Square-to-hex presentation port

Scope and attribution

This port turns an already selected square Wang presentation into a pointy-top hexagonal presentation. It is not a solver, a second solution format, or a new core geometry. The input remains the square-only Wang solution v1 contract, and the hexagonal value exists only in renderer memory.

Section 4 and Figure 13 of Sky Basire’s 2022 report describe the construction principle: transfer the four square edge colors and give the two additional hexagon sides one common color. The report is available through its institutional record, DOI, and PDF. Basire cites Karel Culik II’s Small Aperiodic Sets of Triangular and Hexagonal Tiles for the underlying square-tile simulation result.

The exact edge tuple, coordinate convention, deterministic fresh-color rule, and finite-region bijection below are Tiling Foundry’s formalization of that construction. They are not attributed to either source’s diagram notation.

The visualization component owns the checked transformation animation. The algebra, inverse, and raster-independent checker documented below remain the correctness boundary.

Project convention

The source stores square edges as (N,E,S,W). Its dense row-major coordinates increase to the east with x and to the south with y. The port uses pointy-top axial coordinates

(q,r) = (x,y)

and stores hex edges clockwise as (E,SE,SW,W,NW,NE). The corresponding axial neighbors are:

Let C be the finite set of all colors in the square tile table. The table is nonempty, so the renderer chooses the deterministic fresh color

kappa = max(C) + 1.

Nonnegative integer colors are unbounded in the v1 contract. Therefore kappa is defined and is not in C. Each square tile maps as

H(N,E,S,W) = (E,S,kappa,W,N,kappa)
               E SE  SW    W NW  NE

Tile-table order is unchanged, so tile_id remains the positional index. The boundary tuple maps independently as

B(N,E,S,W) = (E,S,null,W,N,null).

A null boundary entry for a hole remains null. No constraint is invented on the added axis; its tile edges are still both kappa.

Equivalence and inverse

The map has a left inverse on its image:

P(E,SE,SW,W,NW,NE) = (NW,E,SE,W).

For every square tile t, P(H(t)) = t. Thus H is injective, preserves the tile-table cardinality, and introduces no tile choice.

Consider two active source positions. At (x,y) and (x+1,y), square east/west matching is

t(x,y).E = t(x+1,y).W.

Those positions become axial east/west neighbors, and H puts the same two colors on hex E and W. At (x,y) and (x,y+1), square south/north matching becomes hex SE/NW matching for the same reason. Every active pair on the third axial direction compares kappa with kappa, so that axis adds a tautology and cannot remove a square witness.

Conversely, take a valid hex tiling whose tile table is exactly the image of H. Project every tile with P. Hex E/W matching gives square E/W matching; hex SE/NW matching gives square S/N matching; the remaining axis contains no projected information. This recovers one and only one square tiling.

The pointwise maps preserve inclusive coordinates, dense row-major indices, active cells, holes, boundary values, and selected tile IDs. They therefore form a bijection between square witnesses and witnesses over the image table on the corresponding axial region. Existence and nonexistence are preserved, as is the number of witnesses. The serialized v1 format is SAT-only, so an UNSAT result has no document or renderer invocation; UNSAT preservation is a property of the reduction, not a claimed JSON feature.

Independent port checker

renderer/wang_hex_port.py contains the pure reducer and checker. It imports only the Python standard library. The checker does not call the reducer and does not use pixel coordinates. Before hex rasterization it checks:

• exact coordinate, table, assignment, hole, and boundary preservation;
• all six edges of every mapped tile and freshness of kappa;
• the inverse projection for every positional tile ID;
• equality of square and hex matching truth values on both meaningful axes;
• kappa/kappa matching on every active pair along the additional axis;
• equality of boundary-matching truth values after direction mapping.

The truth-value comparison is deliberate. If a structurally present input has an invalid square adjacency, the corresponding hex adjacency remains invalid; the checker establishes translation equivalence without turning the renderer into another solution validator. Contract validation and the independent tiling verifier remain upstream obligations.

Raster boundary

The existing renderer/wang_square.py command remains the only Wang renderer command. Without a geometry flag it follows the unchanged square path. With --hex, it reduces and checks the same in-memory presentation before composing hex pixels:

uv run --locked python wang_square.py \
  ../tests/fixtures/wang_solution_v1_square_sat.json \
  output/wang-hex.png \
  --hex

For hex mode, --pixels-per-cell s is the integer axial raster radius. Centers use

pixel_x = 2*s*q + s*r
pixel_y = (s + floor(s/2))*r.

Integer vertices and a normalized bounding box make negative coordinates and pixel placement deterministic. The same injective palette is extended with kappa; because kappa is greater than every square color, existing logical colors retain their square RGB assignments. Holes use the neutral checkerboard inside the same pointy-top mask. The committed fixture golden is RGB 337×177 at the default radius and margin.

The raster consumes the checked result but proves nothing about SAT, adjacency, boundary validity, or solver behavior. It loads no native library or Z3 module, writes no intermediate hex serialization, and leaves the root python/hex/ placeholders untouched.