InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 597-598
## Section 4.6.2.5. 1D structures with fractal atomic surfaces |

A 1D structure with a *fractal atomic surface* (Hausdorff dimension 0.9157…) can be derived from the Fibonacci sequence by squaring its substitution matrix *S*: corresponding to the substitution rule , as well as two other non-equivalent ones (see Janssen, 1995). The eigenvalues are obtained by calculating The evaluation of the determinant gives the characteristic polynomial with the solutions , with and , and the same eigenvectorsas for the Fibonacci sequence. Rewriting the eigenvalue equation gives Identifying the eigenvectorwithshows that the infinite 1D sequence multiplied by powers of its eigenvalue (scaling operation) remains invariant (each new lattice point coincides with one of the original lattice): The fractal sequence can be described on the same reciprocal and direct bases as the Fibonacci sequence. The only difference in the 2D direct-space description is the fractal character of the perpendicular-space component of the hyperatoms (Fig. 4.6.2.12) (see Zobetz, 1993).

### References

Janssen, T. (1995).*From quasiperiodic to more complex systems.*In

*Beyond Quasicrystals*, edited by F. Axel & D. Gratias, pp. 75–140. Les Ulis: Les Editions de Physique and Berlin: Springer-Verlag.

Zobetz, E. (1993).

*One-dimensional quasilattices: fractally shaped atomic surfaces and homometry.*

*Acta Cryst.*A

**49**, 667–676.