Inflation procedure for a spectre tiling

Such inflation procedure can be iterated infinitely many times and for any given tile of the original tiling we can build the so-called Conway signature, a characteristic infinite-to-the-left sequence that encodes the structure of the tiling.

The inflation procedure undergoes 9 stages that are illustrated in the images below and follows by local properties that any tiling of the plane is forced to possess. We also created an animation that shows the complete sequence of the various stages

• The complete animation

1. Start from a generic tiling (*):
   
2. Isolate groups of three tiles that form two partially superposed mystics pairs.
   
3. The top tile of each group of three is not part of the real mystic and we drop it:
   
4. Each mystic is surrounded by six tiles that are always organized in the same structure to form, with the mystic, a cluster named H₇. These are never superposed and some scattered single tiles are left out:
   
5. Each left-out tile can be associated to an adjacent cluster in such a way that it forms the upper tile of a pair called tower: two adjacent tiles parallel to each other. Thus we enrich some of the H₇ clusters by adding each of these scattered tiles and tranform it to an H₈ cluster. Clusters that are not enriched with a new tile are the 'level 1' "mystic" cluster H₇. The remaining are classified as 'level 1' spectres (H₈).
   
6. Now Place a single reflected, slightly larger and suitably oriented spectre tile on top of each H₈ cluster and a mystic pair of tiles on top of each H₇ cluster:
   
7. We are left with a sparse set of reflected tiles:
8. Finally translate and grow all tiles to form the inflated resulting tiling, click on image for an animation.
9. The last action is a reflection about the x axis in order to obtain a tiling with the correct orientation:

We can now start over again to obtain the next level of inflation

click on image for an animation.

click on image for an animation.