Hypergraphs & Rewriting

The starting point of the Wolfram model — structureless collections of relations, updated by a single rewriting rule.

In the Wolfram model, the lowest level of reality isn’t particles or fields — it’s just a big collection of relations between abstract elements. There’s no built-in space, no coordinates, no geometry. Only a set of connections.

The hypergraph

We record the relations as a hypergraph: a set of hyperedges, where each hyperedge is an ordered list of elements (nodes). Where an ordinary graph edge joins exactly two nodes, a hyperedge can join any number. A state of the universe is written as a set like

\lbrace \lbrace 1, 2 \rbrace, \lbrace 2, 3 \rbrace, \lbrace 3, 1 \rbrace \rbrace

The node labels (1, 2, 3) carry no meaning of their own — only the pattern of connections matters. Relabel every node and you have literally the same universe.

Comparison of an ordinary graph edge linking two glowing points with a single hyperedge — one continuous luminous ribbon threading through four points at once — An ordinary edge joins exactly two nodes; a hyperedge is one connection that can thread through any number of them, in order. (The diagrams below use two-node edges for simplicity.)

One rule, applied everywhere

The universe evolves by repeatedly applying a single rewriting rule. A rule says: wherever you find this little sub-pattern, replace it with that one. For example,

\lbrace x, y \rbrace \;\to\; \lbrace x, y \rbrace,\, \lbrace y, z \rbrace

reads as: “find any edge $\lbrace x, y \rbrace$ ; keep it, and attach a brand-new node $z$ to $y$ .” The variables $x, y$ match existing nodes; $z$ is freshly created each time the rule fires.

At each step we apply the rule to every place it matches at once, producing the next state. Step through it below — press Play, or use Prev/Next to move one generation at a time, and drag nodes to untangle the structure:

Rule: {x, y} → {x, y}, {y, z} — each edge keeps itself and sprouts a new node

gen 0 / 6nodes 2edges 1

Different rules, different universes

The whole project hinges on a striking idea: extremely simple rules can generate endlessly rich structure. Swap the right-hand side and the emergent “shape” of space changes completely. Here each edge instead closes up into a triangle, $\lbrace x, y \rbrace \to \lbrace x, y \rbrace, \lbrace y, z \rbrace, \lbrace z, x \rbrace$ , which grows much faster and curls into a denser web:

Rule: {x, y} → {x, y}, {y, z}, {z, x} — each edge becomes a triangle with a new node

gen 0 / 4nodes 2edges 1

Why this matters

Apply a rule like this billions of times and the hypergraph can start to look like a smooth space with a definite dimension. The conjecture at the heart of the Wolfram model is that the familiar features of physics — the dimensionality of space, curvature and gravity, even the speed of light — are emergent large-scale regularities of this rewriting process, not assumptions baked in from the start.

A few threads to pull on from here (future notes):

Causal graphs — which updates depend on which, and how that gives rise to a notion of time and to relativity.
Multiway systems — applying every possible match in every order at once, and the link to quantum mechanics.
The Ruliad — the entangled limit of all possible rules applied in all possible ways.

Note: the diagrams above use a deliberately simple engine — it matches single-edge patterns and rewrites every match each generation. Rules with larger left-hand sides need genuine subgraph matching, which is a natural next step for this component.

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