1. What is a twin?
No space group
With space group
A twin is a heterogeneous edifice built by homogeneous crystals (individuals) of the same phase in different orientations, related by an operation (the twin operation) that does not belong to the point group of the individual.

2. Point space vs. Vector space
Point space is the ordinary space (direct space, crystal space) where we describe the crystal structure. It has an origin and N basis vectors, N being the dimensions of the space.
Vector space is an abstract space whose elements are vectors. It has no origin but in it the zero vector is defined. Face normals, translation vectors, reciprocal-lattice vectors are elements of vector space.
Obviously, point space and vector space are dual.

3. Point space vs. Vector space
Morphology and crystal physical properties
Crystal structure
Point groups
Site-symmetry groups (Wyckoff positions)
Face normals, translation vectors, reciprocal-lattice vectors
Space groups
Vector space
Point space

4. Space groups, vector point groups and site-symmetry groups
Space groups describe the global symmetry of the crystal structure and are defined in the point space.
The “usual” point groups describe the morphological symmetry, as obtained through the face normals, as well as the symmetry of the physical properties of crystals. The are defined in the vector space and are more rigorously termed vector point groups.
The symmetry of a Wyckoff position is a point group defined in the point space and for this reason it is a “point point group”, usually termed site-symmetry group.

5. Mapping of individuals in twins
Reflection in
(100)
Rotation about
[001]
Reflection in
{031}
(cyclic twin)
Reflection in
{11-22}

6. Twin operation, twin element, twin law
Twin operation: the movement mapping the orientation of one individual onto the orientation of another individual.
Twin element: the geometrical element in direct lattice (plane, axis, center) about which the twin operation is performed.
Correspondingly, twins are classified as reflection twins, rotation twins and inversion twins
Twin law: the set of twin operations equivalent under the point group of the individual, obtained by coset decomposition.

7. Twinning by merohedry All nodes are restored by the twin operation:
we say that the twin index is n = 1

8. Twinning by pseudo-merohedry
All nodes are quasi-restored by the twin operation:
we say that the twin index is n = 1

9. Definition of obliquityw

10. Twinning by reticular merohedry
One node out of three is restored by the twin operation:
we say that the twin index is n = 3

11. Twinning by reticular pseudo-merohedry
One node out of three is quasi-restored by the twin operation:
we say that the twin index is n = 3

12. Zero-obliquity (reticular) pseudo-merohedry
b  c c
90º b d
twin misfit

13. Classification criteria
H = point group of the individual
D = holohedral point group of the individual
D(Lind) = point group of the lattice of the individual
H* = intersection point group of the individuals in the respective orientations
D(LT) = point group of the lattice of the twin
K = (chromatic) point group of the twin
Tind = translation group of the individual
TT = translation group of the twin

14. Tind = TT (I) H < D D(Lind) = D(LT) = D H*=H K > H, K  D
The twin operation belongs to D, not to H
Twinning by syngonic merohedry
ex. H =222, D(Lind) = D(LT) = D = mmm K=m'm'm'

15. Tind = TT (II)
H < D
D(Lind) = D(LT) > D (the lattice has accidentally specialized metric)
The twin operation belongs to D
H*=H
K > H, K  D
Twinning by syngonic merohedry
ex. H = 4, D = 4/mmm, D(Lind) = D(LT) =
m-3m K = 4m'm'

16. Tind = TT (III)
H  D
D(Lind) = D(LT) > D (the lattice has accidentally specialized metric)
The twin operation belongs to D(Lind) = D(LT)
H*=H
K > D, K  D(Lind)
Twinning by metric merohedry
ex. H = 422, D = 4/mmm, D(Lind) = D(LT) =
m-3m K = (4(2,1)3(3)2(2,1))(3)

17. Tind = TT (IV) Quasi-symmetry of D(Lind)
D(LT)  D' > D(Lind) (the lattice is pseudo-symmetric)
The twin operation belongs to D'
H*=H
K > D(Lind), K  D'
Twinning by pseudo-merohedry
ex. H = 2/m, D = 2/m, b = 91º, D(Lind) = D(LT)  mmm, K = 2'/m' 2/m 2'/m'

18. Computation of the obliquityw
[hkl]*
[uvw]
L*(hkl)
L(uvw)
(hkl)
L*(hkl)L(uvw)cosw = hkl|a*b*c*abc|uvw = |hu+kv+lw|
w = cos-1|hu+kv+lw|/L*(hkl)L(uvw)

19. D(Lind)  D(LT), Tind > TT
Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual)
D(Lind)  D(LT) (the individual and the twin have a different point group)
The twin operation belongs to D(LT)
Twinning by reticular merohedry
Rotation about
111

20. Tind > TT, Quasi-symmetry of D(LT)
Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual)
D(LT)  D'  D(Lind) (the twin lattice is quasi-symmetric)
The twin operation belongs to D(LT)
Twinning by reticular pseudo-merohedry
Reflection in
{11-22}

21. D(Lind) = D(LT), Tind > TT
Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual)
D(Lind) = D(LT) (the individual and the twin have the same point group – they differ in the orientation)
The twin operation belongs to D(LT)
Twinning by reticular polyholohedry

22. Tind > TT, Quasi-symmetry of D(LT) (2)
Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual)
D(LT)  D' = D(Lind) (the twin lattice is quasi-symmetric)
The twin operation belongs to D(LT)
Twinning by reticular pseudo-polyholohedry