**Figure 1**
The origin of common lines in the transform of an icosahedrally symmetric object. The common lines arise from the application of a symmetry axis which is not coincident with the direction of view (θ = 89, φ = −1°, shown by the vertical white line). It is illustrated in the figure for a threefold axis (θ = 69.19, φ = 0°). (*a*) The transform of the projection is a central section through Fourier space. (*b*) The application of a threefold symmetry axis generates a new plane from the original. Application of the inverse generates a third plane. The intersections of these two symmetry-related planes with the original yield a pair of lines in the original image transform which must have identical values. These are seen in the stereo pair (*c*). The symmetry axis which created the pair of lines lies at the centre of the surface formed by the three planes. Each pair of symmetry elements and its inverse yield a pair of common lines. For an icosahedral object, this yields 37 pairs (12 from the fivefolds, 10 from the threefolds and 15 from the twofolds) in the transform of the projection. |