Eightfold Rotational Symmetry Research Articles

The octagonal quasilattice and electron diffraction patterns of the octagonal phase

An analysis is given of the dual transformation and also the strip method which can yield the ideal octagonal quasilattice as well as its approximants. An ideal octagonal tiling consisting of 45° rhombi and squares can be derived from the projection of a 4D cubic lattice within an irrational 2D subspace onto an irrational 2D hyperplane, and its Fourier transform matches well the eightfold electron diffraction pattern of the Cr-Ni-Si octagonal quasicrystal. The approximant of an octagonal tiling corresponds to the rearrangement of two kinds of tiles in an ideal quasilattice which destroys the exact quasiperiodic sequence. It is shown that the defects introduced to change the aperiodic order into a regular approximant correspond to a linear phason strain along certain directions, and this will break the eightfold rotational symmetry. The Fourier transform agrees well with the experimental electron diffraction pattern displaying only fourfold symmetry.

Two-dimensional quasicrystal with eightfold rotational symmetry.

A 2D quasicrystal with an eightfold rotational axis has been found in rapidly solidified V-Ni-Si and Cr-Ni-Si alloys by means of transmission electron microscopy. The electron diffraction pattern taken along this axis shows no periodicity but a clear eightfold orientation symmetry. The corresponding high-resolution electron microscopic image agrees well with the 2D eightfold quasilattice consisting of squares and 45\ifmmode^\circ\else\textdegree\fi{} rhombi.

Nonperiodic tessellation with eightfold rotational symmetry

A tessellation with eightfold rotational symmetry was obtained by a self-similar subdivision of unit cells into rhombi and squares.