## tessellation

Here is the abstract of my last publication in International Journal of Architectural Computing. It was a long study of Parquet Deformation for me, partially archived in this blog; Parquet Deformation is an architectural studio exercise introduced by William Huff in 1960s. It aims to improve students’ reasoning of spatiotemporal variation by utilizing sequential shapeshifting of patterns. This article examines the outcomes of this educational research from a perspective of design computing with a purpose to remark its pedagogical significance. A multilayered reading about the exercise will reveal its historical, theoretical, and artistic backgrounds. Then […]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 # Draw centered hexagonal grid # 25.07.2017 www.designcoding.net – Tugrul Yazar import rhinoscriptsyntax as rs import math hexGridCenter = rs.GetPoint("Specify center point") hexGridExtend = rs.GetInteger("Enter the number of radial levels", 3, 2) hexGridClSize = rs.GetReal("Edge size of hexagons", 1.0) hexGridCAngle = rs.GetReal("Rotation angle of hexagons", 0.0) edges = 6 rs.AddPoint(hexGridCenter) hexGrid = [] pythagoras = 2 * math.sqrt((hexGridClSize * hexGridClSize) – […]

This is one of the ideas we’ve tested for the workshop “Animate Patterning“. Inspired from this work, Apart from the pattern that turns around, the torque, rotation radius and the speed of servo, weight, connection detail and 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 perceptual level. However their project was slightly different from this one, I’ll post their results later.

Basic Design I exercise called “Cut and Fold: Deviation” explores diversity within relationships and material behavior. It is initially intoduced by Salih Küçüktuna as a simple but effective exercise platform. Below are some students works of this one week exercise. The term “deviation” has many uses in a range of fields from medicine to sociology. However it is frequently used in statistics with an interesting evocation to design computing: In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, […]

Here I’ve come across to a nice website about the short history of tilings and tessellations: http://gruze.org/tilings/. I’m especially interested in the role of Albrecht Dürer, who seem to pioneer some of the concepts of today’s emerging field of architectural geometry. In this website Kevin Jardine explains mathematical aspects of some of the ancient tilings in a very understandable format. Here is a phrase from his section about Dürer; Like Kepler, Dürer was fascinated with regular polygons and polyhedra, although more for their practical visual possibilities than as the basis for mathematical […]

Kündekari 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 whole structure very durable. Unfortunately very little information could be found on the web about this beautiful technique. Below you see a typical example of Kündekari components and the resulting pattern composition. The subdivision part is out of the scope of this post (as it might be anything). The interesting part is the tongue-and-groove method, something like a puzzle. In […]

Today’s design computing class was about thinking simple. We explored how a small but smart design of a tiling system can generate diversity. Some patterns based on a system called “truchet tiling” are modeled in Rhino using patch and block commands. The example below shows the “prototile” of a hexagonal grid, while each edge is divided into two in order to generate a secondary blobish pattern. As the prototile is symmetric, design possibilities does not allow a hierarchical organization. Instead it refers to a homogeneous construction of minimal variations. Below […]

For the last 10 days I’ve been searching for a proper algorithm in 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 optimization of models to decrease the load of GPUs. In Grasshopper community, this has also been discussed and there is a great implementation at Trada pavilion by Ramboll Computational Design Team (link here). There are […]

It all started with the platonic passion on origami tessellations, not much of the origami, but the tessellation part, as I didn’t want to fold it physically, nor model them using a physical engine such as Kangaroo. That would also be very unnecessary (and yes, very boring) to simulate a folding effort on computer unless we lose our connection with the real world. Instead, I tried to look at a much abstract, silly and basic part of it; the creasing patterns. I found below tessellation named “waterbomb” by the beautiful […]

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. […]

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 […]

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.

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, let’s get faster!” then we can model just the same process in Grasshopper3D to examine varying results in seconds; If we connect any grid of points into this definition, we can clearly see the similar result we obtained by actually projecting the hyperbolic surface on […]

Truncated hexagonal tessellation (or named as 3-12-12) is represented in a hyperbolic space (as far as I understood it). The idea is simple if you don’t mix 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. Here is the Grasshopper3D file containing above idea of hyperbolic representation of a semi-regular tessellation. [GHX: 0.8.0066] (Don’t left click it, right-click and save it to your computer)

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 an more complex tessellations. Here are two examples of Level 1, truncating regular tessellations of squares and hexagons. Then, I tested Level 2, further truncating […]

As far as I understood, it is impossible to physically construct double curved surfaces from quadrilateral and planar faces. This definition tries to find an optimized alternative to this problem. Given any surface, single or double curved, is divided into standard sub surfaces. But this time, those surfaces are treated as planar surfaces, therefore one corner is moved to meet this requirement. The output consists of only planar surfaces ready for fabrication. Here is the initial definition [GHX: 0.8.0066]. There are potential improvements on this definition such as finding the […]

After Puzzling, I tried to establish more of Escher’s basic grid transformations using Grasshopper’s native components. This definition simulates Escher’s transformation of four-cornered grids. Postulate is based on the fact that every quadrilateral (or triangular) planar shape can create regular tessellations without gaps or overlaps. In traditional method, this tessellation is achieved by rotating the shape 180 degrees and copying afterwards. However, in Grasshopper we simply define a fifth point for each shape and divide subsurfaces into four triangular surfaces. There are also more complicated methods of Escher that should […]

Nowadays, I found myself back into the traditional hand sketching. Several failed attempts on Grasshopper led me back there. NURBS (and Grasshopper) somehow limits our conception of surfaces to four-cornered (or two directional) manifolds. Although it sounds like limiting our designs, having four-cornered component spaces has still lots of experimental fields for designers. Escher is a cult person, who transforms euclidean coordinate system to meet his design intentions. There are lots of interesting researches about him, while he shows us how it’s possible to manipulate some fundamental geometric systems, although […]