, a line, or a plane where symmetry operation is done. (E, C n, , i, S n ) Symmetry Operation is an action done on an object (at the symmetry element of the object) and the action leave the object the same as before, without any different, although some movement have taken place. This action includes rotation at an axis, reflection at a mirror plane, inversion at a point of inversion/center of symmetry.

53. Point Group, Determination of Coordinate Axis and Character Table Symmetry elements obey all properties of a group in mathematics and this can be represented in terms of Character Table. Group is a set containing h operation, q 1, q 2, q 3,..., q h which obey the properties of a group with the following condition for each element: Contain identity, E, with Eq i = q i E = q i for all q i. Contain inversion (q i –1 ), with q i q i –1 = q i –1 q i = E Associative Rule which means q i (q j q k ) = (q i q j )q k All symmetry operations fulfilled these conditions and therefore symmetry operation form group.

54. A Point Group means at least a point in the molecule (e.g. the center of gravity of the molecule) do not move during symmetry operation. Vibration and rotation of molecule fall into this category. Determination of x, y, and z Axis in molecule is done as follows: Use right hand rule: thumb = z, index = x, middle finger = y. Place the point of origin at the center of gravity of the molecule

55. Determine z axis as follows: If there is only one axis of rotation, this must be the z axis z axis must be vertical. If the molecule have many axis of rotation, the axis with highest order is the z axis and it is vertical. If the molecule many axis of rotation eith highest order, the axis passing through the most number of atoms is the z axis and it is vertical. Determine the x axis as follows: If the molecule is planar and the z axis is in the plane, x is vertical to the plane. If the molecule is planar and axis z is vertical to the plane, x axis is in the plane (together with y axis) and passing through the most number of atoms. Determine y axis according to the Right Hand Rule.

56. Working Steps in Symmetry Method Identify and confirm the correct structure of the molecule (build 3-dimensional model if necessary) Identify and locate the exact position of all symmetry elements in the molecule Determine the x, y, z axis using the Right Hand Rule and the convention Determine the group of the molecule (see below) Use the Character Table of the group to solve problem (later) Check the correctness of the result against normal method (later)

57. Determination of the Group C 1, C i, and C s Group C 1 No symmetry element other than E C i Only have E and i C s Only have E and  C n, C nv, and C nh Group C n Have E and C n ; eg H 2 O 2 (C 2 ) C nv Also have n  v, so C nv, eg H 2 O (C 2v ), NH 3 (C 3v ) and CO, HCl, OCS (C  v) C nh Also have  h, so C nh, eg trans CHF=CHF (C 2h ), B(OH) 3 (C 3h )

58. D n, D nh, and D nd Group D n Have E and C n and n C 2 vertical to C n D nh Also have  h, eg BF 3 (D 3h ) C 6 H 6 (D 6h ); all diatomic molecules (H 2, N 2, O 2, etc.) and all cylindrical molecules OCO, HCCH (D  h ); PCl 5 (D 3h ); [AuCl 4 ] – (D 4h ) D nd Also have n  d, eg allene H 2 C=C=CH 2 (D 2d ), C 2 H 6 (staggered configuration) (D 3d ) S n Group Molecules not in any of the above group but have one C n axis, eg C(CHCH 3 ) 4 (S 4 ) Cubic Group Tetrahedron Group, T, T d, T h or Octahedron Group, O, O h Having more than one main axis, eg CH 4, SF 6

60. Character, Character Table, Symmetry Species and Reducible Representation Character is a number representing the behavior of properties molecule under the effect of symmetry operation: Character of +1 represent symmetric behavior Character of –1 represent antisymmetric behavior e.g.: if we consider orbital p y and p z in atom O in H 2 O molecule, it can be shown that p z follow A 1 spesies, while p y follow B 2 spesies.