Inertial Confinement Fusion involves the implosion of a spherical capsule con-
taining thermonuclear fuel. The implosion is driven by irradiating the outside
of the capsule by X-rays or by optical laser irradiation, where in each case
the highest uniformity of irradiation is sought. In this paper we consider the
theoretical problem of irradiation of a capsule by point sources of X-rays, and
configurations which maximize uniformity are sought. By studying the root-
mean-square deviation in terms of different order harmonic modes, we ratio-
nalise the dependence of uniformity on distance d of the point sources from
the centre of a capsule. After investigating simple configurations based on
the Platonic solids, we use a global optimisation algorithm (basin-hopping)
to seek better arrangements. The optimum configurations are found to de-
pend strongly on d; at certain values which minimise nonuniformity, these
involve grouping of sources on the vertices of octahedra or icosahedra, which
we explain using a modal decomposition. The effect of uncertainties in both
position and intensity is studied, and lastly we investigate the illumination
of a capsule whose radius is changing with time.