non-euclidean

Unlike classical Pattern Deformation assignment discussed here and here, this time we asked students to explore deformations by using referential systems as a secondary space. They are expected to create variations on a regular pattern only by deforming its underpinning lattice. Below are three examples of this alternative assignment. I’m thinking about improving this exercise to three dimensions, seems very easy to implement by using cage editing commands of Rhino. Selin Işıldar Serra Uludağ Mine Güvenç

In today’s drawing class, we taught methods of drawing basic transformations by hand. Mirror was one interesting subject of that. However, then I opened Grasshopper and Rhino to test the effects of curved mirror planes. Unfortunately I realized that there is already a curved mirror component in Grasshopper :( Here is the Grasshopper definition: [GHX: 0.9.0061] This might be one of the simpliest ways of introducing generative deformations for design geometry.

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 in Rhinoceros, (without any logical purpose). The most important part is the conversion of an Euclidean point into a hyperbolic space. There is no clear formula, directly projecting a point into Hyperbolic space, but the method seems very interesting to me because we can do this […]

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)

[GHX: 0.8.0066] This is my second attempt on getting into non-euclidean representations of space. Althouth it seems easy at first sight, this represents a close point of theory between mathematics and contemporary computational design geometry. As always, architects tend to use mathematical terms such as “non-euclidean geometries” but as far as I saw, most of them have no idea about what it is. So, I’m trying to learn and understand this connection by experiencing small parts of it, sailing at the edges of architectural geometry, but trying not to get into […]

This is my first attempt on representing a non-euclidean space. There are several representations of a non-euclidean space in euclidean means such as Beltrami-Klein or Klein, Poincare , Poincare half-plane and Weierstrass. Here, I tried to understand Poincare’s approach. Random straight lines are drawn on a hypothetical hyperbolic space using a simulation of Poincare’s famous disk representation. Although here is a precise description of the disk and it’s construction, I used ready made arc component of Grasshopper3d, showing start and end points along with start and end tangent vectors pointing at the […]