GCTS / cyclotomic tilings
Markings from tile contacts
Grow a patch. Equate the values where tiles meet. Watch the space of possible markings shrink.
The learner observes contacts in the current patch. Its symbols describe every compatible marking on the chosen finite support. They are not yet a rule that forces a Penrose tiling.
Hover or tap a point to inspect it. Equal symbols share a color; the symbols remain free parameters.
Search and storage details
Logical records; object overhead and temporary copies are excluded.The marking-space experiment
Each oriented tile starts with independent symbolic values at its vertices and edge midpoints. These exact addresses lie in \(\tfrac12\mathcal O_{\mathbb Q(\zeta_5)}=\tfrac12\mathbb Z[\zeta_5]\). The finite support is generated from tile outlines; no planar lattice enumeration, window, bar values, or saved learned assignment is used.
Whenever placed tiles share a support point, their two variables must agree. We impose the same equation under every allowed rotation and reflection. If \(D_P\) records these contact equations, the remaining scalar marking space is \(M(P)=\ker D_P\). Exact equality classes represent this space: the display’s \(u_i\) are independent symbols, not fixed numbers. Equal symbols have the same color; different symbols can still receive equal values. Rotations and reflections permute the address variables and induce an action on the quotient.
The panel reports the original dimension, remaining dimension, and independent equality count. Adding a contact can lower the dimension; removing a tile restores the prior space exactly. Continue retains the current patch and space. Reset restores the unconstrained space. Switching methods pauses and preserves each method; changing tiles resets all methods.
Observation, not an inferred forcing rule. The space describes the current finite patch and its rigid images. Constants always remain possible. Provisional values do not prune candidates or label absent contacts as forbidden. This replaces the former fresh-binary-channel encoding of certified forbidden pairs in this lane. No negative-pair learner, curve fitting, or new aperiodicity claim is involved.
Choose bare geometry to omit boundary labels. With supplied Penrose rules selected, P3 uses edge arrows and P1/P2 and mixed sets use colored boundary ports. Those boundary rules were transferred from the existing decorations; they are problem input, not newly discovered aperiodicity rules. The learner reads only oriented tile outlines and contacts; it has no access to label values, bar interiors or extensions.
Mixed sets are experimental. Tiles are permitted, not required. The existing common scale is preserved: P1 cannot share a whole edge with P2/P3 at this scale, while mixed P2/P3 can be attempted. A run may exhaust its frontier or use only part of the selection.
The Ammann method compares infinite supporting lines directly, without an edge-label check or point-value marking table. Whenever a line meets another tile, that tile must carry the same line in the same family. Lines crossing in different families are allowed. The canvas clips the lines only for display. This method can produce a different patch. Run stops at a completed frontier corona; finite patches do not prove infinite extendibility.
Circular markings on kite and dart
Copies of these circular arcs join across tile edges. In the classical Penrose rule, the crossing positions and colours must agree. A large curve can be assembled from many arcs without being one circle. About curved matching rules.
Illustration only here. The arc switch changes the drawing, not tile placement, matching, or learning. The current searches still use their existing boundary rules or Ammann lines. Rose marks connect the long edges; green marks connect the short edges. The dart’s green arc goes around its concave corner.
Learning curves from bare tilings
The research goal is to infer a reusable marking from tile shapes and unmarked arrangements, without giving the learner these arcs, arrows, or Ammann bars. Repeated contacts can suggest boundary crossing positions and tangent directions; a small library of curves would then have to fit consistently in every rotated or reflected copy of each tile.
Positive examples alone cannot identify a forcing rule: an empty marking also fits them. Candidate rules need independent tests against alternative contacts and larger extensions. A finite failure or a timeout must not be confused with a proof of impossibility. Any proposed marking remains a hypothesis until its stated validation is complete.
Not implemented: this page does not yet learn circular arcs from bare tilings. Its current point learner tracks a symbolic marking space on patches generated with either bare geometry or supplied boundary rules. The two arc drawings above are known illustrations, not learned output.
For the sevenfold rhombs of angles \(\pi/7\), \(2\pi/7\), and \(3\pi/7\), the same experiment could search for compatible curve pieces and their transformation rules. No analogous curved marking is asserted here. The sevenfold demo’s Socolar arrows are a published comparison control, not a discovered marking.
Scope and next experiments
Coordinates use \(\mathcal O_{\mathbb Q(\zeta_5)}=\mathbb Z[\zeta_5]\), with independent basis \(1,\zeta_5,\zeta_5^2,\zeta_5^3\). This module is dense in the physical plane. Finite tile-contact alignments supply the candidate set; no claim of a discrete planar lattice is made.
