I’ve recently been working on a different approach to composing complex topologies using hyperbolic spaces. The motivation for this approach comes from recognizing that biological structures are by and large made up of 5-sided forms and that pentagons can be sued to tile any 2-manifold object. In other words, 2-manifold objects can be viewed through the lens of pentagonal tessellations, however, pentagons cannot tile the Euclidean plane. They do however tile the hyperbolic plane. For some background information, there’s a video on 3-manifolds that is fantastic.
Hyperbolic space has an infinite number of possible planar tessellations whereas spherical and Euclidean planar spaces have a just a small number (5 and 3 respectively). The space of possible tessellations can be mapped out by graphing the number of sides of the basic polygon (p) versus the number of polygons coinciding at a vertex (q).
The idea to use hyperbolic space for composing topology is to color the hyperbolic plane with a periodic pattern of colored tiles where each unique color corresponds to a unique tile. Where the pattern repeats, we can say that there is a loop or hole constructed on a 2-manifold surface. In other words, we break the 2-manifold surface into tiles and map the connections between tiles onto the hyperbolic plane. For example, a torus made of 4 pentagons and a (5, 4) tiling looks like this:
Notice that some tiles connect more than once. This is completely valid as we’re not restricting the way tiles can connect in any way as long as the tiling is a (5, 4) tiling. We could for instance model a sphere as a tessellation with a single tile that repeats in all directions.
In the above diagrams, the (5, 4) tiling wasn’t chosen arbitrarily. While I did restrict myself to pentagons at the start, I could have chosen from an infinite number of values for q. 4 was chosen because of it computational properties. Not only are hyperbolic tessellations useful for topology, but they have some really interesting applications to the design of automata and grammars, particularly with respect to questions concerning the fundamental nature of computation. The main researcher in this area is Maurice Margenstern, who was written extensively about hyperbolic tiling in the context of computer science.
From Margenstern’s work, the Fibonacci trees are of particular interest because they provide a straightforward mechanism to overlaying hyperbolic tessellations with a kind of computational connective tissue in the sense that they become easy to navigate and manipulate since a simple data structure can be used to describe the entire domain including both its connective and hierarchical relationships. The best introduction to this subject is Margenstern’s Cellular Automata in the Hyperbolic Plane: Proposal for a New Environment paper.
The Fibonacci tree follows 2 simple rewrite rules in the process of covering the hyperbolic plane: white -> black, white, white and black -> black, white. The fundamental polygon (root of the tree) in the center has no color and its children are all white nodes.
With the Fibonacci tree in place, and the basic mechanics of working in hyperbolic space worked out, the next step is to work on the navigation of the tiling so that mappings can be made to the pentagonal grid and tiles on a 2-manifold.