A Field Ion Microscope Imaging Atom Probe (FIM-IAP) has been applied to study the atomic structure of the icosahedral phase of the Al-Mn system. A multiple twinning model for icosahedral quasicrystals could be disproved, by calculating its FI image, and comparing it with experimental images. One of the decorations of the three dimensional Penrose packing (3DPP) of which FI images were calculated does compare favourably with experimental images, as far as the relative prominence of the two and fivefold poles, and the interplanar distances along the twofold directions are concerned. In particular an atomic decoration of vertices and two sites in the interior of the thick rhombohedron, which is one of the two building blocks of the 3DPP, turned out to be in agreement with the experimental findings.