An aperiodic tiling is a pattern that covers the plane without ever repeating itself (i.e., it is non-periodic). A few special shapes, arranged according to specific rules, can cover the entire plane, but the resulting pattern never repeats exactly. I embarked on a 3-day coding sprint to create a general-purpose script in Grasshopper that could generate these tilings. I set Socolar Tiling as my first target. This link contains great […]

In large-scale projects, calculating tiling layouts for wall, floor, and ceiling surfaces becomes even more critical. Evolutionary optimization of tilings is a viable approach. Considering how time-consuming and labor-intensive this process can be, I considered using Galapagos, an old but often overlooked feature in Grasshopper, which might be particularly useful here. Galapagos provides an alternative interface for evolutionary optimization within Grasshopper. It works by minimizing or maximizing a fitness value […]

Although it appears simple, Goldberg Polyhedra in Grasshopper became a tough challenge, which I like. I spent about 24 hours attempting to generate these. I started with Goldberg’s original method. This creates an equilateral triangle using coordinates denoted as m and n on a hexagonal grid. Then, we place this triangle onto the faces of an icosahedron. Finally, we project them onto a concentric sphere. We can create many polyhedra […]

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 […]

In Japanese, “kagome” refers to basket weaving, and the name of this tiling derives from the traditional basket-weaving craft of Japanese culture. In geometry, we know Kagome tiling as a semi-regular tessellation, Tri-Hexagonal Tiling. This tiling is composed of regular hexagons and triangles that cover the plane completely without gaps or overlaps. The dual of this tiling is the Rhombille Tiling. In this short tutorial, I am explaining the drawing […]

The Image Sampler has always been a very effective Grasshopper component. Once again, I had the opportunity to use this component for a professional job. As you can see, with the help of a straightforward and short script, we can create a relief of an image to fit the desired panel dimensions on the screen. We can process this onto a suitable material using a CNC mill, laser cutter, or […]

The rhombitrihexagonal tiling is one of the semi-regular tessellations. It is composed of regular hexagons, squares, and triangles. It is a periodic tessellation since you can copy the fundamental unit and move it across the plane to generate the tiling. I use this quality of the tiling to draw and expand it in Rhinoceros software. This is a basic drawing exercise. At the same time, it is a nice exercise […]

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 […]

“In the heart of a sunflower’s embrace, its seeds weave a poetic tale—a dance of two spirals, parastichies they’re called. One unfurls gracefully from the center in the hush of clockwise whispers, while its counterpart whispers secrets in the tender breaths of counterclockwise motion. A subtle ballet unfolds, where the number of these spirals gracefully mirrors the whispers of adjacent Fibonacci numbers, composing a delicate symphony in the sun-kissed fields.” […]

This is an update on my 2012 exercise on rhombille tiling. This reminded me of the old-school subdivisions of surfaces again after 11 years. It was while reading about rhombic dodecahedron (the stackable solid), that I came across this cute tiling. It’s quite simple, just a hexagonal grid with a special reconstruction. I decided to recreate it by using well-known native Grasshopper components. I animated the result with a variation […]

This is the six-pointed star lattice definition in Grasshopper. The definition is generating CNC or laser-ready results as seen below. In this code, first, I developed a common star pattern by exploding a hexagonal grid. The tricky part of this definition is the last part. There, you can see how you can use the Region Union (RUnion) component to add thickness to a pattern we have drawn before. The operation […]

In this Voronoi puzzle exercise, we derive two-dimensional puzzles by using Voronoi diagrams. This is similar to my previous studies here, and here. Part 1: The Random Voronoi Diagram In section 1 of the below image, I use a rectangular region to generate random points. Then, I create a standard Voronoi diagram by using those points. Finally, I explode the Voronoi cells. Part 2: The Voronoi Puzzle Details In the […]

This python code proves how much effort it takes to create a simple hexagonal tessellation. There are, of course, much easier and faster methods than this. But here you see a code that introduces students to Rhino Python. Using this code, a new Rhino command can be generated, and for the first time in Rhino, we can have a command that creates a hexagonal grid. I followed this tutorial to […]

Further studying iteration in Grasshopper, this time, inspired by George Stiny’s “Chinese Ice-Ray Lattice” subdivisions with Aneome, instead of the Hoopsnake add-on I tried in the previous work. As you know, loops add various ways of usage to Grasshopper. In future versions, loops may cease to be just an add-on and become native components of Grasshopper. Until then, loop plugins like Anemone take on this task. In the example here, […]

This quick project was about a special tiling pattern inside a multi-story residential building’s hallways in Grasshopper. While drawing the construction documents of the project, it was necessary to apply some coding here, as each floor had a different shape to be tiled. Although it is not a real Herringbone tiling, I named it because I couldn’t find a better name yet. First, I’ve imported the geometric boundaries and the “middle curves” […]

Again, we revisited the seamless patterns exercise this semester. This is one of the main exercises of architectural geometry class. We expected to improve students reasoning on generative patterning while they explain their processes step by step. The key element of this exercise is the usage of compass and ruler constructions. However, we didn’t keep this rule limiting their creativity too much. In this activity, we tasked students with developing a personalized […]

This is one of the ideas we’ve tested for the workshop “Animate Patterning“. Inspired by this work, Apart from the pattern that turns around, the torque, rotation radius and speed of the servo, weight, connection detail, and a number of foamboards become important inputs for this design. In the three-day workshop, one group of students interpreted this idea of using several moving layers and creating an emergent pattern at the […]

I’ve come across this nice website about the short history of tilings and tessellations. In this post, I am regenerating a tiling of Dürer rhombii. Albrecht Dürer sounds like one of the pioneers of some of the concepts of today’s emerging field of architectural geometry. Here is a phrase from that website about Dürer; “…Like Kepler, regular polygons and polyhedra fascinated Dürer… A large section of the Painter’s Manual is dedicated to […]

Recently, I returned to old fashion RhinoScripts in order to recapture its idea and functionalities again. After almost 10 years, this is my first experiment on creating a custom function that draws hexagonal grids. I tried to implement a fast process for it, however, there could be much faster ones. This script focuses on using functions, variables, and object arrays. I’ll continue to make more of these simple exercises and […]

Kündekâri is an old woodworking technique, composed of interlocking parts without any glue or nail. It is primarily used in wooden doors and minbers inside Mosques. The interlocking system makes the whole structure very durable. I tried to model this technique using the Voronoi pattern, naming it Voronoi Kündekâri. Below you see a typical example of Kündekâri pieces and the resulting pattern. The interesting part of the Grasshopper model is […]