Kolam Tiles
STATUS · ACTIVE NOW
The question
What global forms emerge when binary kolam tiles are assembled under local edge-matching rules?
Current focus
Connecting square-tile puzzles and triangular-tile polyhedra through graphs, symmetry, and interactive constructions.
Latest update
The octahedron explorer and its graph-theoretic article now join the three square-tile sandboxes.
Open the interactive laboratory
Why this project
The square branch begins with the sixteen four-bit words. Each word describes the openings of a square tile in the directions east, north, west and south. Adjacent bits must match, the outer boundary must close, and the fifteen nonzero tiles must form one connected network.
Setting aside the \(0000\) tile turns the same objects into a labelled 15-puzzle. The question changes from which boards are valid? to which valid boards can be joined by legal slides? This brings parity and orbit structure into the picture.
The triangular branch uses the eight three-bit words on the faces of an octahedron. Passing to the dual cube turns each tile label into a vertex degree. The degree sequence forces a seven-vertex tree, and its three possible arm-length patterns give a complete classification without exhaustive computation.
Project strands
Square tiles
Build a connected \(4\times4\) kolam, explore the associated labelled 15-puzzle, and compare configurations in the same sliding orbit.
Triangular tiles
Place eight oriented tiles on an octahedron, reveal the dual-cube tree, and fold each connected net into the completed solid.
Explain
Connect the experiments to binary encodings, graph connectivity, Euler characteristic, enumeration, and symmetry.
Articles
- From Sixteen Tiles to Fifty-One Kolams — a computer-assisted classification of the square-tile boards.
- Kolams on Octahedron — a graph-theoretic hand classification of the triangular-tile surfaces.
Interactive laboratory
- Square Kolam Tile Challenge — use every square tile exactly once.
- Slide to a New Kolam — explore the two sliding-puzzle orbits.
- Move X to Y — transform one valid square kolam into another.
- Kolams on an Octahedron — fold and rotate the three triangular representatives.
Classroom material
- Sixteen tiles, one closed kolam — a classroom-ready investigation.
Next steps
- give the two square sliding-puzzle routes permanent descriptive URLs while preserving the existing paths;
- add downloadable triangular tile sheets and classroom material;
- record a video walkthrough connecting the octahedron, dual cube, and subdivided \(Y\);
- publish reproducibility material for the computational square-tile count.