https://www.novaspivack.com/science/introducing-lace-a-new-kind-of-cellular-automata Nova Spivack Explorer Menu Skip to content * Home * Biography + Nova Spivack - Headshots * Consulting * Contact * Press * Speaking [Realm-of-Lace-Unified_20251015_214315-624x624] Introducing LACE - A New Kind of Cellular Automata This article is about a new kind of simple computational rule ("LACE rules" running on LACE, the Link Automata Computing Engine platform) which, when applied locally on a grid of cells, demonstrates fascinating emergent "artificial life" behavior. For readers familiar with the Game of Life (GOL), this is a next-level class of cellular automata that utilizes neighborhood topology -- the state of the grid is a function of both cell states and their connectivity (links). To quickly get a sense of what LACE does - scroll down in this post to the video Gallery section and take a look at (a) the Game of Life, with links, and then as a comparison, check out (b) the "Amazing Dragons" LACE rule, and the many other "Realm of LACE" (ROL) rules that fully utilize topology. Preamble I have been thinking about cellular automata ever since I read a fascinating and important book, Three Scientists and Their Gods, during the summer of my sophomore year of college. This book changed my life. After reading this book ferociously on the drive home from Oberlin college to Boston, for summer vacation, I became obsessed with the idea of digital physics - and particularly with the work of Ed Fredkin, (and later Stephen Wolfram as well). And so I decided I had to find a way to get into Ed Fredkin's lab at MIT. This became my mission in life Finally, after many attempts, I did managed to finagle a summer internship in his lab at MIT, thanks to the kindness of Professors Norman Margolus and Tommaso Toffolli, two of the great minds in the field. They had authored an MIT Press book that I had found and read countless times, Cellular Automata Machines, which at the time was the bible in the field. In their lab they had built a specialized supercomputer, CAM-6, for digital physics simulations. I spent every minute of that summer in their lab on CAM-6, exploring the computational universe. For years after that internship, I filled dozens of journals with detailed ideas and theories about digital physics. I wrote code, I tested ideas, I learned and explored. But nothing really came close to the richness and complexity of the vision I had experienced, or our actual natural world. These digital models were lacking something, but I wasn't sure what. Then, years later, it became more clear. The topology of spacetime is not merely a fixed set of locations and states, it's a graph. The links between things are important. (Note: As Stephen Wolfram would later point out to me, this idea harkens back to the dialectical debate between the views of Newton and Leibniz). I needed a new kind of model, one where the shape of space could evolve, one where the shape of space could affect matter, and where matter could affect the shape of space, because after all, according to Einstein, they are really the same thing. And from this insight I then undertook decades of exploration into the idea of a new kind of cellular automata -- one in which the cells had links to each other, which could also figure into their states. Both the cell states and link states could interact, forming complex topologies and geometries. Decades went by, along with many experiments that did not bear fruit. But last year, I actually built something that demonstrates this new class of CA rather nicely. Now I'm finally getting around to posting it... Introducing LACE: The Link Automata Computing Engine LACE is a new experimental platform written in python, for exploring a class of cellular automata in which both the states of cells and their connections (links) to each other are subject to the rules. In LACE, the state of a cell is a function of both its neighbors AND the links it has to them, and link states in turn are a function of the cells they connect. LACE rules can use topological properties of cells and neighborhoods, such as number of connections, neighbor degree, and other metrics. This enables rules in which virtual neighborhood topology (the links) affects neighborhood states -- rules in which topology can both evolve and shape the behavior of the system, and in which the behavior of the system can shape topology. The added topological dimension enables rules that can have more interesting behavior than traditional "cells-only" CA rules, opening up a fascinating new computational vista (within Wolfram's, Ruliad) of new kinds of rules that generate new species of stable patterns - oscillators - gliders, puffers, and more. LACE rules range from link-aware variants of Conway's famous Game of Life, but where edges have varying degrees of influence, to completely new kinds of "Realm of LACE" rules that use topological metrics in their computations. In theory, these rules could even be utilized to simulate neural networks. What is fascinating about these new link-aware "Realm of LACE rules" is that they exhibit amazing new forms of stability and periodic structure. They produce new kinds of periodic gliders, oscillators, puffers, and other kinds of phenomena. Some of them even have behaviors that resemble forms of "artificial life." For more 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. The Code You can read more about LACE here. You can get the LACE python code repo here. The Gallery (NOTE - click on the videos and use the settings icon to set quality to 1080p for best visuals) First, let's take a look at Conway's Game of Life, but with links visible. This demonstration simply shows the links and their states, but in this example, the links do not affect the states of cells. This shows that LACE can run traditional 2D cellular automata rules, with the links visible, but without the links changing rule behavior. In more sophisticated LACE rules (scroll down), we will actually modify the behavior of the Life rule based on the links And here are some Life gliders, doing their thing, with links visible... And here is a stable pattern in Life... with links.... Now, here is a version of Life, where edges influence rule behavior... Here's a variant of Life where edges play even more of a role. In LACE we can adjust the influence that links have on rule behavior - from none (links are merely decorative), to strong influence (links condition cell states). Above, we looked at some traditional Life rule cellular automata. These examples illustrated varying levels of link-awareness in rules. But they are still not showing the full capabilities of the LACE platform. Next, we turn to more advanced LACE rules that fully utilize the capabilities of the system... The Realm of LACE: LACE Rules Now, take a look at the "Amazing Dragons" rule in the Realm of LACE (ROL), one of the more interesting rules in this universe. This is not a life rule - it's a LACE rule - a fully topological rule. Here links and neighborhood topology play a major role in the behavior of cell states and cell states modify the states of links as well. (Here's a little more detail on how rules work - see this and this and a longer explanation here) ...and look what happens! LACE also supports an optional high-performance, GPU-accelerated mode using Taichi, for large-scale simulations... And here's another interesting LACE rule... and check this out... and this one.... And now a gallery of many other interesting LACE rules... This has been a preview of the Realm of LACE ... an incredible new class of cellular automata rules, where links matter and topology evolves. Learn more by playing with the repo, and please share your discoveries This entry was posted in Computer Science, Information Physics, Physics, Science and tagged artificial life, emergent, Game of Life, simulation, wolfram on October 15, 2025 by Nova Spivack. 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