Lace: A New Kind of Cellular Automata Where Links Matter
21 comments
·October 16, 2025golol
This is pretty cool. I have several points to make. 1. We all know that Cellular automata or more generally any dynamical system of sufficient complexity (and maybe not too much complexity) will be Turing complete, will have complicated "uncomputable" behavior, will have perhaps pattern formation, or gliders, solitons etc. So what is a valuable addition to this these computational investigations? I think when studying emergent computational behavior we really care about dynamics complexity / rules complexity. It's not impressive to get complicated dynamics out of a complicated system but the simplicity of game of life made it really impressive. I think in that regard LACE is pretty nice: the rule still feels very simple/natural and you can get much more structured/complex behavior with fewer cells.
2. Nevertheless in the end this blog shows mostly pretty pictures of computational, complex, emergent, chaotic behavior, which we've all seen before. And the key features that make the difference go something I would call physics-like are still missing. And I guess that would be complex stable patterns that can have complex stable interactions. Who knows maybe there are 10^16-celled patterns that have this but we don't know.
3. If I were you I would cut the whole preamble. It will make people take you less seriously than they should. You don't want to look like a crank.
sushisource
+1 to this copy being a little bit over-the-top. This is neat, but, as you pointed out at the end of the day this is still computationally equivalent to normal 2d cellular automata. I suspect (not taking the time to prove this) that it's equal in a fairly obvious way, which is that you could just replace "links" with 8*<num link states> additional sub-states per cell. The only real difference is just in how it's visualized.
So, neat, but not exactly mindblowing.
airesearcher
Good feedback -
These rules use very different principles than traditional cell-based rules - for example neighbor degree, number of connections, and eligibility criteria based on connectivity. So the cells are not becoming alive or dead based on the states of their neighbors, but rather on the topology of their neighborhoods.
The details are beyond the scope of a short write up, but are easy to explore in the rule-editor in the GUI of the code.
And preamble pruned of the historical anecdote behind this.
TophWells
So what actually is it? None of the rules in the videos look particularly striking compared to other Life-like cellular automata and 2d cellular automata in general. As you say, their behaviour includes oscillators, spaceships, patterns that grow endlessly... all things that are well-known from other cellular automata. So the videos didn't really show off why they're interesting.
I don't mind the rambling about "planets, galaxies, galaxy clusters, superclusters… and beyond …." but some technical detail would be nice too!
airesearcher
These rules use very different principles than traditional cell-based rules - for example neighbor degree, number of connections, and eligibility criteria based on connectivity.
So in short, the cells are not becoming alive or dead based on the states of their neighbors, but rather on the topology of their neighborhoods.
The details are beyond the scope of a short write up, but are easy to explore in the rule-editor in the GUI of the code.
Here is an example of a rule that is markedly different from a typical "life-like" rule: https://videopress.com/v/lQ5Bghsj
The level of structure and self-organization is striking, to me at least.
Also in all the rules - the links are visible and can have binary or real-valued states as well as the cells. So this enables pretty rich topology which rules can utilize.
TophWells
Could you try explaining it in a comment? Not the general principle, but just the rules for one particular automaton. Whichever one is your favourite. Or Amazing Dragons, if you don't have a favourite.
The amazing part of cellular automata is the emergence of complicated behaviour from simple rules. Life's rules can be written in three sentences, maybe less.
Forgive my quibbling, but I don't understand what this is doing that other projects in this space haven't done before. Adding states and transition rules to edges is new to me...
I did try running your project, but I had to tweak it to get it to work with the instructions in the repo. I seem to be missing a few packages -- mpmath, sympy, typing_extensions. Can you add those to the requirements.txt file?
TophWells
Let's see if I understood this right. For the Betweenness Amazing Dragons rule:
* Compute the "betweenness" of each living cell, which is 1 divided by its degree. Dead cells are not connected to anything, so they have betweenness 0.
* Then, for each cell, sum up the betweenness of its connected neighbours.
