[this is a place holder for the lecture notes]
DeBruijn proposed two methods for realizing quasicrystals as projections of 3D sublattices of the 6D unit lattice that pack 3-space. A packing is the 3D analog of a planar tiling. A 3D packing by two kinds of rhombohedra is quasicrystalline if it displays icosahedral symmetry locally, but is aperiodic globally. That is, the group of isometries that leaves the entire packing invariant is the trivial group.
In this collaboration with mathematical artist, Tony Robbin (New York), we shall develop a construction tool that will enable Robbin, and others, to create arbitrarily large virtual quasicrystal installations in Syzygy driven virtual environments. Preliminary work by past REU students of the PI have demonstrated the feasibility of this project. Another year of work by future REUs or student hourlies can bring this project to fruition.
The quasicrystal tool we shall develop will permit the artist to do the
* Interactively choose particular zonohedra (dodecahedra, icosahedra, triacontahedra) to display or suppress.
* Choose semi-transparent colorings for the individual faces of the zonohedra, or leave them out.
* Interactively change the initial parameters, which changes the 3D sublattive in 6D, thereby changing the packing in 3D.
* Select dome-like subsets of a given packing at a given distance from the observer, and then cause the radius of this dome expand outward. Footnote(The quasi-crytalline dome will change its constituent shapes discontinuously, something like a 3D kaleidoscope.)
* Create a virtual installation, which casts colorful Penrose tilings to the floor as the sun moves overhead. The ability to change the latitude, the season, and the hour of the day automatically will save the artists eons in model building, and guesswork, in the design of physical quasicrystal installations.
We expect that this project will serve as model for other collaborations with working artists who, while not unskilled in the art of programming their graphics tools themselves, would prefer to transfer their knowledge to collaborating IT specialists. But, because the graphics tool resulting from this synergy is totally familiar to the artist collaborator, there is no learning curve to impede their creativity.