Figure 10 The complete set of Φ-lattices for the octagonal (a) and dodecagonal (b) quasiperiodic patterns (see Fig. 11 for a general discussion). In these high-symmetry cases, the prefactors and thus the a unit-cell vectors of the corresponding Φ-lattice (with ) are all half an integer sum of a node of L1 with a node of L2.