This is a small exercise of Grasshopper drawing various archimedean spirals. It is just a polar point construct, mapped onto a range of angles and number of points. Constant a determines how fast the spiral will turn, whereas constant n is the 1/n power of the angle variable that gives unique names to the spirals. According to Woldfram Mathworld (here) constant n = -2 is named lituus, while n = -1 gives a hyperbolic spiral, n = 1 is a regular archimedes spiral and finally n = 2 will give […]

## hyperbolic

After a couple of days with studying the mysterious Doyle spiral, I’ve decided to test an approach of circle packing from conformal geometry. Poincare disk (studied earlier at here, here and here and here) is used as the hyperbolic representation of space. First, I linked a regular hexagonal grid data structure and rebuilt it after the hyperbolic distortion finding this result: Pretty much like a voronoi subdivision, but a very different thing in fact. My second attempt was to create a circle packing out of this: However I couldn’t manage to […]

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 […]

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 […]