OBSERVATION CHAMBER / ACTIVE
Observe an Ammann–Beenker quasicrystal from a 4D cut-and-project window: accepted points, local coordination, eightfold rotation, and S(k) diffraction peaks.
ROUTE 71 / STRUCTURE / QUASICRYSTAL
CUT-AND-PROJECT / STRUCTURE FACTOR
Move the phason and window. Read the accepted points, a finite local patch, nearest-neighbor spacing, and the Bragg-like peaks of S(k).
Canvas 2D is unavailable. The deterministic readings and diagnostics remain available.
POINTS shows the projected set. LOCAL PATCH colors coordination near the origin. DIFFRACTION shows the eight indexed S(k) peaks. Click or touch a point.
Input changes enumerate 194481 candidates.
NUMERICAL READOUT
The local patch uses points with |r| ≤ 15 and a neighbor cutoff of 1.05. The indexed peaks use k = π B∥m and S(k) = |Σ exp(−ik·r)|²/N².
DETERMINISTIC SELF-TEST
CANONICAL M=10 + PHASON Δ / PASS
| CHECK | VALUE | LIMIT | RESULT |
|---|
Fixtures: symmetric window N=1737; phason (0.18, −0.11) N=1760; symmetric difference 385.
OBSERVATION CONTRACT
For n ∈ ℤ⁴, project n to physical space E∥ and internal space E⊥. Keep n only when its internal point lies in the regular octagonal window W. The displayed points are deterministic. No animated tiling, wave field, or physical detector is implied.
WHAT TO WATCH
MODEL NOTE
r∥ = (n₁ + αn₃ − αn₄, n₂ + αn₃ + αn₄)
r⊥ = (n₁ − αn₃ + αn₄, n₂ − αn₃ − αn₄), α = 1/√2. The same point set feeds local coordination and the structure factor.
The window is the projection of the 4D unit cube. A phason shift changes the acceptance test in E⊥: r⊥ + φ ∈ W. The finite generator enumerates [-10,10]⁴ = 194481 candidates and accepts at most 2048 points.
For the eight test indices m, S(k) is evaluated directly from the accepted positions. Rotation is checked by R₄₅(r) against the same finite point set.
OBSERVATION GUIDE
Start at RESET. Use POINTS to see the accepted sites, then LOCAL PATCH to inspect degree. Select DIFFRACTION to read the eightfold indexed peaks.
This is a finite cut-and-project observation model. It is not a wave-interference display, a kaleidoscope, a tiling show, or a physical diffraction measurement.
Runs inside the browser with no upload and no registration.
This chamber runs a deterministic Ammann–Beenker cut-and-project model in your browser. The window, points, local patch, structure factor, reports, and images stay local. It does not measure a physical crystal or visitors.
This chamber belongs to Structures & Symmetries.
STRUCTURE Back to Structures & Symmetries Route.