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.
(c)1991,1993,1995 Chaim Goodman-Strauss
Chaim Goodman-Strauss Dept of Mathematics firstname.lastname@example.org 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!