Appendix..... The problem of quasicrystals

The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier trtinsform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is 

The determination of crystal structure is then immediate, in principle, since any standard diffraction pattern will be related to, e.g., the product of an appropriate combination of three such delta functions (periodic in x,y,z directions), with atomic form factors. Inversion to get the real space atomic positions from the diffraction pattern is then possible via the convolution theorem for Fourier transforms, provided the purely technical problem of the undetermined phase can be solved. 

Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the 

There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of reciprocal space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assigiunent of indices, ambiguous. The features of interest are  

If (1) is taken as the main feature of the structure, the appropriate representation along any axis is [34]  

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