Topological Automata in Hyperbolic Tilings

Hierarchies of Spaces: Building From the Bottom Up

One of the major challenges in building a system that can increase in complexity as it runs is figuring out how to transfer complex structures in a lower level space into simple structures in a higher level space while still maintaining the essential qualities that the complex lower level structure represents.  Without this jump in abstraction, the system will simply asymptotically approach a maximum level of complexity determined by available resources (energy, computation, memory, time, …).

I’ve shown how particles can be modeled to form surfaces and surface patches by endowing them with local surface properties à la differential geometry and using that information to filter how particles connect to each other.  There still remains a question as to what happens next.  Given a million particles, the system (as described in a previous post) can be run in realtime on a decent laptop, which will likely produce beautiful images and structures, but still only reach a particular level of complexity that is all qualitatively the same.  Even with 100 million particles, the complexity threshold will only reach a level or two above where it currently sits.  To really jump ahead, a completely different spatial logic is required that operates on a higher, more abstract level.

The differential particle system deals with local surface properties and local connectivity but can’t say anything about the overall topological properties of the surfaces it’s producing such as genus or 2-manifold-ness.  These kinds of properties are more efficiently calculated at the coarser level of the surface patch.  Thus, if each surface patch produced by the differential particles is itself turned into a “topological” particle, we can start to figure out how the next higher level might work.

A Space for Topological Particles and Automata

The differential particles exist in the usual Euclidean 3D space and interact with each other according to principles derived from this underlying geometry based on distance, local coordinate frames, and local surface properties.  They are discrete entities representing continuous properties.  Topological particles on the other hand are discrete entities representing discrete properties.  They interact with each other in a completely different manner than differential particles.  In the space that topological particles exist in, distance is backgrounded in favor of patterns of connectivity.

In previous posts, I’ve talked about how hyperbolic tilings (particularly the (5,4) tiling) can be used to represent 2-manifold structures.  The structure of hyperbolic tilings provides a natural scaffolding for constructing surface from a topological perspective for a number of reasons.  First, all 2-genus and higher surfaces can be decomposed into pentagonal tiles.  Second, the structure that you get for free with hyperbolic tilings enables the efficient encoding of algorithms encoding how the topological particles interact.

The particles themselves live on the vertices of the hyperbolic tiling and can only interact with other particles on edges that they are directly linked to.  As a result topological particles can only interact with four other particles.  Since topological particles are abstractions of surface patches, this means that surface patches should themselves be quadrilateral in nature.  But what kind of information do topological particles contain aside from connectivity and what rules govern their interactions?

The  Role of Genus

The essential property we’re after with the topological particles is genus.  Once we can talk about genus in an efficient and hierarchically consistent manner, we can start building the next level up from topological particles.  The genus of a network is determined by its Euler characteristic which is given by \(\chi = V – E + F\).  Genus is related to the Euler characteristic by \(\chi = 2 – 2*g\).

If we look at a torus, which has a genus of 1, and Euler characteristic of 0, we see it can be divided up into 4 quadrilaterals.  The quadrilaterals on the outer shell have positive curvature while the inner ones have both positive and negative curvature in orthogonal directions.  By comparison, a sphere has positive curvature everywhere and a genus of 0. The fundamental different between a torus and a sphere is that some surface patches on the torus have negative curvature, inducing a change in genus.  For higher genus structures, it’s the patterns of connectivity between positive and negative curvature regions that determine genus.  All of this points to the idea that the topological properties simply need to know whether they represent surface patches with positive or negative curvature in their two principal directions.

To sum up, topological particles can connect to 4 other particles along 2 principal directions.  Each principal direction has associated with it a value indicating either positive or negative curvature.  For simplicity, we will simply assign +1 for positive curvature and -1 for negative curvature.

Now, if we think about how these particles interact at a very basic level, we can further constrain things by requiring particles that connect to each other to only connect if the principal directions linked by an edge have the same principal curvature.  While this limits a lot of possibilities in principal, it doesn’t actually limit what kinds of topological structures we can make and it vastly simplifies the way the particles operate to great benefit.  Under this constrain as well, there can be no particles with negative curvature in both directions since they would form an unbounded surface whose normals are solely on the interior.  Thus, we can only have particles with (+1, +1), (+1, -1) and (-1, +1) curvatures.

The next concern is how a pattern of topological particles accumulate and transform into different structures.  The most basic structures are particle pairs, of which there are 4 kinds: (+1, +1) x (+1, +1), (+1, +1) x (+1, -1), and two different forms of (+1, -1) x (+1, -1).  These are represented in the image above with black indicating positive curvature and red negative curvature.

When placed into the hyperbolic tiling, the particles look like the above image, which has 4 particles.  In the above image, it appears that there are only 3 edges linking the particles and it raises the question as to how the surface will close since the tiling doesn’t wrap around but continues infinitely in all directions.  The answer comes from looking at the particle pairs.  Whenever an edge is connected, the edge opposite it (in the same principal direction) must also be connected.  In the diagram above, the edges on the outside wrap around to the next particle on the same hyperbolic line.  It’s easiest to think of it as the tiles on a particular hyperbolic line as being cyclical with only one cycle shown in the image.  The one constraint is that there must be at least 2 particles since a particle cannot connect to itself.

Rules of Interaction

After many many hours of playing with topological particle tiles with the goal of having the particles interact in a way that is not deterministic but does favor an increase in genus, I come up with the following rules:

  1. When a particle is place in the hyperbolic tiling, it marks its neighbors with a score indicating how strong a preference that location has for attaching to the next particle.  Neighbors with negative curvature are considered the most “reactive” and have a score of 2 while positive curvature gives a score of 1.  Scoring is only done for unconnected edges.  Edges that connect back around are left out.
  2. All particles are attached to the highest scoring (most reactive) site that they can be matched to.  If there are sites which tie, one is chosen at random

There are a lot of possibilities for further refinement, including using the location of the generators of genus and measures of local complexity to tweak the scoring of reactivity.  Also, there needs to be some rules for determining how things break apart.  Likely this will involve a combination of overall size and genus generators.  The only one of these I’ve explored so far is the measure of complexity

The diagram above shows how this would work.  On the left is is a 4 tile setup.  On the right a 6 tile setup.  The blue lines indicate how the left-most tile/particle gets reflected through the tiling.  The pattern on the left is highly regular.  If the same diagram were made for each particle, they would be identical, indicating that the combinatorics around the tiling are symmetric.  On the right, once the two extra tiles are added, one of the branches projecting from the left-most tile is cut off, generating an asymmetrical combinatorial pattern.  One measure of complexity or asymmetry would be to look at a neighborhood around existing particles and determine their relative combinatorics, which could then be used as an input to determine reactivity.  The input could be used to regulate symmetry versus asymmetry.

An Example Interaction

So what does an example interaction between two networks of topological particles look like? Above are two pairs of topological particles with their reaction sites marked by circles.  Each pair of particles has four open sites that can react with each other.  Since they all have the same level of reactivity, one is chose at random.  One possible result is below:

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