Quasicrystalline projection of chaotic attractors
qca01.png (821k)
The above image is a chaotic attractor in 5-dimensional space with some extra symmtery that has been projected to the plane
using quasicrystalline projection [1]. The above image shows the attractor and the overlying quasicrystalline Penrose tiling.
Penrose tilings themseleves may be constructed via canonical projection of the edges of a hypercube lattice onto a plane. For
details of canonical projection, see [2,3]; however, we note that it is based upon projecting onto the plane the points that
lie in a special "cylinder" near the plane and our quasicrystalline projection generalizes that technique for projection to objects, such as the shown attractors, in high dimensional space. The following four images illustrate examples with other symmetries. Links to a script that could be used
to replicate the above image appear toward the bottom of the page.
qca01.png (2051k) |
qca02.png (1216k) |
qca03.png (932k) |
qca04.png (1957k) |
References
[1] C. Reiter, Chaotic attractors exhibiting quasicrystalline structure, in progress.
[2] M. Senechal, Quasicrystals and Geometry. Cambridge University Press: New York, 1995.
[3] A. Lewis, Some Planar Tilings: A quasicrystal picture gallery, http://penelope.mast.queensu.ca/~andrew/qc/
Scripts for Duplicating the qca00 image above
The J language (Jsoftware) may be downloaded from http://www.jsoftware.com
Copy and save the entirety of following script: quasi_ca_ex.ijs in order to duplicate the top image on this page.
The following two scripts are also required in the "fvj2" subdirectory of the j directory:
· raster5.ijs
· dwin+.ijs
Link to:
Browse Cliff's J materials
The Gallery of Chaos, Fractals and Symmetry
Back to Cliff's Home page
Mathematics Department Pages
