A mathematical group, G
Thirumohoor Kalamegaperumal Temple The Art of Symmetry in Architecture |
- The square of symmetry element or the product of any two symmetry elements should be the symmetry elements of the group, G.
- The commutative law of combination must hold good.
- The associative law of combination should exist.
- There should be an identity element that commutes with all other symmetry elements, leaving it unchanged.
- The product of some symmetry elements nullifies the operation, bringing the orientation to the original.
- Every symmetry element has its own inverse. The product of the symmetry element with its inverse gives the identity element.
- Abelian Group: Here the symmetry elements are commutative i.e., each element commute with every other element in the group.
- Non-abelian group: Here the symmetry elements are not commutative.
- Cyclic Group: Group of symmetries deals with only rotations. All cyclic groups are abelian.
- Dihedral Group: Group of symmetries that deals with rotations and mirror planes or reflections. In addition to this, we should remember for any dihedral group there must be nC2 axes of symmetry perpendicular to Cn, the principal axis. Designated as 'D'
- Identity Element (E): Doing nothing or leaving the molecule unchanged or a rotation through 360°
- Axis of symmetry (C2): Rotation through 180° producing an indistinguishable configuration.
- Two mirror planes of symmetry (2𝜎v): Plane containing the principal axis of rotation.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
E |
C2 |
𝜎v |
𝜎v' |
E |
E |
C2 |
𝜎v |
𝜎v' |
C2 |
C2 |
|
|
|
𝜎v |
𝜎v |
|
|
|
𝜎v' |
𝜎v' |
|
|
|
|
E |
C2 |
𝜎v |
𝜎v' |
E |
E |
C2 |
𝜎v |
𝜎v' |
C2 |
C2 |
E |
|
|
𝜎v |
𝜎v |
|
E |
|
𝜎v' |
𝜎v' |
|
|
E |
Results |
Infers |
Focus
! |
E=
C22 = C2C2 =
C2C2-1 |
The
inverse of C2 is C2-1 |
C2-1= C2 |
E=
C33 = C3C32 =
C3C3-1 |
The
inverse of C3 is C32 |
C3-1= C32 |
E=
C44 = C4C43 =
C4C4-1 |
The
inverse of C4 is C43 |
C4-1= C43 |
E=
C66 = C6C65 =
C6C6-1 |
The
inverse of C6 is C65 |
C6-1= C65 |
|
E |
C2 |
𝜎v |
𝜎v' |
E |
E |
C2 |
𝜎v |
𝜎v' |
C2 |
C2 |
E |
|
|
𝜎v |
𝜎v |
|
E |
|
𝜎v' |
𝜎v' |
𝜎v |
C2 |
E |
|
|
|
C32 |
|
𝜎v'' |
𝜎v''' |
E |
|
|
|
|
𝜎v'' |
𝜎v''' |
|
|
|
E |
|
𝜎v' |
𝜎v'' |
C32 |
|
E |
C32 |
|
|
𝜎v' |
|
𝜎v' |
|
|
E |
|
|
𝜎v'' |
𝜎v'' |
𝜎v''' |
𝜎v' |
C32 |
E |
|
𝜎v''' |
𝜎v''' |
𝜎v' |
|
C3 |
C32 |
E |
Comments
Post a Comment