Non-Euclidean Geometry

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…

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…

[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…

This is my first attempt at 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…