set to zero. The background intensities, $B(\mathbf{q})$, are replaced by the spherically symmetric version i.e. the intensities in each radial bin are replaced by their average. The derivation that both these operations are projections is given in Appendix B. Further details regarding masking and alignment of reconstructions from different random starting models are discussed in Appendix C.

$P_M(P_S(x))$

$\beta = 0.7$ $f_M(x) = (1-1/\beta)P_M(x) + (1+1/\beta)x$ $f_S(x) = (1+1/\beta)P_S(x) + (1-1/\beta)x$ $x + \beta\left[P_M(f_S(x)) - P_S(f_M(x)) \right]$

$\Psi_i \gets$ Uniform Random $\Psi_i \gets$ ER($\Psi_i$) (100 times) $\Psi_i \gets$ DM($\Psi_i$) (200 times) $\Psi_i \gets$ ER($\Psi_i$) (100 times) Align all $\Psi_i$ Calculate Phase Retrieval Transfer Function (PRTF) Average over all aligned $\Psi_i$

\[sec:phased\_all\]Reconstruction from the full data set
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The results of applying the above two-step reconstruction method to all patterns are shown in Fig. \[fig:pipeline\]. The 3D intensity shows strong icosahedral symmetry even though this constraint was not enforced during the reconstruction. The resolution corresponding to the edge of the spherical volume of intensities is . After iterative phasing, the electron density shown in the bottom row was obtained. The contour plot shows an icosahedron with bulges at each vertex while a slice through the object centre shows the presence of 