Pattern Field Theory

Canonical morphogenesis sequence

The paper defines a five step sequence on a finite connected region \(H \subset A\) of the hexagonal Allen Orbital Lattice, where each site carries position \(r_{i,j}\), phase \(\varphi_{i,j}\), and coherence weight \(w_{i,j}\).

1. Permission (admissibility)
   The region \(H\) is admissible when it satisfies the PAL coherence condition \[ \forall u,v \in H : \cos(\varphi_u - \varphi_v) \ge 1 - \frac{1}{p(u)p(v)} \] with \(p(\cdot)\) the prime index mapping. The permission operator \(P(H)\) acts as a binary gate for morphogenesis.
2. Resource allocation
   A resource vector \(R(H) = (E(H), M(H), C(H))\) is defined for tension, material, and coherence bandwidth. Each lattice site receives a local triple \((E_{i,j}, M_{i,j}, C_{i,j}) = f(p(i,j), w_{i,j})\) such that the sum of local resources equals the global budget over \(H\).
3. 2D program (geometric specification)
   A fold path \(\gamma : [0,L] \to H\) is defined with curvature seed \(\kappa_0(s)\), torsion seed \(\tau_0(s)\), and phase gradient \(\Delta\varphi(s)\). Extrudability is controlled by \(\Psi(s) = \Delta\varphi(s) C(s)\) with a threshold \(\Psi_{\min}\), producing a 2D program \(\Pi_{2D} = (\gamma, \kappa_0(s), \tau_0(s), \Delta\varphi(s))\).
4. Extrusion into 3D
   Vertical displacement along the path is given by \(z(s) = F(\Psi(s), \kappa_0(s))\). Rail pairing and crosslinking are permitted when positional, phase, and material thresholds are met, producing duplex or higher dimensional structures as the path propagates.
5. Stabilisation (Equilibrion)
   A tension functional \[ T[\Sigma] = \int_0^L \left(\alpha \kappa(s)^2 + \beta \tau(s)^2\right) ds \] is minimised to obtain a stable configuration \(\Sigma^{*}\). Unspent resources are returned to the surrounding field.

Biological systems mapped by the sequence

The same structural sequence is shown to apply across multiple biological morphogenesis systems on planar substrates:

• Dictyostelium aggregation – cAMP spiral wave collisions generate polygonal domains with average sixfold coordination.
• Penicillium and filamentous fungi – early hyphal networks approximate 60 degree branching and quasi hexagonal expansion.
• Physarum polycephalum – optimised transport meshes form local hex like networks on planar substrates.
• Bacillus subtilis colonies – nutrient depletion and branch angles yield hexagonal segmentation patterns.
• Epithelial monolayers – isotropic tension drives hexagonal cell packing during early curvature.
• Reaction diffusion tissue folding – uniform gradients form hex like domains before 3D elevation via differential growth.

Across all systems, hexagonal or quasi hexagonal residues appear as planar coherence partitions prior to elevation. Extrudable paths follow differential tension lines, with final 3D configurations arising from tension minimisation.

Relation to the Allen Orbital Lattice and PAL

The morphogenesis operator is defined directly on the Allen Orbital Lattice and uses PAL coherence as the permission gate for structural evolution. The sequence

\(H \xrightarrow{\text{PAL}} H_{\text{adm}} \xrightarrow{\text{alloc}} R(H) \xrightarrow{\Pi_{2D}} \text{extrusion} \xrightarrow{\text{pair+crosslink}} \Sigma_{3D} \xrightarrow{\min T} \Sigma^{*}\)

shows how planar domains on the lattice convert to stable 3D forms while remaining consistent with PFT and the broader AOL equilibrium framework. Hexagonal residues are interpreted as direct signatures of lattice driven morphogenesis under uniform or isotropic coherence fields.

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The full paper includes the formal definitions, equations, and references underlying this summary.

Download PDF: Two-Dimensional to Three-Dimensional Morphogenesis

Citation

James Johan Sebastian Allen, Two-Dimensional to Three-Dimensional Morphogenesis: A Unified Structural Sequence Across Biological Systems, Pattern Field Theory Papers, November 2025. Available at patternfieldtheory.com.