In October verified function , which returns six variants of twinning in the parameterization Rodriguez by calculating the six- point normal to the plane of twinning , for example, 101 or 102 for the input vector Rodriguez. If n is a vector normal to the plane hkl crystallographic system , and g is the passive rotation matrix of the S- > C , all twinning planes normal to get the formula: n_i = g ^ - 1.OperatorSymetry_i.n . Being in possession of six planes normal to the twinning in the crystal hexagon , we can transform the orientation g to find the orientation of the twin . Having done this by the formula g ' = g.Transpose [ RotationMatrix [ Pi, Gn [ [ i]] ] ] I found that I made a mistake , because , although confusion with the orientation g was confusion twin , the sense of displacement systems gig ' was unfounded. It is assumed as follows from the formula , the orientation of the coordinate system that rotates around g n_i normal through an angle of 180 degrees, this movement takes place by way of rotation around the line perpendicular to the n_i (lying in the plane of twinning ) and at the same time parallel to the unit cell of a suitable vector a_i . Simply put this is a rotation around a_i about 86 degrees in the case of twin 102 and 56 degrees in the case of twin 101 ( shortest turnover ) . Confirmation of these relationships were based on an analysis of the orientation of the twin on the experimental map. First, select a place on the map where the confusion found twin 102 two grains. Then selected two orientations forming the above confusion , and the first of them were propagated rotation relative to the symmetry operators a total of six (because each is repeated twice , so not twelve ) crystallographic directions a_i rotated additionally actively transposing the matrix g according to : a_i = g ^ -1. Hex [ [ i]] . and , where a marked unit vector [ 1,0,0 ] . To find variants of twinning should rotate 102 g with respect to the input orientation a_i an angle of 86 degrees according to : g.Transpose [ RotationMatrix [ angle , Ga [ [ i]] ] ] .