### A revised catalog theorem

As mentioned in the section on the forbidden tilings, the Catalog theorem is a central result of regular polygon tiling theory:

### Catalog theorem

To construct a catalog of all edge-to-edge regular polygon tilings, it is necessary only to start with the 4.8.8 uniform tiling and add all possible tilings with equilateral triangle and square prototiles.

The remaining tilings can be constructed by replacing some of the hexagon shaped patches of triangles with hexagons, and replacing some of the dodecagon shaped patches of triangles and squares with dodecagons.

The universal tiler and decomposition theorem presented in Part 3 allow a beautiful generalisation of this result if rhombs are allowed in the prototile set. Since we know that any composite order regular polygon can be decomposed, we have:

### Catalog theorem (revised)

To construct a catalog of all edge-to-edge regular polygon and rhomb tilings, it is necessary only to start with all possible tilings with prime order regular polygon and rhomb prototiles.

The remaining tilings can be constructed by replacing some of the composite order regular polygon shaped patches of rhombs and prime order regular polygons with composite order regular polygons.

In the general result, prime order regular polygons replace triangles, and rhombs replace squares. No special case like the 4.8.8 tiling is needed for the general theorem.

### Vertex figure extension

The "forbidden" tilings presented in Part 2 allow a slightly improved version of the vertex figure extension theorem for regular polygons, replacing "a finite set of" with the more specific "at most 8":

### Vertex figure extension theorem (revised)

Any vertex figure of regular polygons can be extended to a periodic edge-to-edge tiling of the plane using *at most 8* regular polygon or rhomb prototiles.

The tilings from Robert Fathauer for the 3.10.15 vertex figure require eight prototiles. All of the other vertex figures can be extended to periodic tilings using fewer than eight prototiles.

There are many mysteries about these tilings. I've mentioned a number of questions in the section for each tiling. A larger mystery is the nature of the "jewels" that occur in these tilings. Why do these specific irregular polygons (often with rhomb shaped chunks cut out of them) appear in the tilings?

More generally, is there an interesting way to characterise polygons that can be decomposed into smaller regular polygons and rhombs? Does this have a larger significance for tiling theory?

Then there is the generalised vertex figure extension conjecture I described towards the end of part 3.

Establishing the 5 claims mentioned in that section would not be trivial.

Of course, a counter example would perhaps be even more interesting!

### Penrose patches

In Part 4 I describe a set of tile patches extended from the 5.5.10 vertex figure that can convert any Penrose rhomb tiling into a quasiperiodic 5.5.10 tiling with each vertex in the original Penrose tiling replaced by a decagon.

Is it possible to find a similar set of of tile patches extended from the 4.5.20 vertex figure that can convert any Penrose rhomb tiling into a quasiperiodic 4.5.20 tiling with each vertex in the original Penrose tiling replaced by a icosagon (20-gon)?

In Part 4 I point out a possible approach to find such a set of patches but was unable to complete it.