Here is the step-by-step generation of the old Snub Square Tiling. Frankly, this is the first step in the generation of Cairo Pentagonal Tiling I generated with Grasshopper earlier. Because Cairo pentagonal is the dual of a snub square. The first step was easy. Just dispatch cells of a square grid, then evaluate them according to the ratio of 0.366 approx. which is derived from the bisector of an equilateral triangle. Now, we have a snub square tiling, composed of tilted squares, but to process it further and explore different potentials, I had to tell Grasshopper about the equal triangles also. So that made the definition a little bit more crowded because I had to connect proper vertex IDs of different grid cells and join them together to emerge new shapes:

Then, I played with these vertices to produce the following results. Note that you can map these definitions onto any surface as we did before. This set of Grasshopper definitions re-generates the Snub Square Tiling along with its variations. The inputs are the x and y sizes of the lattice (grid). The output of the definitions are sets of lines of the tessellations. You can reverse-engineer these definitions and learn a lot about the geometric constructions. The output is ready for further investigation and development.

You can rebuild the Grasshopper definitions by checking the diagrams on this page. I used native Grasshopper components while developing this design. Thus, you don’t need any add-on to run it. However, if you like my content and want to support me, would you consider being one of my Patreons? Here is the link to my Patreon page, including my working Grasshopper files for Snub Square Tiling and more.