GCTS / aperiodic laboratory

Penrose
Cyclotomic Lattice

Four integer coordinates. Two complex embeddings.

Generating cyclotomic frontier proposals…

The host set

Search in the ring of integers ℤ[ζ₅] ≅ ℤ⁴. Rhombs attach at exposed edges; exact corner loads and the internal-window predicate prune proposals before placement.

z = a₀ + a₁ζ₅ + a₂ζ₅² + a₃ζ₅³
z⋆ = a₀ + a₁ζ₅² + a₂ζ₅⁴ + a₃ζ₅

rank 4 lattice2D physical viewexact arithmetic

Point-capacity rule

Each corner value is its angle measured in 36° units. A complete turn therefore contains ten units.

1value unit= 36°

thin rhomb1 / 4

thick rhomb2 / 3

Mixable catalog

P1, P2, and P3 follow the standard Penrose naming. A family button preselects its tiles; every individual tile remains independently toggleable before Run. P1 is the original six-tile set: three marked pentagons, star, boat, and diamond.

Fixed Ammann markings

show the fixed stripes on every rhomb

Checking rigid prototile orientations…

Each catalog rhomb has one permanent five-stripe decoration. Copies only rotate or reflect; no stripes are removed at the patch boundary. GCTS rejects a placement when its complete edge markings cannot match its neighbors.

These are the classical Ammann templates ↗, supplied as constraints. They are not a learned rediscovery.

Older midpoint learner (diagnostic only)

Ammann bars do not generally live on the vertex set. Their ports lie inside tile edges at exact algebraic positions. Learn a sparse equivariant tensor over those support ports, then test whether its induced strands are genuinely straight.

θ[tile, orientation, exact port, channel] ∈ ℝ

Ammann audit: waiting for a marking.

rank—

vertex stars—

bar straightness—

tensor slots—

No marking learned yet.

Held-out branch workwaiting

filternodesbacktracksearly prunesfalse prunes

capacity only————

rank-five only————

compatibility tensor————

The same ordered proposals will be replayed against all three filters.

show learned edge marking show midpoint baseline strands

Experiment

Growth target120 tiles

MAXplayback12 ms / frame

Window phase0.17 show reachable exact support show conjugate embedding

Why this is the proper generalization

Each individual Penrose vertex set is Delone, while the common cyclotomic module is dense. The universal module remains the exact ambient address space, but the canvas now draws only the finite support reachable by the active patch.

All model coordinates are exact integer coefficient vectors in the basis 1, ζ₅, …, ζ₅⁴, with an exact integer denominator. Pentagrid floors and radius comparisons are performed in ℚ(√5). The P1 patch is reconstructed from unit-edge direction steps in ℤ[ζ₅]. Conversion to floating-point occurs only when a coordinate is projected to a canvas pixel.

Ammann-bar crossings require an augmented support set on tile edges, for example exact parameters involving φ/2 = (1+√5)/4. These are algebraic points, but usually not Penrose vertices. The current markings use exact edge parameters 1/(4φ), 1/(2φ²), 1/2, and their complements. The same complete template is used for every copy of a given rhomb, including tiles on the outer boundary.

The P2 view follows the standard mutual-local-derivation rule illustrated in Figures 30–32 of Bettencourt’s Penrose-tiling thesis ↗.

P3 generates lattice translations on exposed edges, checks capacity and fixed Ammann marking compatibility, then applies an exact de Bruijn pentagrid predicate. No precomputed target patch is queried. P2/mixed catalogs retain the bounded model-set solver. Original 1981 paper ↗

2026 cocycle paper ↗ explains why matching rules, bar continuity, cycle closure, and height functions are related but distinct tests. 2022 Ammann-bar paper ↗ gives the geometric straight-line viewpoint and its equivalence to edge matching.

LIVE SEARCH / EXACT P3 RHOMBSinitializing the reachable support

ready

reachable thick thin trial rollback

reachable sites0

placed0

nodes0

backtracks0

saturated sites0

max value10