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Receptive-Field Tiling Grid

Structural tiling — instantiates Neighborhood-Preserving Substrate Mapping

Carves the substrate into a grid of local fields, each owning one patch of the source, so every input has exactly one home and coverage is complete by construction.

Before anything is placed on a map, the substrate has to be carved into local units, each owning one patch of the source, with the patches together covering the whole domain. Receptive-Field Tiling Grid is that carving: a designed lattice of receptive fields — jurisdictions — each responsible for the inputs falling in its region, laid out so that neighbouring fields own neighbouring source patches. Its defining idea is that coverage and locality are guaranteed by construction, up front: the tiling is a fixed structural commitment, not a pattern learned from data or measured after the fact. Every input has a defined home, and a failure is contained to a tile, because the tiles were drawn that way before the first datum arrived.

Example

A carrier planning coverage for a new city divides it into a grid of cells, each served by one tower — the textbook receptive-field tiling. Every location falls in exactly one primary cell (its jurisdiction); adjacent cells cover adjacent ground; the hexagonal layout keeps gaps and overlaps controlled. A phone anywhere has a defined home tower, and as it moves it hands off to the neighbour that owns the next patch. The whole grid is fixed before a single call is placed: coverage is complete because the tiles tile the plane, and locality holds because neighbouring towers own neighbouring territory. Every real design question is a question about the tiling — cell size, cell shape, and where to shrink cells for dense downtown blocks.

How it works

  • Fix the substrate geometry and a tile shape — square, hexagonal, or Voronoi.[1]
  • Assign each tile a receptive field — the source region it owns and is answerable for.
  • Guarantee a partition — the tiles cover the whole domain with single ownership, save for a deliberate overlap band where handoff needs it.
  • Order the tiles so adjacency in the grid mirrors adjacency in the source.

It declares the jurisdictions and the coverage plan; it does not learn placements from data, adapt magnification on the fly, or grade the map's fidelity afterward.

Tuning parameters

  • Tile shape — square (simple indexing), hexagonal (uniform neighbour distances), or Voronoi (fits irregular density). Shape sets how clean adjacency comes out.
  • Tile size / resolution — small tiles give fine locality but many units; large tiles are cheap but coarse. It can vary across the map.
  • Overlap band width — a zero-overlap partition (crisp ownership, hard handoff) versus an overlap margin (smooth handoff, some redundancy).
  • Boundary handling — how edge tiles that would spill past the domain are clipped or extended.
  • Uniform vs. graded tiling — equal tiles everywhere, or finer tiles where the source is dense or important — a static magnification baked into the grid.

When it helps, and when it misleads

Its strength is guaranteeing complete coverage and single, interpretable ownership up front: no learned surprises, every input has a defined home, and a local failure is contained to its tile by construction — the cleanest way to make locality a property you can prove rather than hope for.

Its failure mode is rigidity. A fixed grid is only as good as the density assumptions baked into it; if the source's density shifts, uniform tiles over-serve empty regions and starve dense ones, and the grid cannot adapt without a redesign. The classic misuse is choosing a uniform tiling for convenience over a genuinely non-uniform source, then blaming downstream processing for poor resolution exactly where it mattered. The discipline is to size tiles to the actual source density and revisit the tiling when that density drifts — or hand adaptivity to a remeshing mechanism.

How it implements the components

  • local_receptive_field_or_jurisdiction — each tile is a receptive field: a bounded jurisdiction that owns one patch of the source and answers for it.
  • coverage_and_tiling_plan — the partition itself, the plan that guarantees complete, single-ownership coverage of the domain.

It does not learn its unit placements from data — that is Self-Organizing Map Training — and it does not adaptively re-size tiles as density drifts (Adaptive Remeshing and Reallocation, Magnification Function).

References

[1] A Voronoi tessellation assigns every point of the domain to its nearest site, producing exactly one owner per location — the general form of a receptive-field tiling, and what a fixed grid approximates with a regular lattice of sites. ↩