* If the total betweenness of a dead cell is in the range [(1.3, 3.6)], it is born and becomes alive at the next generation.
* If the total betweenness of a living cell is in the range [(0.9, 2.6)], it survives and remains alive to the next generation.
* Exception: any cell with 0, 1, 7 or 8 neighbours (in total, ignoring betweenness) dies anyway after the rules above were applied.
... That's not quite right, there's some references to "eligibility" that I can't make sense of. What else am I missing?
bee_rider
Game of life on a sphere could be nice. I never really liked how the gliders would just go off into nowhere.
Although I guess if we play with this too much there is a risk of inventing something like… Bloch's Game of Life or something.
nh23423fefe
probably torus. could you actually discretize the sphere properly? isnt this like tessellating a soccer ball?
wat10000
You can get the topology of a sphere with holes at the poles by wrapping left/right but not up/down, like the Civilization games. Or you can do it with no holes by having left/right wrap, and top/bottom do a different wrap where you come back from the same direction but halfway around horizontally. (Imagine the path crossing over the poles projected onto a Mercator map.)
TophWells
The Game of Life implementation in this post is based on a torus. Watch the gliders when they go off the edge of the screen: they return from the other side!
dvt
This is really cool, I wish they explained what the rules were; for example, the "Amazing Dragons" seems to self-organize very neatly, but I'm not really sure why it has that behavior.
airesearcher
Yeah thanks for the feedback - explaining the rules would require a longer write-up so I opted for the late binding path... in the repo there is a rule editor, and it explains the parameters of every rule in the GUI. It's a lot easier since there are many different kinds of rules and many parameters to explore.
airesearcher
Birth conditions: Based on sum of neighbor degrees
Survival conditions: Based on sum of neighbor degrees
Death conditions: Specific degree counts that cause node death
Eligibility: Nodes must meet conditions to form connections
user____name
Did you end up simulating the mouse brain after the LSD trip?
airesearcher
hahaha ... I actually wondered if someone dosed me... but no that was not possible in that situation at that time (and it would have had to be an elephant dose to last a week). Yes, I did try to simulate a mouse brain... but on a Mac 128 that was rather intractable at the time (actually it still is...)
airesearcher
LACE - Link Automata Computing Engine
(written in python, with optional taichi GPU-powered mode for large-scale simulations)
LACE is a new kind of cellular automata where rules operate on cell states and their links to other cells.
Check out the Gallery in https://www.novaspivack.com/science/introducing-lace-a-new-k... to see the familiar Game of Life rule, but with links.
* Quick Examples **
Game of Life, with links: https://videopress.com/v/lTZ8e4hD
Amazing Dragons (LACE rules): https://videopress.com/v/lQ5Bghsj
** MANY more examples in the Gallery (in the blog post cited above)
Rules can use topological properties of cells and neighborhoods, such as number of connections, neighbor degree, and other metrics.
The added topological dimension enables rules that can have more interesting behavior than traditional "cells-only" CA rules, opening up a fascinating new computational world of new species of stable patterns - oscillators - gliders, puffers, and more.
For details on how these rules work, get the repo and open various rules in the rule editor, where all their parameters are explained. There are many new classes of rules to experiment with.
** You can get the repo and learn more at: https://github.com/novaspivack/lace
analog8374
That's pretty darn cool.
How might you increase the overall order and decrease the chaos?
airesearcher
Realm of Lace Rules The signature "Realm of Lace" family of rules uses:
Birth conditions: Based on sum of neighbor degrees Survival conditions: Based on sum of neighbor degrees Death conditions: Specific degree counts that cause node death Eligibility: Nodes must meet conditions to form connections
airesearcher
Good question... and I explored this a lot in the rules - some of the parameters are extremely sensitive... so it takes a lot of trial and error...
Have they proven their computational model is inequivalent to cellular automata? It is possible that the link rules could be translated into cell rules in regular cellular automata.
That said a different representation can always reveal new phenomena about an old model.