The very simplest possible rules are ones that transform a single unary relation, for example the 1_{1} 2_{1} rule:

RulePlot[ResourceFunction["WolframModel"][{{x}} -> {{x}, {y}}],
"RulePartsAspectRatio" -> 1.99]

This rule generates a disconnected hypergraph, containing 2^{n} disconnected unary hyperedges at step *n*:

Framed[#, ImageSize -> Tiny, FrameStyle -> LightGray] & /@
ResourceFunction["WolframModel"][{{x}} -> {{x}, {y}}, {{1}}, 6,
"StatesPlotsList"]

To get less trivial behavior, one must introduce at least one binary relation. With the 1_{1} 1_{2}1_{1} rule

RulePlot[ResourceFunction["WolframModel"][{{x}} -> {{x, y}, {x}}]]

one just gets a figure with progressively more binary-edge “arms” being added to central unary hyperedge:

Framed[#, ImageSize -> Tiny, FrameStyle -> LightGray] & /@
ResourceFunction["WolframModel"][{{x}} -> {{x, y}, {x}}, {{1}}, 6,
"StatesPlotsList"]

The rule

RulePlot[ResourceFunction["WolframModel"][{{x}} -> {{x, y}, {y}}]]

produces a growing linear structure, progressively “extruding” binary edges from the unary hyperedge:

Framed[#, ImageSize -> Small, FrameStyle -> LightGray] & /@
ResourceFunction["WolframModel"][{{x}} -> {{x, y}, {y}}, {{1}}, 4,
"StatesPlotsList"]

With two unary relations and one binary relation (signature 1_{1} 1_{2}2_{1}) there are 16 possible rules; after 4 steps starting from a single unary relation, these give:

GraphicsGrid[
Partition[
Labeled[ Framed[
ResourceFunction["WolframModelPlot"][
ResourceFunction["WolframModel"][#, {{1}}, 4, "FinalState"],
ImageSize -> 140], FrameStyle -> LightGray],
RulePlot[ResourceFunction["WolframModel"][#], ImageSize -> Tiny,
"RulePartsAspectRatio" -> 1]] & /@ {{{x}} -> {{x,
x}, {x}, {x}}, {{x}} -> {{x, x}, {x}, {y}}, {{x}} -> {{x,
x}, {y}, {y}}, {{x}} -> {{x, x}, {y}, {z}}, {{x}} -> {{x,
y}, {x}, {x}}, {{x}} -> {{x, y}, {x}, {y}}, {{x}} -> {{x,
y}, {y}, {y}}, {{x}} -> {{x, y}, {y}, {z}}, {{x}} -> {{y,
x}, {x}, {x}}, {{x}} -> {{y, x}, {x}, {y}}, {{x}} -> {{y,
x}, {y}, {y}}, {{x}} -> {{y, x}, {y}, {z}}, {{x}} -> {{y,
y}, {x}, {x}}, {{x}} -> {{y, y}, {x}, {y}}, {{x}} -> {{y,
y}, {y}, {y}}, {{x}} -> {{y, y}, {y}, {z}}}, 4],
ImageSize -> Full, Spacings -> {20, 10}, Alignment -> Bottom]

Many lead to disconnected hypergraphs; four lead to binary trees with structures we have already seen. ({{*x*}}→{{*x*,*y*},{*x*},{*y*}} is a 1_{2} 1_{2}2_{1} rule that gives the same result as the very first 1_{2} 2_{2} rule we saw.

Rules for a single unary relation can never give structures more complex than trees, though the morphology of the trees can become slightly more elaborate:

GraphicsGrid[
Transpose[{Show[
ResourceFunction["WolframModel"][#1, #2, #3, "FinalStatePlot"],
ImageSize -> 180],
RulePlot[ResourceFunction["WolframModel"][#1],
ImageSize ->
180]} & @@@ {{{{x}} -> {{x, y}, {x}, {x}, {y}}, {{0}},
4}, {{{x}} -> {{x, y}, {y}, {y}, {y}}, {{0}},
5}, {{{x}} -> {{x, y, z}, {y}, {z}}, {{0}}, 7}}],
Alignment -> {Automatic, {Bottom, Top}}, ImageSize -> 600]