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1911-12.] The Geometry of Twin Crystals. 
point, and in them every plane may be regarded as a twin plane 
(§ 10, ii.). 
iv. A rotation with cyclic number 4 (that is, a rotation through a 
quarter of a complete turn) round a crystallographic axis in the cubic 
system is a co-linear and co-directional operation in the galena ( GBc ) 
and cuprite ( GUh ) classes ; a co-linear and contra-directional operation in 
the diamond ( CBc ) class ; and a co-spatial operation in the pyrite ( CUc ) 
and ulmannite ( GUu ) classes. In the case of the last-mentioned class 
there is no twin plane, but every plane with rational indices is a co-spatial 
plane. The compound crystal thus formed will be strictly analogous to 
a supplementary twin crystal. There is no other system in which a 
rotation with cyclic number 4 can be co-linear in one class and co-spatial 
in another. 
v. A co-spatial operation may also correspond to a co-directional 
operation belonging to the symmetry of a class of a system other than that 
to which the crystal belongs. The only important case is that of a reversal 
relatively to an axis with co-linear and co-directional cyclic number 3. 
Such a reversal, R h is equivalent to a rotation with co-linear and 
co-directional cyclic number 6; for: — [S]Ri = [$]Oi = ( = ) [$](7i 
(where Cl is a rotation with cyclic number n). It is of course a twinning 
operation. 
Thus a reversal of a rhombohedral crystal relatively to the principal 
axis, or of a cubic crystal relatively to the normal to an octahedron face, 
will be a co-spatial operation. For both these crystal structures can be 
referred to hexagonal axes. 
In the first case the corresponding positive and negative rhombohedra 
coincide, and the same in the case with the scalenohedra (or such faces of 
the latter as the symmetry retains). The co-spatial planes formed by the 
coincidence of the positive and negative rhombohedral faces with each 
other contain two common lines — the edges between the rhombohedra of 
different sign, but as these are not at right angles, and the planes which 
coincide are not equivalent, the co-spatial planes which they form by their 
coincidence are not cross planes (§ 21, iv. and v.). In the second case 
the cube faces will coincide with certain of the faces of the trigonal triakis 
octahedron {122}, and other faces will coincide at the same time. The 
co-spatial planes formed by the coincidence of the faces of the cube and 
trigonal triakis octahedron will each contain two common lines correspond- 
ing to the intersections of the cube and trigonal triakis octahedron, but 
they will not be cross planes, for the reason already given. 
Similar co-spatial operations may be conceived as occurring in other 
