18.701 2010 Plane Crystallographic Groups with Point Group D 1 . This note describes discrete subgroups G of isometries of the plane P whose translation lattice L contains two independent vectors, and whose point group G is the dihedral group D 1 , which consists of the identity and a reﬂection about the origin. Among the ten possible point groups C n or D n with n = 1 , 2 , 3 , 4 , 6, the analysis of D 1 is among the most complicated. There are three different types of group with this point group. Let G be a group of the type that we are considering. We choose coordinates so that the reﬂection in G is about the horizontal axis. As in the text, we put bars over symbols that represent elements of the point group G to avoid confusing them with the elements of G . So we denote the reﬂection in G by r . The lattice L consists of the vectors v such that t v is in G , and we know that elements of G map L to L . If v is in L , rv is also in L . I. The shape of the lattice Proposition 1. There are horizontal and vertical vectors a = ( a , t 1 0) t and b = (0 , b 2 ) respectively, such that, with c = 1 2 ( a + b ) , L is one of the two lattices L 1 or L 2 , where L 1 = Z a + Z b, is a ‘rectangular’ lattice, and L 2 = Z a + Z c, is a ‘triangular’ lattice . Since b = 2 c − a , L 1 → L 2 . The lattice L 1 is called ‘rectangular’ because the horizontal and vertical lines through its points divide the plane into rectangles. The lattice L 2 is obtained by adding to L 1 the midpoints of every one of these rectangles. There are two scale parameters in the description of L – the lengths of the vectors a and b . The usual classification of
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