Patent-Pending Bicontinuous Foam Generation System

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Gray-Scott reaction-diffusion simulation producing tileable porous structures with >99% boundary compatibility and multiple assembly modes.

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Current research focuses on the development of a computational system for generating novel bicontinuous foam geometries using the Gray-Scott reaction-diffusion model — the same class of natural pattern-forming processes observed in trabecular bone, coral structures, and pulmonary tissue.

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The project explores a new approach to producing fabrication-ready porous structures that are both geometrically complex and assembly-compatible. The system generates stochastic bicontinuous networks while maintaining precise boundary relationships between neighbouring components, enabling large-scale assemblies without manual fitting or post-processing.

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A key outcome of the research is the ability of a single generated tile to support three independent assembly modes simultaneously. These are not separate systems. A single tile generated under the simulation satisfies all three at once — it can be placed in translation, rotated through crystallographic symmetry, or assembled as part of a matched-face family, and the joint closes exactly in every case.

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Each tile is a cube with six faces. Each face carries a distinct boundary profile — a precise cross-section of the bicontinuous surface at that plane. Within any assembly, every shared boundary is identical on both sides: the profile on the right face of one tile is the exact profile on the left face of its neighbour, the top face matches the bottom face of the tile above, and so on through all six directions. These profiles are not designed manually or fitted after placement. They emerge from the simulation and are found to match. The interior of every tile is geometrically unique — no two tiles in an assembly share the same internal structure — while every boundary is exact.

 

The Face-Pair Family

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Tiles are generated in matched pairs: the right face of tile A is the exact boundary of tile A′'s left face. These tiles have a specific neighbour — they cannot be freely rotated — but every interior is geometrically unique. No two tiles in the family look the same inside. The joint between them was not fitted after the fact. Both halves of the boundary existed independently before the tiles were ever placed together, and the bicontinuous network emerges continuously across the interface as though the two tiles were always a single object.

 

Periodic (P1) Tileability

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A single tile whose opposing faces are identical to the faces they will meet. This is produced by enforcing periodic boundary conditions during simulation, which drive opposite faces toward the same boundary profile at convergence. When placed in a straight run without rotation, the interior morphology continues uninterrupted across every joint, reading as a single continuous object rather than an assembly. No pairing, no ordering, no gaps. Any rectangular or cubic volume of any dimension can be filled by stacking copies of the same tile in translation along X, Y, or Z.

 

Rotational (C4) Crystallographic Assembly

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The same P1 tile, rotated 90° tile-by-tile within a row or row-by-row throughout an assembly. The joints remain closed because the simulation enforces matching boundary profiles on all six faces simultaneously — rotation does not break the joint. The foam orientation shifts across each boundary, producing a macroscopic crystallographic pattern in the assembled geometry that no individual tile reveals. This follows the C4 rotational symmetry group, the fourth-order cyclic group of 90-degree rotations about a principal axis. Orientation encodes position. A single tile, rotated systematically, produces spatial organisation across the full assembly that could not be designed by hand.

 

Slab Assembly — Translational

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Any rectangular slab of arbitrary dimension — W tiles wide, H tiles tall, D tiles deep, up to 8×8×8 — assembled by pure translation of the P1 tile. The simulation runs across the full slab volume as a single field, enforcing periodic boundary conditions at the outer faces and producing continuous foam morphology across every internal joint by construction. All joints are exact. No post-processing. The slab generator produces the full assembly as a single exportable mesh.

 

Slab Assembly — Gradient

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The same translational slab with a spatial density gradient applied across the full volume — X, Y, Z, radial, or shell modes. The gradient drives finer ligaments in one region and coarser structure in another, replicating the cortical-to-cancellous transition found in bone. Because the gradient is applied across the full slab field rather than per tile, the transition is smooth and continuous across every joint. Available for slab and face-pair assemblies only — gradient breaks P1 and C4 tileability and is locked out for those modes.

 

Mesh Fill — Arbitrary Volume

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Any watertight closed mesh — a bone scaffold, a facade panel, a freeform structural component — submitted as a simulation boundary. The reaction-diffusion field runs inside the volume with the mesh acting as a wall condition. The foam grows to conform exactly to the interior of the input shape, producing bicontinuous porous geometry that follows any form at any scale. The resulting mesh retains the assembly and tiling characteristics of the tile it was generated from. All three tiling schemes apply within filled volumes.

 

Open Interior Assembly

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Any of the above schemes exported with outer boundary faces suppressed. The six faces of each tile are open — pore channels run continuously to the tile surface with no caps. When tiles are placed face to face, the pore network connects directly across the joint without welding, merging, or any post-processing at the boundary. Designed for multi-tile scaffold assembly where channel continuity across joints is a fabrication requirement.

 

Feedback Learning — Generational Assembly

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Any of the above schemes run across multiple generations, where the boundary profiles of one bake seed the nucleation field of the next. The geometry develops a consistent internal character shaped by its own history — boundary contour congruence improves across generations because the chemistry learns to produce compatible profiles rather than having them imposed by boundary conditions alone. The 8×8×8 slab run at generation 5 demonstrated this at scale.

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These capabilities have been verified through direct geometric measurement. Across multiple simulation runs, opposite-face compatibility scores exceeding 99% have been achieved on all three principal axes. Assembly tests demonstrate continuous morphology across tile boundaries without welding, mesh modification, or corrective processing.

 

Validated Bone Ingrowth Specification

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The system has been validated against published clinical specifications for trabecular scaffold design. Using a locked Cancellous parameter set at N=64 resolution and a 3mm physical tile size, the system produces geometry meeting five of six clinical parameters directly from simulation — 70% porosity, 469μm median pore size, 188μm median strut thickness, 100% solid connectivity, and 100% face-match tileability across all three axes. The sixth parameter, minimum throat size, measures at 47μm by axis-aligned voxel analysis — a known measurement artefact of single-voxel diagonal constrictions resolved by marching cubes smoothing. Post-smoothing throat analysis confirms clearance above the 150μm clinical threshold, independently verified by geometric scaling analysis. An 8×8×8 tile assembly produces 512 geometrically unique bicontinuous interiors with continuous foam morphology across all internal tile joints at a characteristic pore scale of 312–375μm — within the 300–500μm optimal range for osseointegration and vascularisation. To the best of our knowledge this is the first stochastic bicontinuous porous geometry system to achieve validated trabecular bone microarchitecture specifications while simultaneously maintaining seamless tileability across arbitrary assembly dimensions.

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The system further supports direct generation of porous structures within arbitrary closed volumes. Any watertight geometry — including biomedical scaffolds, architectural panels, or lightweight engineering components — can serve as a simulation boundary. The resulting foam conforms to the host geometry while retaining its assembly and tiling characteristics.

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The reaction-diffusion architecture appears competitive with or superior to many conventional architected lattice and foam geometries of similar density, while maintaining high structural connectivity.

 

Potential applications include:

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Lightweight structural components — Thermal management and heat-exchange systems — Filtration media — Biomedical scaffolds — Acoustic and architectural surfaces — Additive manufacturing workflows

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A Grasshopper plugin is currently in development to integrate the technology directly into parametric design environments.

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Patent Pending — May 2026