The complex atomic structure of a high-order approximant to face-centered icosahedral quasicrystal in AlPdTM (TM = transition metal) systems can be deconvoluted into two kinds of atomic clusters pinned at the vertices of a tiling composed of four basic polyhedra, called the canonical cells. As a result, thousands of atoms per unit cell can be registered in fifteen orbits associated with vertices, edges, faces, and cells of the relevant canonical-cell tiling. This geometrical framework facilitates a rational guess of an atomic jungle in an unknown approximant structure, could an underlying tiling be postulated. A novel approximant phase in the AlPd(MoFe) system is discussed within the present framework.