Tiling families

Exact geometry is proven. Symmetric child frames follow the attributed Ginden reconstruction; the original raster cannot independently certify Paz’s decorated hierarchy.

Review scope: All eight dissections have exact oriented boundary cancellation. Every one of fourteen states is checked through two native generations, including ordered promotions and rotated/reflected frames.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Exact rules reproduce the Encyclopedia’s convex green variant and its complete visible vector patch.

Review scope: All 273 source children; every kind through four native applications. A sixth-application patch matches 15,888 color-consistent source paths covering the visible patch and all 299 visible green marks.

Exact rules follow the published standalone construction. Three inconsistent tiles in a later source illustration are documented.

Review scope: Every kind through two native generations, including source corner frames, transformed and reflected seeds, and partition checks.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Exact ten-state rules match Pautze’s 2019 construction. The source patch comparison checks tile shapes, not global recursive placement.

Review scope: All 585 source children and every kind through four native applications, including ordered promotion frames, reflected descendants and partitions.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Exact girih rules include explicit boundary halves. The decagon contains 211 physical pieces, equal to 196 full tiles by area.

Review scope: All five rules through two native generations, including exact shapes, partitions, transformed and reflected frames, and an adjacent decagon–bowtie seam.

Exact plastic-number triangles and the eleven-child split match independently extracted source regions.

Review scope: Every kind through fourteen native applications, enough for promotion descendants to split twice. Source comparisons cover eleven first-application and 41 fourth-application triangles.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

All six rules use exact shared-edge geometry from the original vector patch, removing the traced gaps and overlaps.

Review scope: Every root through three native generations, including exact shapes, partitions and reflected frames. Source comparisons cover six rule regions and an independent 148-tile second-generation heptagon; other recursive source markings remain outside that comparison.

Nine exact marked rhomb rules match the source. Their jagged replacement boundaries intentionally cross the parent outline.

Review scope: Every root through two native generations; all 793 polygons of the source’s second iterate and a neighboring parent, including ordered corners and reflected frames.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Both color states use nine exact triangular-lattice children, independently transcribed from the original source.

Review scope: Both colors through three native generations under translated, rotated, scaled and reflected seeds; source coordinates, color counts, shapes and partitions checked.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Exact notebook geometry restores the Encyclopedia’s four-state rule graph. The full source patch’s seed, depth and boundary remain unestablished.

Review scope: Every kind through six native applications with ordered frames and partitions. Independent raster comparisons cover all four rules and a fifteen-tile later motif.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

All eight geometric rules are exact. T5 and T6 follow marked patch occurrences. Two T4 corner choices and a conflict in the inferred larger hierarchy remain unresolved.

Review scope: All 35 T5 shapes and frames checked in five regions, all 30 T6 shapes and frames in four regions. Native checks cover two generations and transformed poses. Full-patch recursion is not certified.

All ten geometric rules are exact. Nine apex choices in T4 and T7 remain unresolved; the higher-resolution source patch also omits their marks.

Review scope: Exact first-generation graphs and partitions; every parent through two native generations in transformed and reflected poses. Selected recursive frames are not independently source-certified.

Seven exact marked rhombi and fourteen boundary half-rhombi follow the author’s construction.

Review scope: All 21 computational rules through two native generations, source counts and a literal source landmark, exact shapes, partitions, rotations and reflections.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

Eight marked quadrilateral rules reproduce the source. Boundary protrusions are part of the published replacement convention.

Review scope: Every kind through two native generations, a 149-polygon source patch, ordered transformed/reflected descendants and neighboring patches.

Exact shared edges preserve all 129 source placements. Two independent patch regions confirm the half-turn where the rule sheet disagrees.

Review scope: All roots through two native generations, including ordered and reflected frames and neighboring clusters. Two held-out 46-tile regions resolve the disputed corner; complete whole-patch compatibility is not certified.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

The 16 and 19 unique source children retain full SVG precision and ordered recursive frames, including intentional boundary spill.

Review scope: Both roots through two native generations, reflected frames and neighboring clusters; source SVG placement chains agree through four applications. Source color reproduction and arbitrary depth are not certified.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.

The corrected January 2023 source rules and their corner marks are reproduced. Children intentionally extend beyond the straight parent.

Review scope: All three rules through three native generations, ordered transformed/reflected frames and neighboring patches. A fourth-generation square matches all 69,446 source polygons and their marked order.

The variant shares the corrected base geometry but preserves 39 independently read source corner marks that change later recursion.

Review scope: All three rules through three native generations, ordered transformed/reflected frames and whole-patch seams.

Exact source clusters use explicit half-squares to share boundary tiles between neighboring parents.

Review scope: All three rules through three native generations, marked rhomb corners and rotated/reflected frames; two generations across a shared seam.

No source review is recorded for this implementation.

Review scope: Geometry and recursive markings have not been assessed in this catalog.