Evolved Girih Tiles - Adapters - Φ - Non-Equilateral
This document presents an evolved Girih tile set beyond the five Girih Tiles popularized by Lu and Steinhardt.

Additional tiles have been presented in multiple papers before, but here, a wider set is officially defined. I refer to this as the evolved tile set or evolved Girih tile set. It includes the adapter tiles, which also have been presented before, but not defined as a group or part of a tile set.

Adapter tiles differ from the five Girih Tiles as the latter only have equilateral sides. This paper presents one category of adapters containing tiles having at least one side with the length of Φ (phi).

The aim is to establish non-equilateral sides as a necessary concept to enable tiling of a greater number of existing Islamic geometric patterns. This opens up for future introductions of other categories of non-equilateral tiles.

The video covers 90 percent of the paper, and have some extra clarifications and imagery. Enjoy! If you have any comments, let me know.

/Lars Eriksson

A note about Tiles vs. Shapes
I've noticed that when I read about tessellation of tiles, the word "tile" often refers to the ceramic tiles used to construct Islamic geometric patterns. The same word are used when talking about the Girih Tiles, that is, when a sub-grid is used to show the puzzle pieces on which the pattern resides and that are governed by edge rules for how the lines cross over to the other tiles.

This is bound to create confusion, so I think it is important to make a distinction between the two. Tiles are the modular puzzle pieces that carries the motif that continue over to other tiles by the guidance of the edge rule(s). Each tile have a positive and a negative side of the motif. When tessellated they form the positive and negative shapes.