Developmental Dimensionality and Morphological Geometry: A Mathematical Framework for Plant Evo-Devo

Why are ferns fractal? A gene regulatory network is not an algorithm computing morphology—it is a low-dimensional constraint manifold through which high-dimensional molecular dynamics must flow. A narrow channel cannot transmit complex morphological signals, regardless of the underlying dynamics.

The central equation bounds morphological complexity by channel dimension:

$K(M) \leq O(D \log \frac{1}{\varepsilon})$

where $D$ is the developmental dimensionality (measured by participation ratio) and $K(M)$ is the Kolmogorov complexity of the morphology.

Key empirical finding: Duckweed (Spirodela polyrhiza), despite being an angiosperm, has developmental dimensionality $D_{\text{dev}} = 1.99$—matching ancient liverworts and 3.7× lower than Arabidopsis ($D_{\text{dev}} = 7.30$).

This confirms the central prediction: morphological complexity, not phylogenetic age, determines developmental dimensionality. Low-dimensional GRNs force recursive self-similar morphologies because the same narrow channel constrains growth at every scale.