#semi-regular

Snub Square Tiling is a semi-regular tessellation of the plane, composed of two squares and three regular triangles at each vertex. The arrangement of these two regular shapes seamlessly covers the plane without any gaps or overlaps. It is one of the eight semi-regular tessellations known in geometry. The dual of Snub Square Tiling is Cairo Pentagonal Tiling, which we will explore later. In this short tutorial, I am drawing and constructing the Snub Square Tiling by using an abstract…

The snub square tiling is one of the semi-regular tessellations, where regular triangles and squares match perfectly to fill the plane without gaps or overlaps. The Euclidean construction of Snub Square tiling is possible by utilizing the basic compass and straightedge operations. I made this construction in Rhinoceros to show that there is no need for any numerical input to locate the points and draw the tiling. There are two major techniques explained there. One of them is the basic…

For the last 10 days, I’ve been searching for a proper algorithm for representing surfaces using planar shapes. It is obvious that triangulation is an answer but there is an interesting research topic of planar remeshing using shapes other than quads, hexagons, or any other regular polygons. Especially in computer graphics, such things refer to the optimization of models to decrease the load of GPUs. In the Grasshopper community, this has also been discussed and there is a great implementation…

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…

The intricate harmony of the Islamic Patterns is amazing. The geometry of this and other Islamic pattern designs are explained in the 3rd chapter of Craig S. Kaplan’s Ph.D. dissertation. I constructed a semi-regular tessellation, particularly the 4.8 because it seems to open interesting explorations that mostly emerge from truncated squares. We know equilateral triangles and hexagons are also fundamental shapes for this task. However, the dual nature of the semi-regular tessellations makes a difference in most compositions. First, I…

Truncated hexagonal tessellation (or named 3-12-12) is represented in hyperbolic space (as far as I understood it). The idea is simple if you don’t mix it with complex equations. Below is the 2-dimensional representation of hyperbolic projection. Paper space is defined by the thick line there. Projection is based on a two-sheet hyperboloid surface. Euclidean version of this tessellation is described here.

Creating and handling new types of grid configurations might be an important topic, as Grasshopper is not supporting them natively (yet). I tried to create some semi-regular tessellations based on regular grids. It is actually truncated versions of regular grids, but it slowly becomes interesting as I realized that I may further truncate emerging grids to create Level 2 and Level 3 grids with more complex tessellations. Here are two examples of Level 1, truncating regular tessellations of squares and…

Back in 2011, one of the first posts on designcoding explored how to divide a surface in alternative ways. I called this Grasshopper definition a semi-regular surface tessellation. Although the term isn’t entirely accurate -since the pattern is neither planar nor made of equilateral shapes- it still reflects what I had in mind. Dividing surfaces into fabricable parts remains a key concept in learning parametric modeling, so I decided to revisit and update this old definition. There must be many…