Vertices and edge midpoints are a first finite marking support. Further experiments could compare supports, accumulate independently validated positive patches, introduce justified negative examples, and seek curve realizations. The present equality space always admits constant assignments and does not prove aperiodicity.
The graph checks global dead ends, propagates forced moves, then branches at the earliest generation. Polygon overlap and boundary checks remain geometric controls; this is not yet the pure point-value reference engine. Finite patches and corona milestones do not establish infinite extension.
Prototile reference · Earlier Ammann experiment · Experiment documentation
Three unit-edge rhombs in \(\mathbb{Z}[\zeta_7]\), where \(\zeta_7=e^{2\pi i/7}\). Their acute angles are \(\pi/7\), \(2\pi/7\), and \(3\pi/7\). Explore unmarked growth or Socolar’s published aperiodic edge-matching rule.
Learning curved markings · research direction
Could these three bare rhombs reveal a reusable system of curves? The intended experiment would fit curve pieces to unmarked contacts and test their continuation across independent patches and alternative arrangements. Supplied Socolar arrows would be kept out of the learning input. No sevenfold curved marking or curve learner is implemented here; the current arrows remain a published control.
Known aperiodic edge markings
Socolar’s Weak matching rules for quasicrystals (1990), Section 5 and Figure 4, gives an explicit edge-arrow construction for these three rhombs. There are four arrow types, shown here with one to four arrowheads. Shared edges must agree in both arrow type and direction. Each shape has eight decorated variants up to rotation, counting reflected variants separately: \(24\) decorated prototiles altogether. The solver includes all variants automatically.
The arrows enforce Socolar’s alternation condition: along a row sharing edge direction \(e_m\), tiles of type \((m,m+k)\) and \((m,m-k)\) alternate, for each \(k\in\{1,2,3\}\), ignoring the other shapes in between. Equal-shaped neighbors are permitted when the particular sides and arrows match. The previous blanket ban on equal-shaped neighbors has been removed.
For sevenfold tilings these rules enforce aperiodicity while allowing bounded deviations from a canonical projection and local tile flips. “Weak” refers to that remaining freedom, not to permission for periodic tilings. The full-plane result is Socolar’s theorem; a finite search patch alone does not prove infinite extension. The optional unmarked mode permits periodic tilings.
GCTS marking and propagation
The GCTS method stores Socolar’s supplied labels as a global point marking at edge midpoints. A conflicting assigned value removes a candidate from the frontier graph. It then checks all possible ways to fill each candidate’s corners using the selected rhombs and their labels. An impossible completion removes that candidate before branching; cached results are reused across backtracking.
This mode uses a known marking; it does not claim to learn or rediscover Socolar’s arrows. The corner cache contains exact local feasibility results, not learned marking values. Learning additional restrictions would require the repository’s separate unmarked pair-corona classification and validation workflow. The baseline uses the same point obligations and pairwise edge checks, without corner-completion propagation.
Each oriented rhomb has sixteen arrow assignments. Two attempts with the same outline can therefore be different decorated candidates. The activity line shows branch choices and backtracking, and the peak tile count records progress even while the current patch shrinks.
Point model and implementation scope
Coordinates use the independent basis \(1,\zeta_7,\ldots,\zeta_7^5\), with \(1+\zeta_7+\cdots+\zeta_7^6=0\). Edge directions are \(e^{j\pi i/7}=(-\zeta_7^4)^j\). A rhomb of acute angle \(k\pi/7\) contributes corner values \(k/14\) and \((7-k)/14\); a complete corner has total \(1\). Each edge midpoint contributes \(1/2\), requiring a matching neighbor. The point domain is \(\tfrac12\mathbb{Z}[\zeta_7]\), with translations restricted to \(\mathbb{Z}[\zeta_7]\).
At each edge midpoint in \(\tfrac12\mathbb{Z}[\zeta_7]\), a three-bit label encodes one of four arrow types and its direction. Traversing a rhomb between parallel edges flips the bit for its own shape and preserves the other two. Adjacent rhombs compare labels for equality. The full decorated catalog is used; the optional smaller, phase-specific set discussed in Section 6 is not imposed. Published markings are problem input, not learned constraints.
This is a geometric search control: exact coordinate identities, integer corner capacities, and exact marking equality are combined with floating-point polygon separation and edge-to-edge checks. The complete point–candidate graph checks global dead ends, then forced moves, then the earliest generation, with reversible updates. Exposed vertices and edge midpoints are activated. Budget exhaustion is unknown, and no numerical finite patch is presented as an exact infinite-tiling certificate.