PolyFoam

POLYFOAM!

How to Fill up space completely with a fractal froth of polyhedra! In essence, one uses the stellations of a polytope to construct a substitution system, and then use this system to produce similarity tilings. There are three main categories of foams that can result: periodic, aperiodic but periodic in scale, and aperiodic in scale. Algrebraic invariants can be found describing these possibilities more fully.

The first class, while producing periodic structures in the limit, is often quite interesting before the limit is reached. Click for a sample of periodic octahedral foam. Click for periodic hexa-foam.

The second class produces structures that can be parsed into traditional aperiodic tilings. Pentafoam

and Dodecafoam are notable examples.

It seems though that every convex polytope does lead canonically to a foam, and of course this usually is of the third, most degenerate kind. Any set of standard tiles decorated by foam in the third class would need tiles of arbitrarily many scales and arbitrarily many local configurations to actually produce the global foamal structure.

Follow this link to get to a slew of dodecafoam pictures.


Chaim 
Goodman-Strauss

Dept of Mathematics          strauss@geom.umn.edu
Univ Arkansas                http://www.geom.umn.edu/~strauss
Fayetteville  72701          The 
Geometry Center

A hand scrawled set of lecture notes (Smith 1993) is available if you write me. Thanks to Mathematica, Christopher Carleson's DiscoverForm, Geomview and Dan Krech for the software to make these pictures.

If anyone out there knows how to reach Carleson, please write!