Documentation: https://bright-fakl.github.io/isotile/

Isohedral tilings in NumPy: the 81 isohedral tiling types, vectorized tile generation and exact tile adjacency from a lattice table. It replaces tactile for these tasks and keeps its tiling type numbers (IH1 to IH93) and parameter format.

• Tiles are generated for a rectangle or for a target tile count, as arrays of cells, aspects and affine transforms.
• Adjacency (shared edge, shared vertex, vertex only) comes from a table of the tiling type, as a scipy sparse boolean matrix. It does not depend on the parameters, the scale or the position of the tiling.
• Squares, hexagons, any triangle and any simple quadrilateral are named constructors of the same class.
• A parameter set that makes the prototile self-intersecting raises ValueError; check_prototile reports the edges at fault without constructing a tiling.

Install

pip install isotile            # numpy and scipy
pip install "isotile[shapely]" # adds TileSet.polygons()
pip install "isotile[plot]"    # adds isotile.plot (matplotlib)

Python 3.10 or newer.

Quickstart

from isotile import IsohedralTiling, check_prototile, find_types

tiling = IsohedralTiling(7)  # type IH7 at its default parameters
print(tiling.describe())

tiles = tiling.fill_region((0, 0, 10, 6), n_tiles=200)  # tiles that cover the region
print(len(tiles), tiles.cell.shape, tiles.aspect.shape, tiles.transforms.shape)

polygons = tiles.vertices()  # (n, n_vertices, 2) tile outlines
edge = tiles.adjacency("edge")  # scipy.sparse.csr_array of bool, (n, n)
print(edge.sum(axis=1).max(), tiles.neighbors(0, "vertex"))

hexagons = IsohedralTiling.hexagon(pointy_top=False)
triangles = IsohedralTiling.triangle([(0, 0), (3, 0), (1, 2)])  # any triangle
print(find_types(num_vertices=3, all_edges="S"))
print(check_prototile(7, [0.25, -1.0]))  # PrototileReport(valid=False, ...)

Concepts

A tiling is one of 81 types with free parameters that change the shape of the prototile. The lattice is spanned by two translation vectors (translation_u, translation_v). A tile is identified by its cell (u, v) and its aspect, the index of the tile within the cell. A tile set holds cell (n, 2), aspect (n,) and the affine transforms (n, 2, 3) of the prototile onto each tile.

IsohedralTiling(type, parameters, frame=...) takes an affine frame that maps the type's native coordinates to world coordinates (scale, rotation, reflection, position). fill_region(bounds, n_tiles=... | tile_size=..., margin=...) computes the frame from a region and a tile count or size.

Documentation

https://bright-fakl.github.io/isotile/: quickstart, catalog of the 81 types, adjacency, edge curves, benchmark, migrating from tactile and the API reference.

Background and citation

The 93 isohedral tiling types were classified by Branko Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman, 1987). The type data and the parameter format come from Tactile / tactile-js by Craig S. Kaplan; if you use isotile in published work, please also cite Tactile and Grünbaum and Shephard.

License

MIT. The tiling type data is derived from Tactile / tactile-js by Craig S. Kaplan (BSD 3-Clause); see NOTICE.