Lace: A New Kind of Cellular Automata Where Links Matter

So, neat, but not exactly mindblowing.

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.

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.

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

Consider e.g. that Game of Life has had folks build universal Turing machines for it, and those machines can be programmed to run a Lace automaton.

However, your Shadertoy port doesn't produce equivalent behavior to the original rule—I've tried several corrections in your shader and none match.

While you've demonstrated the theoretical point, this actually highlights why the abstraction matters: implementing these dynamics correctly is non-trivial even when the algorithm seems straightforward. The three-phase execution, metric calculations, and mutual eligibility create subtle interactions that are easy to get wrong.

Why the shader port doesn't work:

I've successfully implemented this same rule in Taichi (also GPU-based), and comparing the two reveals the issue.

The three-phase execution model requires intermediate storage between phases that Shadertoy's ping-pong buffer architecture can't provide.

In my Taichi implementation, each phase has its own buffer:

- Phase 1 reads node_degree (previous state) and writes to node_next_eligible

- Phase 2 reads node_next_eligible and writes to node_next_degree

- Phase 3 reads node_next_degree and writes back to node_degree

Shadertoy only has two buffers (previous frame read-only, current frame write-only), so the shader tries to collapse all three phases into one pass. But this breaks the execution model: when Phase 2 checks if a neighbor is eligible, it needs the neighbor's just-computed eligibility from Phase 1 of the current step, not the previous frame's data.

To properly implement this in a shader, you'd need either:

- A multi-pass setup with three separate render passes and intermediate textures

- Clever state packing to encode multiple values (eligibility + degree) in one buffer

The fact that the algorithm works correctly in Taichi but requires careful buffer management even on GPU demonstrates that "theoretical equivalence to traditional CA" doesn't mean "trivial to implement."

The three-phase execution model with intermediate states is a real architectural requirement, not just an abstraction.

Regarding edge rule tables: LACE has complete UI infrastructure for true edge-to-edge dynamics (edge states evolving based on connection patterns, independent of node eligibility).

The Rule Editor even has a full tab for it (which appears if a rule implements rule tables). But no rule implements the execution logic yet—it's a dormant feature that would provide the dynamic topology you're suggesting.

Thanks for pushing me to be more precise about all this though :)

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