Colour Groups

Let G be a crystallographic group with elements $g \in G$ and P is an arbitrary group with elements $p \in P$. In the general case, a crystallographic colour group G^P belonging to the family of G and P is defined as a set of ordered pairs $\langle p;g \rangle$ where:

• all elements $g \in G$ appear as right-hand-side components of the pairs and all elements $p \in P$ appear as left-hand-side components;
• a composition law (group multiplication) is defined such that the product of any two pairs from the set is contained in the set, and that the group axioms are satisfied.

The various types of crystallographic colour groups can be distinguished according to the specific of composition law. Depending on latter it is reasonable to distinguish four types of colour symmetry groups (Table 1).

 

P-type direct

Q-type semi-direct

W_P-type wreath

W_Q-type generalized wreath

Let consider an example. The symmetry groups of the cube (as a geometrical figure) is the point group G=O_h=m3m, the subgroups of full orthogonal group O(3). If an arbitrary point on the surface of the cube is taken as $\vec r_1$, one can generate an orbit.

$$O_{O_h}(\vec r) = O_h \vec r_1 = \left\{ \vec r_i \\ gt \bigl\vert \\ gt\vec r_i = g_i \vec r_1, \forall g_i \in O_h \right\}$$ (1)

The whole cube is an unification of such orbits.

Let us consider now coloured cube, where each of its six faces is uniformly painted with one of the six different colours, $\{c_1, \ldots, c_6\}$. It is easy to check that under the action of $g \in O_h$ no one of the rotated cubes coincides with the initial one. It follows that this cube has to be treated like as a totally asymmetric object! So, for adequate symmetry description, it is necessary to extend the framework of symmetry operations: each geometrical operation (defined as acting on the coordinate of points only) should be combined with a suitable permutation of colours. In this case the real symmetry is restored to the isomorphic to O_h colour group O_h^P₆, a subgroup of the direct product group $P_6 \times O_h$. Here P₆ is one of the transitive groups of permutation of 6 colours, a subgroups of the symmetric group S₆. If we take an arbitrary coloured point on the surface of the cube, say $(c_1,\vec r_1)$, the coloured orbit can be generated as follows:

$$O_{O_h^{P_6}}(c; \vec r) = O_h^{P_6} (c_1, \vec r_1) = \\ \... ..._j \vec r_1), \langle c_i, g_j \rangle \in O_h^{(P_6)} \right\}$$ (2)

The whole coloured cube can be generated as unification of such orbits.