Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Local Configurations of the Tilings
A few statements about the local configurations of the tilings can also be made.
The small face of the O 1 tile is always joined to a T 1 . This rule is inviolable. There is not a one-to-one correspondence between an O 1 and the T 1 it is joined to, however, for two O 1 s may be joined to the same T 1 across opposite faces that share a long 5-fold edge forming an assembly of three tiles. No T 1 is ever joined to more than two O 1 s, and the two free faces of the T 1 joined to O 1 s in this way are joined to other T 1 s. In fact, any T 1 is joined to two more T 1 s either across opposite faces as when the T 1 connects two O 1 s, or across adjacent faces in other situations forming a chain of T 1 tiles.
These chains and the T 1 s that comprise them do not account for all of the small faces in the tiling, however, for there are configurations where two T 2 tiles will match up along one of their small faces or where two O 2 tiles will line up in this way. These configurations are comparatively rare compared to those of the T 1 tetrahedra, but they indicate, for instance, that one cannot generate all the tile rearrangements possible with these tilings by propogating vertex flips along chains of T 1 s. The large faces that contain 3-fold and 5-fold edges are much less discriminating in who they contact. Any configuration possible of large faces exists in these tilings.
The only other face necessary for completing the tiling is the largest face of the O 1 tile which contains two long 5-fold edges and a long 2-fold edge. This face is also formed by two O 2 tiles that meet along their longest two-fold edge. An O 1 and two O 2 s may meet along this face and form an asymmetric pyramid. These joining faces share no short 5-fold or 3-fold edges, so this meeting does not constrain the way either of the joined tiles must inflate.
Chris Kimmer 2011-06-01