As I wanted to learn to use our new CNC cutter, I needed a dummy object to test it. However this became more interesting than I expected. It is moving so sexy! First, I tried to experiment a regular egg-crate definition (explained here and here) last year. I chose Moebius strip as the underlying surface (also experimented here and here before) However, Fulya warned me about their weak relationship. So I’ll work on a more conformal egg-crate definition in order to match the properties of a Moebius surface. This is […]

## October 2012

This is not to explain the method, but to see the potentials. After we’ve studied regular, semi-regular, dual and truncated tessellations with students, Architectural Geometry course is now expecting them to develop a pattern deformation such as these shown below. These samples are taken from this website if you are also interested in other topics of tessellation. They can all be drawn with simple commands, line, control point editing, trim, extend, and rotate. Their firing rule is so simple that maybe they don’t need a complex algorithms to model such structures. […]

Nowadays I plan to enter Rhinoscript, Pyton and DesignScript back again. However, I can’t leave Grasshopper3D without mentioning the “cognitive shift” it pioneered in design computing community. Here is a phrase from famous special issue of “Computer” Journal, published in 1982 with Tilak Agerwala and Arvind’s editorials; Data flow languages form a subclass of the languages which are based primarily upon function application (i.e., applicative languages). By data flow language we mean any applicative language based entirely upon the notion of data flowing from one function entity to another or […]

After Snub Square Tiling, I found out that it might be impossible tell Grasshopper about the dual of it because I had no idea on how to connect those area centroids appropriately in creating pentagons. That is where “Proximity 2D” component came to aid. I know it is not the best solution for a tessellation analysis, but this component saved my day. Or, maybe this is also a property of such semi-regular organizations that duals of polygonal centroids are always creating other regular tessellations if connected by using a proximity algorithm. […]

This was the first step to the generation of Cairo Pentagonal Tiling. It is the dual of a semi-regular tiling of 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. Here is the definition: [GHX: 0.9.0014] Now, we have a snub square tiling, composed of tilted squares, but in order to process it further and explore different potentials, I had to tell Grasshopper about the equal […]

After “parquet deformation of islamic patterns” post, I decided to improve that by adding a semi-regular tessellation. This and other designs are explained in 3rd chapter of Craig S. Kaplan’s phD dissertation (here). This time I tried to construct a semi-regular tessellation, particularly the 4.8 one, because it seems interesting explorations mostly emerge from truncated squares and their patterns. We know equilateral triangle and hexagon are also fundamental shapes for tessellations, but square is somehow makes difference in most compositions. Here is my working file if you are interested to check […]

In 1982, it has been more than 15 years since the dataflow approach to algorithm designing are discussed in computer science. Computer journal publishes a special issue with the foreword of Tilak Agerwala; he says; …We have discussed two characteristics of the von Neumann model of computation: global updatable memory and a single program counter. It will become clear shortly that the data flow model has neither of these. First, the data flow model deals only with values and not with names of value containers (i.e., addresses). This concept is also fundamental to purely functional or […]

This was my old plan to work with images in Grasshopper. Certainly that was not the result I expected, but this could be counted as a starting point. After seeing beautiful circle packing compositions here, I decided to program Grasshopper, so that it’ll create a subdivision, based on an image data. This was the initial version, just subdividing a plane with voronoi points and visualizing it according to image’s color values of proper UV coordinates: [GHX: 0.9.0014] There is no interpretation on the points, they are just “random”. (Although I’m looking […]

This is a small exercise, to remember old-school tessellation of surfaces. While reading about rhombic dodecahedron (the stackable solid), I’ve come by this tiling. It’s quite simple, just a hexagonal grid, animated by a variation of Breststroke surface function (described here), then reconstructed as three quadrangles with proper vertex id. [GHX: 0.9.0006] here is the Grasshopper definition. You may subdivide any surface to create such tessellations, this time I chose to rebuild the surface from hexagonal cells.

This is the set of all Julia Sets (studied here). Here is a simple explanation without math. According to this website: The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers. […]

I’m completely stuck with fractals nowadays, especially famous Mandelbrot and Julia sets. Here is my first definition which estimates Julia sets. There are lots of applets about this fractal calculations on the internet because the computational method is very simple, (and generally people love them because of this dilemma between simplicity, chaos and infinity). It is the iteration of a single function over and over again and placing points on the complex plane (not the cartesian plane). Here, it was my first recall of complex numbers, now I’m happy I’ll […]

Studying circle packing led me back at the highschool days. First, I’ve tried to write a vb.net component so that I would say Grasshopper to place circles and check lots of things iterating again and again. Then I felt that this was not my real interest in circle packings. After finding an old post of Daniel Piker (here), I’m truly enlightened about an old topic of our highschool education: The complex numbers! Then I found this link, explaining the short history and meaning of complex numbers for geometry. My first […]

When I was younger, among the branches of philosophy, I had studied a little logic and, among the subjects of mathematics, geometrical analysis and algebra, three arts or sciences which looked as if they ought to contribute something to my project. But in looking at them, I took care, because, so far as logic is concerned, its syllogisms and most of its other instructions serve to explain to others what one already knows or even, as in the art of Lully, to speak without judgment of things about which one […]

Since last week, I’m very curious about circle packing. There are a couple of complete solutions on the internet. I’m still at early steps of such a solution yet. A full circle packing means that it does not include any gaps and each circle is tangent to all possible neighbors. Sounds easy in Grasshopper but I couldn’t see any solution yet. There a some circle packing attempts but they have gaps. Also I don’t want to use an evolutionary solver (Galapagos) or physical engine (Kangaroo) because I believe there is […]