© Steffen Weber, August 1997
Projection Method
In the Crystallography of Quasicrystals a quasiperiodic structure is described in terms of a higher-dimensional (nD) periodic crystal. Suitable projections from the latter then yield the quasiperiodic arrangement. A textbook example for an analogon in one-dimensional space (1D) is the Fibonacci chain, which consists of two segments of different lengths (L-long and S-short). Their ratio L/S is tau=(SQRT(5)+1)/2=1.61803...
This applet demonstrates the projection algorithm. The solid line represents the 1D arrangement which is obtained from the the higher-dimensional space (2D periodic square lattice) under the given slope. If the slope is irrational eg.: 31.7..degree (corresponds to ARCTAN(1/tau) [rad]) a quasiperiodic sequence like the Fibonacci chain is obtained. For a rational slope the arrangement will be periodic, whereby the period may be large if the slope is close to an irrational one. In Quasicrystallography such arrangements are called approximants.
The 2D lattice points will be projected onto the 1D line if they fall within the yellow stripe. This stripe serves as a projection window and its width is the projection of the 2D square along the solid line.