Elements of symmetry

Importance

The symmetry of molecules is essential in →      stereochemistry Lattice symmetry is essential in →     crystallography

Elements of symmetry

Two figures are mutually symmetrical , if they can be co-ordinated by a geometric operation carried out using operators called elements of symmetry:

 
Element of symmetry
Notation
Effect
   Point I    Central symmetry
   Plan M still designated σ    Mirror
   Axis of rotation * A2 still designated C2 or 2
A3 still designated C3 or 3
A4 still designated C4 or 4
A6 still designated C6 or 6
   Rotation by 180°
   Rotation by 120°
   Rotation by 90°
   Rotation by 60°

* The rotation-inversion axes are characterized by the combination of an axis and a center of symmetry. The rotation-reflection axes are characterized by the combination of an axis and a plane of symmetry.

Examples of centers of symmetry. I (sometimes designated C) The entire frame has the center of symmetry I :         A' is the image of A , B' that of B The $ C_8H_8 $ cubane molecule has the center of symmetry I This pyrite crystal $ FeS_2 $ has inside the center of symmetry I

Examples of symmetry planes M (still called σ) The frog has the plane of symmetry M (still named σ) :         A ' is the image of A , B' that of B , C ' that of C The formaldehyde (methanal) molecule $ CH_2O $ has two planes of symmetry M1 (still designated σ 1) and M2 ( still designated σ 2) . This vivianite crystal $ Fe^{2+}_3(PO_4^{3-})_2· 8H_2O $ has the plane of symmetry M (still called σ) .

Examples of axes of symmetry A2 (still designated C2 or 2) The old amphora has four handles two to two opposite. Even though the handles are not quite right-angled, it has an axis of symmetry of order 2 A2 ( still designated C2 or 2) . This boat conformation of 1,3-dichlorocyclohexane has an axis of symmetry of order 2 A2 ( still designated C2 or 2):           All the atoms are deduced from each other by a rotation of $ 180^o $ around this axis. The selenite $ Ca^{2+}SO_4^{2-}·2H_2O $ has a second-order symmetry axis:         A2 ( still designated C2 or 2).

Examples of axes of symmetry A3 ( still designated C3 or 3) The tetrahedron at Pen-Bron Beach has a symmetry axis of order 3:           A3 ( still designated C3 or 3) . A' is the image of A . The ammonia molecule $ NH_3 $ has a symmetry axis of order 3:             A3 ( still designated C3 or 3) . The $ SiO_2 $ quartz has a 3rd order symmetry axis:     A3 ( still designated C3 or 3).

> Examples of symmetry axes A4 ( still designated C4 or 4) The Cheops pyramid has a fourth-order axis of symmetry:                                       The pentaborane (9) $ B_5H_9 $ molecule has a fourth-order symmetry axis:                               A4 still designated C4 or 4 . Cassiterite $ SnO_2 $ has a fourth-order symmetry axis:     A4 ( still designated C4 or 4) .

Examples of axes of symmetry A6 ( still designated C6 or 6) The old Church of Our Lady of Nazareth has a line of symmetry of order 6:                               The benzene molecule $ C_6H_6 $ has an axis of symmetry of order 6:     A6 ( still designated C6 or 6). The emerald $ Be_3(Al,M)_2(SiO_3)_6, \; (M \; = \; Cr, Fe, V) $ has an axis of symmetry of order 6:       A6 ( still designated C6 or 6) .

Combinations of elements of symmetry

Rotation-inversion

The rotation-inversion axes are characterized by the combination of an axis and a center of symmetry. The cube has an axis of rotation-inversion of order 3::       $ \bar{A3} $ again designated $ \bar{C3} $ or $ \bar{3} $ : The point $ A $ gives by rotation of $ 120^o $ the point X (1), which, in turn, by central symmetry, gives the image point $ A'$ (2 ). Note: The rotation-inversion of order 1 is the composition of a 360° angle rotation and an inversion: $\bar{1}= \;i $ . The rotation-inversion of order 2 given as an example is in fact a reflection with respect to the mirror plane perpendicular to the axis of rotation and passing through the center of inversion: $\bar{2}=\sigma $ .

Examples of rotations-inversions

Rotation-reflection

The axis of rotation-reflection are characterized by the combination of an axis and a plane of symmetry perpendicular to this axis. The tetrahedron has a rotation-reflection axis of order 4: :    $S4$ : The point $A$ gives by rotation of $90^ o$ the point X (1), which, in its turn, by reflection on the plane, gives the image point $A'$ (2). Note: As the mirror (reflection from the plane) is equivalent to a rotation of π followed by an inversion, a rotation-reflection (by definition a rotation of θ followed by a mirror) is therefore a rotation-inversion of angle θ+π The rotation-reflection of order 1 is therefore a mirror: $S1= \sigma $ . The rotation-reflection of order 2 is therefore an inversion: $S2= \;i $ .

Exercise

Find the vertical rotation axes (click)

Find the number of mirrors(click)

Find the number of rotation-reflection axes (click)

Click on the following figures to identify the represented element of symmetry: