hat-amp 0.1.0

Aperiodic hat monotile tiling and percolation library

hat-amp

Python tools for aperiodic tilings and percolation experiments.

hat-amp generates exact hat monotile coordinates from Craig Kaplan's hatviz substitution system, 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.

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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

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

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 Hat And Penrose SVGs

from hat_amp.penrose import generate_penrose_tiling
from hat_amp.tiling import generate_tiling
from hat_amp.viz import 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")

Features

• Hat H/T/P/F metatile substitution and full patch generation.
• 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

• 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
• Aperiodic-Monotile-Percolation — github.com/aaryashBharadwaj/Aperiodic-Monotile-Percolation for reference percolation workflows and numerical cross-validation in tests.
• Smith, Myers, Kaplan, Goodman-Strauss. An aperiodic monotile. Combinatorial Theory 4 (2024). arXiv:2303.10798

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Acknowledgments

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