Architecture

Core data structure: Half-edge graph

Pleat is built around the half-edge data structure (also known as DCEL — doubly-connected edge list). Every undirected edge is stored as a pair of directed half-edges linked via rev. Navigation around a face uses nex/pre; navigation around a vertex uses v.outgoing_iter().

HalfEdgeGraph          Topology only (vertices, halfedges, faces as sets)
  └─ InAngleHEG        + interior angles and edge lengths
      └─ GeometricHEG  + pluggable geometry backend
          └─ EuclideanPositionHEG  + 2D vertex positions

All graph elements (HalfEdge, Vertex, Face) inherit from AttributeObject, providing dict-like attribute storage via obj['key'] = value.

Tiling construction pipeline

1. ProtoTiles define tile shapes via angles, edge lengths, and gluing labels
2. Tile sets group prototiles into Archimedean or other tilings
3. Instructions on border edges describe how to attach new tiles
4. Growth iterates over border vertices, executing their instructions

tiles = platonic(6)          # hexagonal prototile set
G = from_tiles(tiles, rings=5)  # grow 5 rings

Conway operators

Topological operators that transform tilings: dual, ambo, truncate, kis, join, gyro, starify, and more. Each operator defines a fundamental domain as a small half-edge graph that gets substituted into each face triangle.

G2 = ambo_graph()(G, delete_on_border=True)

Geometry backends

Three pluggable backends in pleat.geometries:

Backend	Model	Use case
EuclideanGeometry	Flat plane	Standard tilings
PoincareDiskModel	Hyperbolic disk	Hyperbolic tilings ({7,3}, {5,4}, ...)
SphereModel	Unit sphere (stereographic)	Platonic solids, spherical tilings

Origami pipelines

Multiple pipelines for turning tilings into origami tessellations are implemented, see the example notebooks for details.