Research

Can we go back to the beginnings of algorithmic design tools, when it was still as simple as possible (not to the binary level of course)? Most of the theorists agree about the fact that contemporary parametric design tools sometimes provide needlessly many possibilities that…

I’ve been searching for a method to study the Voronoi subdivision in order to manipulate it. There are well-known algorithms for that. But I thought it would be better if I use a projective approach just as I did in studying hyperbolic space (here). This…

Today’s Architectural Geometry course was about platonic solids and different attractor objects in introducing component-based design systems. Benay’s idea was both pedagogical and interesting to test in Grasshopper. I searched for the most fundamental type of attractor solid in creating a composition such as this;…

In this exercise, we asked students to develop a method to produce custom tessellations. This is based on the analysis of what is called “Islamic patterns”. We have discussed Eric Brough‘s famous book “Islamic Geometric Patterns”, regarding geometric relationships and linear connectivities via underlying tessellations…

The first-year Architectural Geometry course includes Euclidean constructions as a study of associative geometry. We have exercised the below questions to study this topic. These are three mutually tangent circles, that can be drawn using only a compass and ruler, without built-in tangency functions in…

Last week, the first-year architectural geometry course was about pattern deformations. Students are expected to familiarize themselves with 2d drawing, transformation, and control point editing commands while trying to design a deformation. After studying regular and semi-regular tessellations of the plane, they are expected to…

The Möbius strip is a famous mathematical object. Although being in three-dimensional space, it is a closed-loop of only one surface and only one edge. This quality alone makes the object an interesting study for computational design. I aimed to create an object to test…

This is not to explain the method of the Parquet Deformation but to see the potential. After we’ve studied regular, semi-regular, dual, and truncated tessellations with students, the Architectural Geometry course expects them to develop a Parquet Deformation handmade such as those shown below. I…

This is a late update for my 2012 study on Cairo Pentagonal Tiling (or Cairo Tessellation). Originally, it was an exercise of dual tessellations. Because this tiling is the dual of the famous semi-regular tessellation of Snub Square. After coding the Snub Square tiling, I…

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…

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…

Today’s fractal is the famous Mandelbrot Set. The Mandelbrot set is a well-known and complex mathematical set often associated with fractals and chaos theory. Named after the mathematician Benoît B. Mandelbrot, it’s a set of complex numbers defined by a simple iterative process. The Mandelbrot…

Today’s fractal is the Julia Set, the amazing simplicity of chaos. There are lots of applets and articles on the internet about this fractal. You can generate this with the iteration of a basic function many times and placing points on the complex plane. I…

After a couple of days of studying the mysterious Doyle spiral, I’ve decided to test an approach of circle packing from conformal mapping. First, I tried to understand the Poincare disk (earlier at here, here, and here and here). I used it as the hyperbolic representation…

We can create tessellations of outer points in a Poincare Disk, using the manual method explained in the last post (here). But repeating that compass and straightedge process is becoming a little useless after a couple of repeats. If you say “ok. I understood the concept,…

The poincare disk is still an interesting representation of hyperbolic space for me, full of mysteries. I’ve had several attempts to understand it previously (here and here). Finally, I found a resource* explaining basic concepts about it. I tried to repeat some of the constructions…

This was before Spherical Fantasies, while I was trying to update my surface equation definition. In between designerly intentions and mathematical facts, it’s hard to maintain a process, while keeping the definition yet simple and open to exploration. Grasshopper definition is here: [GHX: 0.8.0066] A little…

This is about conforming distortions on surfaces and creating imperfect (say ugly) surfaces. I started with planar surfaces, however, I continued with spherical ones. There are interesting results when applying trigonometric functions to spherical surfaces. Example surface equations: W=(sin(x*y)) / 2 and W=(cos(x)+sin(x-y²)) / 2…

The Truncated Icosahedron (5,6,6) is an Archimedean Solid we often recognize as the iconic soccer ball. This geometric structure, also affectionately known as the “Buckyball” in honor of the visionary architect Buckminster Fuller, has gained significant popularity and recognition both within the realms of mathematics…

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…