A structure model, based on a rhombohedral random tiling, is proposed for the ‘perfect’ Al-Cu-Fe and Al-Pd-Mn icosahedral phases. Locally, the model has much in common with six-dimensional models, with over 78% of all atoms belonging to an edge-sharing network of Bergman dodecahedra. The size of the tiles in the model is sufficiently small that relatively little atomic motion is required to implement an elementary rearrangement of tiles. Periodic arrangements of the tiles are consistent with known approximant phases. © 1996 Taylor & Francis Group, LLC.