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D. S. VLACHAVAS 221superposition of any two identical point groups can be treated in a similar way. Table 3 gives the antisymmetry groups created by the superposition of a white and a black point group of crystallographic symmetry.A number of interesting conclusions can be obtained from the application of the proposed algorithm. These are expressed in the following rules.Rule 1: Rotations R being isomorphic to a symmetry operation of the white point group Gw yield a dichromatic composite with symmetry described by the grey point group D = Gw + Gw 1', where 1' is the anti-identity operation.Let R = g,, 1' = l'g,,, then the relation RgiR-~ = gj becomes g~ l'&l'g~ ~ = l'g~gig~ ~1'= & and, thus, it holds for all the elements of the white point group. Also, R 2 = g~ l'g,, 1' = g~ ~ G~. Consequently, this case corresponds to complete coincidence of the white and black point groups and, hence, the dichromatic point group is a grey point group isomorphic to the white point group.Rule 2 A special case of this rule is the following principle given by Pond & Bollmann (1979): 'colour-reversing rotation axes, u', can only be evenfold, and arise when two ordinary u/2-fold rotation axes coincide and 0 is 27r/u'.Rule 3: For a mirror plane any rotation 0 ~ 180 ° along a direction on the plane results in a colourreversing mirror plane (or, in the case of improper rotation, in a twofold colour-reversing rotational axis), whereas for 0 = 180 ° an mm'2' composite group is created.Rule 4: In the case of two-, four-and sixfold ordinary rotational axes, rotation about a direction perpendicular to these axes results in a twofold colour-reversing rotational axis (or to a colourreversing mirror plane in the case of improper rotations) except for some special rotation angles for which higher symmetry results due to the particular symmetry.Rule 2 implies that in the particular case of a fouror sixfold ordinary axis special rotations (i.e. 0 = 2~r/u, u =8 or 12, respectively) create an eight-or 12-fold colour-reversing axis, respectively. Therefore, the superposition of ordinary point groups may result in noncrystallographic point groups and such groups are discussed in the following paper (Vlachavas, 1984). Here it is sufficient to notice that the symbolism of these groups follows the notation scheme of the senior crystallographic point groups. Also, we must mention that the 12-fold rotation and rotoinversion axes are designated for clarification by a line underneath their symbols. AbstractLists of 8-and 12-fold two-coloured groups consistent with zero-and one-dimensional periodic objects are given. These groups are derived as extensions of the corresponding crystallographic two-coloured groups * Present address: 3 K.ristalli Str., 111 41 Athens, Greece.0108-7673/84/030221-05501.50and are of particular interest because they are the only non-crystallographic groups obtained by the appropriate superposition of crystallographic point or rod groups.

371in reciprocal-space components; 0, 0, 1 in the case of K2PtCI4) lies in the plane of scattering or perpendicular to it. The angles and coordinate system used here are defined as for the Enraf-Nonius CAD-4 diffractometer; this coordinate system is not the same as that used in the main text. First calculate Eulerian setting angles ~b and X for 0 = 0. For these angles h is along the y axis and the direction of d is s = XZRd, where R is the reciprocal-space orientation matrix and AbstractA methodology is described that enables the determination of symmetry groups of bicrystals manufactured from a given dichromatic complex. The dichromatic symmetry group is sectioned by a unique twosided plane corresponding to the planar grain boundary and the symmetry elements of the bicrystal are established as those symmetry elements of the dichromatic group that leave the sectional plane invariant. The procedure is employed for investigating generic relations for bicrystals whose components have a given misorientation relationship. It is demonstrated that, for a dichromatic complex with point symmetry * Now at National Defence Research Center, Athens, Greece.0108 -7673 / 85/040371-06501.50 higher than 1, bicrystals of identical symmetry can be created by more than one crystallographically equivalent interfacial plane. Finally, a new scheme is proposed for the classification of grain boundaries. This scheme provides a comprehensive framework for describing the variation of bicrystal symmetry due to changes in the orientation and/or position of the associated interfacial plane.

The determination of the full space symmetry of two interpenetrating lattices in a coincidence-site lattice orientation is discussed. The considered coincidencesite lattices are formed by two primitive cubic, facecentred cubic or body-centred cubic lattices. The two interpenetrating lattices form a dichromatic pattern and its symmetry is investigated by combining the three-dimensional periodicity of the coincidence-site 0108-7673/85/060530-11501.50 lattice with the point-symmetry operations of the motif characterizing the particular dichromatic pattem. This provides a very concise formulation for treating this subject, especially if antisymmetry (twocoloured symmetry) is used. The translational symmetry of coincidence-site lattices with 2<50 is specified by determining the finest common sublattice of the two interpenetrating lattices. The point symmetry is determined by using the principle of the symmetry of composites and it is shown that the O 1985 International Union of Crystallography D. S. VLACHAVAS 531 permissible point groups of dichromatic patterns in the cubic system are 4/mm'm ', 6'/m'mm', 3rn', mm'm', 2'/m', 1.

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