Neutron diffraction data were obtained from a single-phase icosahedral powder of the system AI,,Si,Mn,, and its modification by isomorphous substitution on the Mn sites. Amplitudes and phase differencesof the partial structure factors (FA], FMn) were determined. From their Q--dependences within a strip-projection approach, phases were reconstructed. Atomic densities were then calculated in the physical space and in the six-dimensional periodic lattice, resultingin the first experimentally deduced decoration of the quasi-periodic network. The Mackay icosahedron no longer appears as the necessary basic structural unit. In six dimensions the structure has a simple CsCI-like space-group symmetry.