hat-amp

Python tools for aperiodic tilings and percolation experiments.

hat-amp generates exact tile coordinates for the Hat (Tile(1,0)) and Spectre (Tile(1,1)) aperiodic monotiles via substitution systems ported from Craig Kaplan's reference implementations, plus Penrose Robinson-triangle patches. It also provides vertex and tile-dual graph builders, square-window cropping, site/bond percolation helpers, .npz result persistence, and SVG/PNG rendering.

The Spectre is the strictly chiral companion to the Hat — no reflected copy appears in any valid tiling, making it the first truly chiral aperiodic monotile. The Mystic is a fused Spectre pair (at 30°) that plays the physics role of an "anti-hat" and is the minority tile in every Spectre tiling. Vertex-10 of each Spectre polygon is the Singh–Flicker gold vertex: its inclusion makes the vertex graph bipartite and enables exact dimer/zero-mode analysis.

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Installation

uv add hat-amp

Graph and percolation helpers need SciPy:

uv add "hat-amp[graph]"

PNG export needs CairoSVG:

uv add "hat-amp[viz]"

Requires Python >= 3.13.

Quick start

Hat:

from hat_amp.tiling import generate_tiling

hats = generate_tiling(level=3)   # list of 1156 arrays, each shape (13, 2)

For the larger finite-window patch used by percolation workflows:

from hat_amp.tiling import generate_patch_tiling

patch_hats = generate_patch_tiling(level=3)  # 3603 hat polygons

Spectre:

from hat_amp.spectre import generate_spectre_tiling, generate_spectre_tiling_labeled

# 559 polygons at level 3, each shape (14, 2)
spectres = generate_spectre_tiling(level=3)

# same polygons with per-tile labels: 'S' (standalone) or 'M' (Mystic component)
polygons, labels = generate_spectre_tiling_labeled(level=3)

Vertex-10 of each polygon is the Singh–Flicker gold vertex. Remove it for natural (non-bipartite) graph analysis:

from hat_amp.spectre import strip_gold_vertex

polys_13v = strip_gold_vertex(spectres)   # each shape (13, 2)

Full API documentation: docs/API.md.

Example: Graph Crop And Site Percolation

from hat_amp.graph import build_vertex_graph, crop_square
from hat_amp.percolation import BoundarySets, Criterion, run_site_trials
from hat_amp.tiling import generate_patch_tiling

polygons = generate_patch_tiling(level=3)
graph = build_vertex_graph(polygons)
cropped = crop_square(graph, L=100.0)
boundaries = BoundarySets.from_cropped_graph(cropped)

thresholds = run_site_trials(
    cropped,
    boundaries,
    trials=200,
    seed=123,
    criterion=Criterion.INTERSECTION,
)

print(thresholds.mean())

Example: Render SVGs

from hat_amp.penrose import generate_penrose_tiling
from hat_amp.tiling import generate_tiling
from hat_amp.viz import render_patch_svg, render_svg, save_svg

hat_svg = render_svg(generate_tiling(level=2), stroke="#222222", fill="none")
save_svg(hat_svg, "hat_level_2.svg")

penrose = generate_penrose_tiling(divisions=4, scale=200.0)
penrose_svg = render_svg(penrose.polygons(), stroke="#3366aa", fill="none")
save_svg(penrose_svg, "penrose_level_4.svg")

# Spectre: S tiles blue, Mystic tiles red
spectre_svg = render_patch_svg(3, source="spectre")
save_svg(spectre_svg, "spectre_level_3.svg")

Features

• Hat (Tile(1,0)) H/T/P/F metatile substitution and full patch generation.
• Spectre (Tile(1,1)) 9-type substitution system (Delta…Psi + Mystic/Gamma compound).
• Per-polygon chirality labels ('S' standalone / 'M' Mystic component).
• Gold-vertex helpers (strip_gold_vertex / add_gold_vertex) for bipartite graph analysis.
• Penrose Robinson-triangle subdivision.
• Vertex graphs and tile-dual graphs from polygon tilings.
• Square-frame cropping with boundary node sets.
• Site and bond percolation with intersection/union crossing criteria.
• Finite-size weighted least-squares extrapolation for p_c.
• .npz result persistence with pydantic metadata.
• SVG output and optional PNG export.

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Reference

Tiling sources

• Kaplan's hatviz — hat.js · geometry.js
  • geometry.js: affine transforms as 6-element arrays [a,b,tx,c,d,ty]
  • hat.js: metatile definitions, 28 substitution rules, recursive inflation
• Kaplan's Spectre app — cs.uwaterloo.ca/~csk/spectre/
  • 9-type substitution system; tile polygon coordinates and placement geometry
  • spectre.js: buildSupertiles() transform rules ported to spectre.py

Percolation reference

• Aperiodic-Monotile-Percolation — github.com/aaryashBharadwaj/Aperiodic-Monotile-Percolation — reference percolation workflows and numerical cross-validation in tests.
• Bhola, Biswas, Islam, Damle. Site percolation on the Hat tiling. Phys. Rev. X 12, 021058 (2022). [arXiv:2108.12440]
• Bhola, Damle. Percolation on aperiodic tilings. arXiv:2311.05634 (2023).

Primary papers

• Smith, Myers, Kaplan, Goodman-Strauss. An aperiodic monotile. Combinatorial Theory 4 (2024). arXiv:2303.10798
• Smith, Myers, Kaplan, Goodman-Strauss. A chiral aperiodic monotile. (2023). arXiv:2305.17743
• Singh, Flicker. Exact solution of the dimer model on the spectre tiling. Phys. Rev. B 109, L220303 (2024). arXiv:2309.14447

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Acknowledgments

Tiling generation ported from Craig Kaplan's hatviz (BSD-3-Clause).